Smoothings of Piecewise Linear Manifolds. (AM-80), Volume 80 9781400881680

Table of contents :
CONTENTS
PREFACE
SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS 1: PRODUCTS
SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II: CLASSIFICATION

Citation preview

Annals of Mathematics Studies Number 80

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS BY

MORRIS W. HIRSCH AND

BARRY MAZUR

PRINCETON UNIVERSITY PRESS

AND UNIVERSITY OF TOKYO PRESS

PRINCETON, NEW JERSEY 1974

Copyright© 1974 by Princeton University Press ALL RIGHTS RESERVED

Published in Japan exclusively by University of Tokyo Press; in other parts of the world by Princeton University Press

Printed in the United States of America

Library of Congress Cataloging in Publication Data will be found on the last printed page of this book.

PREFACE The triangulation theorems of Whitehead [12] show that to every smooth manifold V there is associated a PL manifold M and a homeomorphism f: M-> V, having particularly nice local properties. Moreover M and f are unique up to very strong equivalence relations. When M is triangulated, f is called a smooth triangulation of V. Whitehead's results suggest the inverse problem: is a given combinatorial manifold M the domain of a smooth triangulation of some smooth manifold V? This is possible if and only if M has a smoothing, i.e., a differential structure which is compatible with its piecewise linear structure. Smoothing theory is concerned with the problem of finding and classifying smoothings of PL manifolds. The equivalence relation we use for the classification is not the obvious one of diffeomorphisms but the stronger relation of concordance: two smoothings of M are concordant if they extend to a smoothing of M x I. This relation has nicer functorial properties than diffeomorphism. The diffeomorphism classification of smoothings is still unknown. It turns out that concordance is the same as the relation of isotopy: two smoothings a, f3 of M are isotopic if there is a diffeomorphism Ma -> Mf3 which is PD isotopic to the identity as a map M-> Mf3. This book is divided into two parts. Part I is devoted to proving that every smoothing of M x I is isotopic to a product smoothing. This is a very useful stability result; it implies for example that smoothings of M x Rk correspond bijectively to smoothings of M, modulo concordance. (In Part II this is extended to smoothings of vector bundles over M.) Another consequence is that concordance and isotopy are the same equivalence relation. The methods used in Part I are entirely elementary.

v

vi

PREFACE

Part II builds on the results of Part I to classify concordance classes of smoothings of M. For each PL manifold M, there is a "PL tangent bundle" tM over M. Such bundles are classified by homotopy classes of maps M-> B PL where PL is a certain semisimplicial group. There is a natural map BO -> B PL. If M is smoothable then tM has a compatible vector bundle structure which we call a linearization; and isotopic smoothings define equivalent linearizations. The main result of Part II is that smoothings of M are completely classified by linearizations of tM. This basic fact can be rephrased in

terms of classifying maps: M is smoothable if and only if the classifying map fM : M -> B PL of tM lifts to a map M-> BO; and every homotopy class of liftings can be realized by a unique concordance class of smoothings.

If M is smoothable, it turns out that concordance classes of smoothings also correspond to homotopy classes of maps M-> PL/0, where PL/0 is the fibre of BO-> B PL. The set [M, PL/0] of such homotopy classes has a natural abelian group structure. See in this regard [8]. The classification of smoothings has been a model for several other classification theories. For example, Kirby and Siebenmann [2,13] have shown that the problem of putting a PL structure on a topological manifold M is equivalent to putting a PL structure on the "topological tangent bundle" of M - except perhaps in dimension 4. Sullivan [3,4], Browder [1] and Novikov [9] have analogous (but more complicated) results for the problem of classifying manifolds of a given homotopy type. The present study is entirely theoretical in that no computations are carried out. The task of actually calculating the set of concordance classes of smoothings of a given manifold is still formidable. As a consequence of Part II, however, it "reduces" to standard problems in homotopy theory. Some results of this type are contained in the references.

REFERENCES [1].

Browder, W., Manifolds and homotopy theory, in ManifoldsAmsterdam 1970. Lecture notes in mathematics 197, Springer-Verlag (Heidelberg) 1971.

[2].

Kirby, R., and Siebenmann, L., Some theorems on topological manifolds, ibid.

[3].

Sullivan, D., Geometric periodicity and the invariants of manifolds, ibid.

[ 4].

, Triangulating and smoothing homotopy equivalences. Mimeographed notes, Princeton University, 1967.

[5].

Munkres, J., Concordance of differentiable structures - two approaches. Michigan Math. Journal 14 (1967), 183-191.

[6].

, Obstructions to imposing differentiable structures. Illinois J. of Math. 18 (1964), 361-376.

[7].

27-45.

, Higher obstructions to smoothing. Topology 4 (1965),

[8].

----,Concordance inertia groups. Advances in Math. 4 (1970!, 224-235.

[9].

Novikov, S., Homotopy equivalent smooth manifolds I. lzv. Akad. Nauk. SSSR. Ser. Mat. 28 (1964), 365-474 = Amer. Math. Soc. Translations series 2, vol. 48, 271-396.

[10]. Scharlemann, M., and Siebenmann, L., The Hauptvermutung for smooth singular homeomorphisms, to appear. [11]. Wall, C., an extension of results of Novikov and Browder. Amer. J. of Math. 88 (1966), 20-32. [12]. Whitehead, J., On

ci

complexes. Annals of Math. 41 (1940), 809-832.

[13]. Kirby, R., and Siebenmann, L., On the triangulation of manifolds and the Hauptvermutung. Bull. Amer. Math. Soc. 75 (1969), 742-749.

vii

CONTENTS PREFACE ...................................................................................................... v SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS 1: PRODUCTS .............................................................................................. 3 SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II: CLASSIFICATION ...... .. ....... ....... .... ............ ...... .. .................... ............ .... 77

ix

Smoothings of Piecewise Linear Manifolds

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS I: PRODUCTS Morris W. Hirsch § 1. Introduction

The object of this paper is to give an elementary proof of the fundamental theorem of the theory of smoothings of PL manifolds: MAIN THEOREM.

Let a be a smoothing of

M x

I, inducing the smooth-

ing {3 on M x 0; let A C M be a closed set such that a is a product smoothing in a neighborhood of A x I. Given

E :

Mx I

-->

R+ there is a

diffeomorphism from (Mx I)a to the product smoothing M{3 x I which is PD E-isotopic to the identity rei M x 0 U Ax I. A more elaborate theorem is stated in 4.1. Versions of this theorem have been proved by Munkres [14] and independently, the author. Both of these proofs were based on Munkres' ingenious obstruction theory. The proof presented in this paper is independent of obstruction theory, and will be used in subsequent articles as the basis for the classification of smoothings; the classification has an obstruction theory as a corollary. The origin of the main theorem was the announcement by R. Tj10m [20] in 1958 that every smoothing of M x I is diffeomorphic to product. Munkres' obstruction theory [12] has Thorn's theorem as an immediate consequence. A theorem weaker than the "M x I theorem" is the "M x R theorem" 7. 7: Every smoothing (M x R)a is isotopic to a product M{3 x R and {3 zs unique up to concordance. This was proved independently by Mazur3

4

MORRIS W. HIRSCH

Poenaru [11] and the author [3]. Using the main theorem, it follows that

f3

is in fact unique up to isotopy.

Therefore: for every n

f

Z+, isotopy classes of smoothings of M

correspond bijectively to isotopy classes of smoothings of M x Rn. This is the first step in the isotopy classification of smoothings.

§2.

Structure of the paper

In essence our method is that of Cairns [2] (see also [23]) who used fields of transverse planes to smooth submanifolds of Euclidean space.

If a is a smoothing of M x I, we would like to simultaneously smooth all the level surfaces M x t of the projection

11 2

:

(Mx I)a ->I by

a small isotopy. This can be done provided there exists a vector field which is everywhere transverse to the level surfaces. By uniqueness of collaring we may assume that a is a product near M x 0 and M x 1. A small PL isotopy allows us to assume that (M x I)a has a smooth triangulation such that

11 2

is simplicially affine and separates vertices in

M x (I- dl). This ensures that every point other than a vertex has a tangent vector transverse to nearby level surfaces. The most delicate part of the proof involves a close examination of the behavior of

11 2

in the star of a vertex v. Everything is essentially

determined by the level surfaces of

11 2

small smooth ball B around v. If

11 2 (v) =

PL sphere whose dimension is m -1

=

restricted to the boundary of a y 0 then M x Yo n dB is a

dim M -1, and its bounds a PL

m-ball D in dB. We can make D a smooth submanifold of B. By applying the main theorem one dimension lower as an induction hypothesis, we can assume D is a smooth m-disk. We then make D coincide with the northern hemisphere of dB by a diffeotopy. Let us identify B with the unit ball om+l C Rm+l. By uniqueness of collaring of sm-l in dDm+l

=

agrees with the projection

: Rm+l -> R on a neighborhood E of

11 m+l

sm, we can assume the map 172

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS I

5

sm-l in sm. Some careful smooth triangulations and the Alexander trick permit us to assume that

1T 2

and

1T m+l

One more simple radial isotopy makes hood of 0. Since

1T m+l

agree on the cone on E from 0.

1T 2

and

1T m+l

agree in a neighbor-

: Dm+l --> R is a smooth map, it follows that we

have isotoped the smoothing a until the vertex v

f

(M x I)a has a tangent

vector transverse to the level surfaces near a. The delicate part of the above argument lies in making sure that after each isotopy, we still have vectors transverse to the level surfaces at all points of the star of v other than v. By a partition of unity we can find a globally defined vector field X in (M x I)a which is transverse to the level surfaces M x t. By convolution we approximate

1r 2

by a smooth map f: (Mx I)a--> I whose level

surfaces are also transverse to X. We then push each M x t onto C 1 (t) along integral curves of X. Finally if M is compact, we push each arc x x I onto the integral curve of X that starts at x x 0. If M is not compact, the last step is not always possible, for there is no guarantee that every trajectory of X passes from M x 0 to M x 1. We are forced to prove the theorem in stages: first the compact case, then a globalization to the non-compact case. The globalization is not trivial because PD homeomorphisms do not form a category: in general they can be neither composed nor inverted. However, they "almost" form a category: a PD homeomorphism can be approximated by a PL homeomorphism. By using a refined version of

J. H. C. Whitehead's theory of smooth triangulations, we obtain the globalization. It is remarkable that no change is required in Whitehead's proofs of thirty years ago; it is only necessary to examine them closely and extract their full strength.

"Relative" and "absolute" In many statements of results to be proved there appears a manifold M and two closed subsets K C N C M. Their significance is the following. We want to improve some object (e.g., a map or a smoothing) in some way,

6

MORRIS W. HIRSCH

in a neighborhood of K, without changing the object outside of N. A theorem that such improvement is possible we call strongly relative. If the theorem is in such a form that K = N, we call it relative, while if K = M = N it is absolute. Strongly relative theorems are the most useful although relative theorems often suffice to produce interesting theories. Absolute theorems are not as useful; they produce interesting theorems, but not theories. To prove a strongly relative theorem, it usually is enough to prove it in a "local" form, where e.g., N is compact, or lies in a coordinate chart. Many results below follow this pattern. The globalization is ordinarily quite easy; occasionally we leave it to the reader. Most, if not all, examples of strongly relative theorems can be placed in the context of sheaf theory: the "improved object" is a section of a sheaf of

1, and the strongly relative theorem says simply that every section

1 over a closed subset of the base extends to a global section. We

have, however, resisted the temptation to introduce sheaves. Nor has any attempt been made to formalize the notions of relative, strongly relative, etc.; rather they are used informally in discussing the logical pattern of the paper. The main theorem, even in local form, seems to resist a direct proof. We first prove a local relative theorem, 4.2, which we refer to as the simplified form of the main theorem; it involves a weaker approximation. Then a local strongly relative theorem is proved, and finally an easy globalization yields the main theorem. Contents of sections Section 3 contains basic definitions and a brief discussion of diffeomorphism, isotopy and concordance. Section 4 contains statements of the main theorem (4.1) and its simplified form (4.2). Regular maps are studied in Sections 5 and 6. ln Section 7 we digress from the proof of the main theorem to apply the preceding material to smoothings of submanifolds; some of these applications are used in Section 10. It is shown that

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS I

7

smoothability depends only on PL tangential homotopy type. The proof of the simplified main theorem is completed in Sections 8 and 9; Sections

10 and 11 finish the proof of the main theorem itself. In Section 12 we study smoothings of M x Rn. The last section is an appendix in which we have collected various facts about PL and PD homeomorphisms; in particular the refined forms of Whitehead's smooth triangulation theorems are discussed.

§3. Basic definition Differential structures An atlas on a topological space M is a set = {(¢i, Ui)!i

R+" is an equivalence relation.

It is clear that isotopy implies diffeomorphism. It is almost as obvious

that isotopy implies concordance. To see this, suppose ft: a ~ f3 and define F: M x I--> Mp x I by F(x, t) be the smoothing induced from f3 x

(ft(x), t). If aM

=

l

by F, where

t f

=

0, let y f S(Mx I)

S(I) is the stan-

dard smoothing, so that Mp x I = (M x I)Y" Then y is a concordance from f3 to a. It was stated by Thorn [21] and proved by Munkres [14] and the author

(unpublished), that concordance implies isotopy. This is a corollary of the main theorem, whose essence is that every concordance y

f

S(Mx I)

from a to f3 is isotopic

rel M x 0 to the product smoothing a x

Therefore if ht :y ~ax

rel M x 0, then ht I M x 1 :{3 ~a. As rela-

l

L

tions on S(M), we have: concordance

=

isotopy

diffeomorphism .

~

In proving the main theorem, however we may only use the obvious relations:

concordance

0. If a

f3

f

f

S(M xI) is a product in a neighborhood of (bdN)x I there exists

S(Mxi) such that a

e!!

d+€{3

rel M x 0 U (M-N)xi where d=-diam N.

We prove 4.2 in Section 9, along with 9.1, by double induction on m. We shall use the following consequence of 4.2(m-1). 4.3(m) THEOREM (Munkres). Let B be a smooth manifold which is PD

homeomorphic to an m-simplex. Assume 4.2(m-1). Then B is diffeomorphic to the unit bail om c Rm. Proof. Let C be a triangulation of an m-simplex !l and let T : C .... B be a smooth triangulation. Let a

f

C be an m-simplex, and let D 0 C a-au be a smooth convex

n-disk. Put T(D 0 )

=

DCB -

aa.

Then D is a smooth submanifold of

B diffeomorphic to Dm. It is easy to see that there is a PD homeomorphism

(all) x I .... B- int D .

MORRIS W. HIRSCH

14

By 4.3(m-1) with M = a/1;., it follows that B- int 0 ~ (aO) x I. Therefore B is diffeomorphic to om with a smooth collar sm- 1 attached to aom

X

I

sm- 1 by a diffeomorphism. Hence B ~om.

=

§5. Regular maps Let E C Rn be a set and f: E--. R a map. For each Y! Rn, thought of as a vector, we define a map

by Lyf(a)

=

Lim inf(x,Z,t)-.(a,Y,O)C 1 [f(X+tZ)-f(x)]

with the understanding that (x, Z, t) varies over the subset of Ex Rnx R such that x and x + tZ are in E, and t ~ 0. If Lyf(a) > 0, we say a is a regular point for f and that Y is transverse to f at a. This is

devoted by Y (j1 a£. An equivalent formulation is: there exist positive numbers

E,

w such that

C 1 [f(X+ tZ)- f(x)] > w if 0 < \t\ < E, \x-a\ < E, \Z-Y\ < E, and the left side of the inequality is defined. Any other point is singular. The reader can easily supply proofs for the following fact. 5.1 LEMMA. Let f, g: E--. R be maps defined on a subset E C Rn. Then the following relations are true if the indicated sums and products exist(thatis, (+oo)+(-oo) and O·(±oo) areexcluded)

(a) Ly+zf2: Lyf+ L 2 f (b) Lcyf

=

cLyf if c > 0

(c) Ly(f+ g) 2: Lyf + Lyg (d) Ly(cf)

=

eLy£ if c > 0

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS I

15

(e) Ly(f ·g) 2: f · Lyg + g · Lyf if f and g are continuous and nonnegative

c1

(f) If f is

then Lyf(a) = Dfa(Y)

(g) Lyf(a)S lim inf(x,Z)->(a,Y)Lzf(x). Thus Ly f(a) is lower semicontinuous in (Y, a). It follows from (g) that the set I (a, Y)

f

Ex R n\ Y rjl afl is open in

Ex Rn. From (a) and (b) the set IY f Rn\Y rjlafl is a convex open cone in Rn. From (c) and (d) we find that the set of f: E-. R such that a is a regular point of f is a convex cone. The following two lemmas are exercises in Taylor expansions. 5.2 LEMMA. Let E, E'C Rn be subsets and U, U'C Rn neighborhoods of E, E' respectively; let ¢: (U, E)-> (U', E') a f: E-> R be a map. Let a Then

Ly'(f¢- 1 )(a')

f

E, Y

f

d

diffeomorphism. Let

Rn, and put a'= f(a), Y' = D¢a(Y).

= Lyf(a). Therefore Y'rjla,f¢- 1 if and only if

y rna£. 5.3 LEMMA. Let a E = El C1

f

E C Rn. Let f: E -> Rn be a map such that

u ... u Er, where a f El n ... n Er, and f IEi extends to a

map fi defined on a neighborhood of Ei. Then for every Y Lyf(a) = min !Dfi(a)(Y)\i= 1, ... ,rl

f

Rn

.

In particular Y rjlaf if Dfi(a)(Y) > 0 for i = 1, ... ,r .

The proof of the following technical result is complicated by the fact that the Lipschitz constant Lip (¢h- ¢) does not necessarily approach 0 as Lip (h-1)-> 0. See Section 13; and (7], Remark 1 of Section 1.

16

MORRIS W. HIRSCH

5.4 LEMMA. Let E C Rn be a set and ¢:E .... R a map. Let F C E be compact such that ¢ is regular on F. For every neighborhood U C E of F there exists 8 > 0 such that if a map h: E .... E satisfies

(a) Lip(h!U-1u) < 8 and lh!U-1ul < 8, then ¢h: E-> R Is regular on F. Proof. By compactness it suffices to prove the lemma for the case where F is a point x 0 .

Let Y0

•

Rn,

E

> 0, w > 0 be such that

(1) C 1 [¢(X+tY)-¢(x)]>w if O R a continuous map, and K C U a compact set. Let Y: K .... Rn be a vector field. Given E

> 0, there exists a smooth map g: K -> R such that

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS I

17

and

Ig- (f IK)i < e: . In particular, if Y rDKf we can choose g so that Y rDKg. Proof. A convolution kernel is a smooth nonnegative map A: Rn ... R such

f

A= 1, and there exists r(A) > 0 such that A(x) = 0 if Ixi~ r(A). Rn The convolution that

f *A : x

1-->

is defined on the set

J

f(x-s)A(s)ds

R0

It is well known that f *A is smooth, and as r(A)-+ 0, f *A -+ f uniformly on compact sets. Suppose r(A) so small that K C U'. If

e: > 0, by compactness there exists 8 > 0 such that (1) C 1 [f(x+ tX)-f(x)] ~ Ly(a)f(a)- e: where x

f

U, a

f

K, X

f

Rn are such that the left side of (1) is defined

and 0< ltl 0 or xn < 0. Suppose then that u c V

n Rn- 1

LY(x)f(x) 2': a for all x c V, given

E

so that un_ 1 = 0. Since

> 0 there exists 8 > 0 such that,

forall x, acV. (4) 0 < ltl < 8, lx-al < 8 and IZ- Y(a)l < 8. Choose r > 0 so small that r < 8, and if lx-ul :S r and IX-Y(u)l :S r, then

19

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS I

(5) xn > {3 and (6) the interval [x, X+ rX] lies in V and in the ball of radius

o

around u. Now assume (7) u

f

V

n Rn- 1 ,

\x-u\ < r, \X-Y(u)\ < r, and 0 < It\ < r.

If [x, x+tX] C R~ then

since xn > {3. If [x, X+ tX] c R~. then

Consider the case Xf

v n R~

Then there exists 0 < s

and X + tX

f

v n R~.

0 0 then x is a regular point for f; and Y is transverse to f at x denoted by Y rDxf. If every point of M is regular for f, then f is a regular map, If Y is a vector field on a subset K C M and Y(x) rDxf for every x f K, we say Y is transverse to f, or f is Y regular and write

Y rD f or Y rD K f. 5. 7 LEMMA. Let M be a smooth manifold and f: M -> R a regular map.

Then: (a) There exists a vector field on M transverse to f. (b) If Y is a vector field on M transverse to f and Y is

smooth in a neighborhood of a closed set X C M, then for every

E:M

->

R+ there exists a smooth vector field Z

transverse to f such that IZ- Yl < neighborhood of X.

E

and Z

=

Y in a

21

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS I

Proof. (a) Given x < M, let Y be a vector field defined on a neighborhood of

such that Y(x) rnxf. Since this mean LY(x)f(x) > 0 it

X

follows that Y rtluf for some neighborhood U of x. Let !u.!. A=U 1 lf be an open covering of M such that for each i < A there is a vector field Yi on Ui with Yi rtJ (f!Ui). Let !Pi!i O!. From 5.1(a,b) we have LY(x)f(x)

~

2, Pi(x) Ly.(x)f(x) > 1

0

Hence Y rtJ f. Part (b) follows from the approximation of continuous vector fields by smooth ones and 5.1(g). The global approximation theorem for regular maps is the following strongly relative theorem. 5.8 THEOREM. Let Y be a vector field on a smooth manifold M,

transverse to f: M -> R. Let N C M and K C N be closed sets such that f is

Given

cr E:

in a neighborhood W C M of K

n

bdN for some 1 :::;

r:::;

oo.

M -> R+, there exists a map g: M-> R such that:

(a) g is Y -regular

(b)

Lyg~

(c) g(x) (d)

=

Lyf-

E

f(x) for x f. M - N

lg-fl < E

(e) g is (f) f

Ic =

cr in a gl c for

neighborhood of K every component

c

of

aM

on which f

IS

constant; (g) giQ is smooth for every subset Q C M on which f is smooth. In particular if f is smooth on each simplex of a given smooth triangulation of M, the same is true of g. Proof. Let !(Ui,¢i)!icA be a C 00 atlas on M having the following properties:

22

MORRIS W. HIRSCH

(1) ¢i(Ui)

=

Rn or R~, according to whether Ui n

aM

is empty

or nonempty; (2) the open covering {UilifA is locally finite; (3) the sets lint DilifA cover M where Di (4) if Din K n bdN (5) if Din (K-W)

t-

0, then Ui

t- 0

=

¢i 1 Dn;

C W, and hence f lUi is cr;

then Ui C N.

Choose a C"" partition of unity ! Pili(A such that int Di

=

!xfMIPi(x) > Ol. We may assume 0 < Choose

o.1

e < Lyf.

such that 0 < oi < min{Lyf(x)-

Put A'= {i(AIDin(K-W)t-

~(x)ixfDi}.

01.

By 5.5 and 5.6 there exists C"" maps gi: Di-> R for if A', such that, for x f Di (6) Lygi(x) > Lyf(x)- oi < e (x)/2 (7) lgi(x)-f(x)i < oi and (8) gi(x)

=

f(x) if

X (

Di n c

for each component c

of

aM

on

which f is constant. If i (A -A', define gi

Observe that gi

=

=

fiui.

fi on M- N, and gi is cr in a neighborhood of

K. Moreover giiQ n Di is smooth if f IQ is smooth. Define g: M .... R by x

t->

I

If the

Pi(x)gi(x) where A(x)

=

{ifAixf int Dil.

ifA(x)

oi

are sufficiently small, g will obviously satisfy all parts

of the theorem except perhaps (a) and (b). To obtain (b) and hence (a), observe that since the Pi are smooth, ~ Ly Pi

=

~ DPi(Y) = (~ DPi) (Y) = D(~ P) Y = 0. Therefore (~

and we have, summing over A(x) for each point x f M:

Ly Pi) f = 0

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS I

23

Lyg- Lyf = LyiPigi- LyiPif- ILy Pi > I(Ly Pi) gi +I Pi Ly gi- Lyi Pif- (I Ly Pi)f > I(Ly Pi)(gi- f)+ I Pi(Ly gi- Lyf) > -I\LvPi\\gi-f\ + IPi(Lygi-Lyf) Put A(x) = sup l\DPi Y(x)\ :if A(x)l. Then Ly Pi(x) :S A(x), and hence from the above,

we obtain (b) by choosing the 8 i so small that

for all x f M. This completes the proof of Theorem 5. 7. §6. Regular maps and isotopics

Geometric properties of regular maps 6.1 LEMMA. Let U C Rn be an open set, f: U ... R a map, and Y

f

Rn

a vector transverse to f at x 0 f U. Then the level surface C 1 f(x 0 ) through x 0 is locally homeomorphic to Rn- 1 at x 0 . Proof. Without loss of generality we may suppose that x 0 = 0

f

Rn, f(x 0 )

= 0 f R, and that Y is the unit vector (0, ... , 0, 1). Since Y rD 0 f, there exists numbers E > 0, u > 0 such that (1) C 1 [f(x+tY)-f(x)]>u if 0 0. Similarly (4) f(y,-E) < 0. Therefore there is a unique point z f Iy such that f(z) = 0. Therefore the set C 1 (0)

n IV

has only one element. Put Bn- 1 = {yf Rn- 1 : IYI

:sol.

The projection

maps the compact set

bijectively onto Bn- 1 . By compactness, "1 IE : E ....

Bn- 1

is a homeomorphism. Since E is a neighborhood of 0 in f- 1 (0), the lemma is proved. 6.2 LEMMA. Let U C Rn be open, f: U--+ R a map, and Y: U-+ Rn a

vector field transverse to f. Then f is strictly increasing on each integral curve of y. Proof. Let J C R be an interval and A : J --+ U an integral curve of Y. It suffices to prove that fA : J --+ R is locally increasing. Let b f set x = A(b) f U. Since Lyf > 0, there exists

E

J

> 0 such that:

(1) f(X+tZ)- f(x)~ te if 0< ltl < E and IZ-Y(x)j R is X-regular.

o: K-> R+.

Proof. Let

By 5.8 there exists a map g:

K-. R

such that

g is X-regular; g simplicially smooth; g is smooth in a neighborhood of M0 ; g

=

Ig- fl Let r./1 t:

K- U and < o; g and f

f on

K-> K be the

in a neighborhood of A; are X-coregular.

X-homotopy relating f to g. Extend r./1 t to be

the identity outside IKl. If

o

is small enough r./1 t will be an E-isotopy,

prismatic by 6.4b. The following variation of 6.11 will be used in the proof of 8.1. To state it we define the link L(v) of a vertex v of a complex C to be the subcomplex:

L(v)

=

{aESt(v)l v/ al.

Thus ISt(v)l is the join of v and IL(v)l. 6.12 LEMMA. Let v be a vertex of a complex C and g, f: St(v)-. R simplicially affine maps. Suppose g(v) = f(v) = 0 and g(x) > 0, f(x) > 0, if x E L(v). (Hence g- 1 (0)

=

C 1 (0)

=

v.) Given a neighborhood V of

v m ISt(v)\, there exists an isotopy r./Jt of ISt(v)\ such that r./1 0 = 1 and:

(a) r./Jt isfixedon \St(v)\-v andalsoon {xEIS(v)\:f(x)=G(x)l; (b) r./Jt maps each simplex of St(v) into itself by a diffeotopy; (c)

gr./1 1

=

f in a neighborhood of v.

Proof. Let < 0,1 > C R be the open interval {xE R \ 0 < x < 11. Let .\(s, t): I-. I be a family of diffeomorphisms indexed by (s, t)

< 0, 1 > such that:

f

< 0, 1 > x

34

MORRIS W. HIRSCH

A(s, t) maps [0, s] linearly onto [0, t]; A(t, t)

=

1I;

A(s,t):x

1-->

x if max ls,tl

the map (s, t, x)

1-->

:S 2x- 1;

A(s, t) (x) is smooth in (s, t, x).

Choose €f sothatif XfL(v) then [0,2e] C f(Vn[v,x]) n g(Vn[v,x]) Let Yx: [0, 1]--+ [v, x] be the affine map such that 0

1-->

v and 1

1-->

x.

Define ax, bx f < 0, 1 > by

Define

Then

r/J t

r/1 t

by

has the required properties.

§7. Smoothing submanifolds We digress from the proof of the main theorem in order to prove some applications of the results of the last section. They will be used in Section 10 for the proof of the main theorem and again in Section 12. If M' is a PL manifold and M C M'- aM' a PL submanifold of co-

dimension k we call M flat if M has a PL collar in M'; that is, M has a neighborhood E C M' such that (E, M) = (Mx Rk, Mx 0). If M has an open cover by sets U i C M which are flat submanifold of M', then M is flat in M. Theorem 7.4 below says, in a strongly relative form, that a locally flat submanifold M of codimension 1 in a smoothed PL submanifold M'a can be pushed onto a smooth submanifold by arbitrarily small PD isotopies of M'a. In 7.5 a similar result is proved for flat submanifold of arbitrary codimension. Because the theorems are relative in form, it follows that the induced smoothing of M is unique up to concordance and therefore, after the main theorem is proved, it is unique up to isotopy.

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS I

35

First, to explain the main ideas, the absolute theorem is proved.

7.1 THEOREM. Let M' be a PL manifold and M C M'- CIM' a flat PL submanifold. If a is a smoothing of M', there is a PD isotopy

ht: M'-> M~ such that h 1 (M) C M~ is a smooth submanifold. Moreover, if M is compact, given any neighborhood N C M' of M and any

E

> 0,

ht can be chosen to be an E-isotopy fixed on M'- N. Proof. By induction on k

=

dim M'- dim M. The last statement of the

theorem will be left to the reader, since 12.1 below includes all of 7.1 as a special case. We may replace (M', M) by (Mx Rk, Mx 0) since M is flat in M'. Suppose k

=

1. Give (Mx R)a a smooth triangulation C so that the

projection rr 2 : M x R -> R is simplicially affine. Let y

f

R be such that

M x y contains no vertices. There is a PL isotopy ft: M- R-> M x R taking M x 0 onto M x y, for example, ft(x, s)

=

(x, s+ ty). By 6.8

(applied to the complex K = laECianMxy,£01) rr 2 :(MxR)a-> R is regular on a neighborhood of M x y. By 6.5 there is a PD isotopy gt: M x R -> (M x R)a such that g1 (M x y) is a smooth submanifold. The PD isotopy ht

=

gt ft takes M x 0 onto a smooth submanifold. If k > 1, write M x Rk

for k

=

=

(M x Rk- 1 ) x R. Since the theorem is proved

1, we may assume M x Rk- 1 is a smooth submanifold of

[(M x R k- 1 )x R]a. Moreover by uniqueness of collaring we can assume that a is a product smoothing:

By the induction hypothesis there is a PD isotopy ht: M x Rk- 1 -> (Mx Rk- 1 )a such that h 1 (Mx 0) is a smooth submanifold. Then

is the required PD isotopy. This completes the proof of 7 .1.

36

MORRIS W. HIRSCH

The PD isotopy ht constructed in the proof is not generally prismatic, since the PL isotopy ft taking M x 0 onto M x y is not generally prismatic. 7.2 THEOREM. Let M and N be PL manifolds, with aN=

0.

If

M x N has a smoothing, then both M and N have smoothings.

Proof. First suppose aM=

0.

Let U C N be PL homeomorphic to Rn.

Then M x Rn has a smoothing; by 7.1, M x 0 has a smoothing. Thus we find that M has a smoothing. If aM=

0, let M1

C M- aM be a PL homeomorphic to M.

Then

M - aM has a smoothing a. Since aM 1 is collared in M - aM, we apply 7.1 to find a PD homeomorphism f: M- aM .... (M- aM)a such that f(aM 1 ) is a smooth submanifold. Therefore f(M 1 ) is a smooth submanifold. Hence M is smoothable. To see that N has a smoothing, observe that N

X

(M- aM) is

smoothable; hence N is smoothable by what has been already proved. This completes the proof of 7.2. We can now prove that the smoothability of a PL manifold depends only on its PL tangential homotopy type. It is known (see e.g., [6]) that smoothability is not an invariant of homotopy type. 7.3 THEOREM. Let M0 and M1 be PL manifolds without boundaries,

having the same proper PL tangential homotopy type. If M0 has a smoothing, so has M1 . Proof. The hypothesis means there is a proper homotopy equivalence f: M0 .... M1 such that f*T(M 1 ) ~ T(M 0 ) where T(Mi) is the PL tangent bundle. In [4] it is proved that this implies M0 x Rq

=M1 x Rq

for some

q;::: 0. Therefore if M0 is smoothable, so M1 x Rq, and M1 is smoothable by 7.2.

37

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS I

7.4 THEOREM. Let M' be a PL manifold and M C M' a locally flat PL

submanifold of codimension 1. Let a be a smoothing of M'. Let N C M' and K C M n N be closed sets such that K n bd N has a neighborhood in M which is a smooth submanifold of M~. For every Is a PD E-isotopy ft: M'->

M~

E:

M'-> R+ there

such that

(a) K has a neighborhood WC M such that f 1 (W) C M~ Is a

smooth submanifold

In other words a is e-isotopic rel M'- N to a smoothing {3 of M' having W as a smooth submanifold. Proof. We shall use the following lemma; it says, in a strongly relative form, that the graph of a PL map can be pushed onto the graph of a smooth map.

7.5 LEMMA. Let U be a PL manifold and a a smoothing of U. Let N1 C U and L C N1 be compact sets. Let 0 < b < a. Suppose A: U-> [-b, b] is a PL map which is constant on each component of L

Given

E

> 0 there exists: a PD E-isotopy ht: u X R->

ua

X

n bd N1 .

R; a neigh-

borhood A C N1 of L; and a smooth map A0 : A->< -a, a>, such that (a) h1 (graph AlA)= graph A0 ; (b) ht is fixed on U x R- N 1 x [-a, a]; (c) ht preserves the projection rr 1 : U x R-> U.

The proof of the lemma, based on a smooth approximation to A, is left to the reader. Now consider the special case of 7.4 where M is flat in M', N is compact, and

aM = 0

so that M C M'-

aM'.

Call this the first special

case. We may replace (M',M) by (MxR,MxO) and N by N0 x [-a, a] where N0 C M is a compact set containing K and 0 < a< E/8.

38

MORRIS W. HIRSCH

By uniqueness of collaring (13.10) we may assume that a IU x R is a product smoothing. Give (Mx R)a a smooth triangulation such that rr 2 : M x R-+ R is simplicially affine. Let 0

in the

notation above.) Let L 1 C L 0 C L be open sets such that cl(L 1 ) C L 0 , cl(L 0 C L. Put

40

MORRIS W. HIRSCH

Then M* C M'* - aM'*, N* is compact, and K*

n bd N* has a

neighborhood in M* which is smooth in (M'*)a. Also M* is flat in M'*· We apply the first special case to (M'*, M*, N*, K*) and find a small PD isotopy g*t of (M'*)a, fixed on M'*- N*, taking a neighborhood W* C M* of K* onto a smooth manifold of (M'*)a. Extend g*t to a PD isotopy gt of M'a fixed outside N*. Then gt is fixed on M'- N C M~-N*. Put

W = W* U (L n M) C M

This is a neighborhood of K in M, and g1 (W) is the union of two open subsets, g*l (W*) U (L 0 n M) each of which is a smooth submanifold of M'a. This proves the third and the second special cases of 7.4. The general case is proved by triangulating M' so finely that each simplex a lies in a neighborhood of M which is flat in M'. Assuming inductively that the (i-1)-skeleton Ki of M has a neighborhood Ai C M which is a smooth submanifold of M::Z for each i-simplex a, let MaC M be a flat neighborhood of K 0 =a- Ai so that M0

n

Mr = 0 if a and r

are disjoint i-simplices. Then apply the special case simultaneously to all the neighborhood M0

.

The details are left to the reader. (Compare

Section 11 for a similar globalization.) 7.6 THEOREM. Let M' be a PL manifold with a smoothing a. Let

M C M'- aM' be a closed PL submanifold; let N C M' and K C M n N be closed sets. Suppose: (a) K has a neighborhood N 0 C M such that N 0 is flat in M'

and (b) K

n bd N has

a neighborhood U C M such that U C M::Z is a

smooth submanifold. Then for any

E:

M'-> R+ there is a PD

E-isotopy ft: M'-> M::Z such that: (c) K has a neighborhood W C M such that f 1 (W) C M::Z is smooth; (d) ft is fixed on M'- N.

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS I

41

Proof. By induction on k = dim M'- dim M, using 7.4 to start the induction if k = 1. The details are left to the reader. Remark: Notice that 7.6 is not a generalization of 7 .4. In the latter the submanifold was assumed only locally flat, whereas in 7.6 we assumed the submanifold was flat where we wanted to smooth it. In codimension 1 we have uniqueness of collaring at our disposal. We could not expect a straightforward generalization of 7.1 to higher codimensions, since it would prove that every PL manifold could be smoothed. (For counterexamples see Smale [19], Kervaire [8].) The essence of uniqueness of collaring is that a submanifold of codimension one has a PL normal bundle which can be given a compatible vector bundle structure. The proper generalization of 7.4 is to PL submanifolds having such normal bundles: they can be isotoped onto smooth submanifolds. (See LashofRothenberg [10]. Rourke-Sanderson [18].) The proof of this follows from 7.3 as soon as the proper definitions are made. (See Part II.) We are now in a position to prove a weak form of the M x R theorem referred to in the introduction: 7.7 THEOREM. Let a be a smoothing of M x R. Then there exists a smoothing f3 of M such that (Mx R)a is concordant to Mf3 x R. Such a smoothing f3 is unique up to concordance. Proof. By 7.4 we may assume that M x 0 is a smooth submanifold, say Mf3xOC(MxR)a. Thenormalbundleof Mf3x0=Mf3 in (MxR)a is trivial; hence Mf3 has a smooth product neighborhood. Let f: Mf3 x R

-->

(M x R)a be a smooth embedding which is the identity

on M x 0. By uniqueness of PD collars (13.10) there is a PD isotopy of f to the identity; let Ft : Mf3 x R .... (M x R)a be such an isotopy. Then F;a = f3 x p and Fta =a. Therefore G*(a x t) is a concordance from f3 x p to a where

42

MORRIS W. HIRSCH G : M x R xI .... (MxR) xI

is given by

G(x, y, t)

=

(Ft(x, y), t)

This proves the existence of {3. To prove f3 unique, suppose {3 0 x p and {3 1 x p are concordant. Let y be a smoothing of M x R x I which is a concordance from {3 0 x p to {3 1 x p. Extend y to a smoothing of M x R x R so that Mf3 1 x R x x R x and Mf3

0

MxOx(). By 7.4 this partial smoothing can be extended to a smoothing

Tf

of N.

If we restrict Tf to M x 0 x [-2, 2] we obtain a concordance between

Mf3 0 x 0 x (-2) and Mf3 1 x 0 x 2. Therefore {3 0 and {3 1 are concordant.

7.8 COROLLARY. The natural map from concordance classes of smoothings of M to concordance classes of smoothings of M x Rn is bijective.

§8. Isolating singular points In this section we prove a result which shows that a smoothing of M x I is isotopic to a smoothing for which the projection

1T 2

:Mx I

->

I

has an isolated singular set. PL nonsingular maps Let P be a PL manifold. A PL map ;\ : P

->

R is called PL non-

singular if each point of P - dP belongs to a coordinate chart ¢: U .... Rm- 1 x R such that ;\\ U is the composition U

L

Rm- 1 x R ~ R .

We call the pair (¢, U) a local representation of A. 8.1 LEMMA. Let P be PL manifold and ;\: P .... R a PL nonsingular map which is constant on each component of dP. Let C be a triangula-

43

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS I

tion of P such that f is simplicially affine in C. Given E: P .... R+'

o

there exists a positive function

A.':

c . . R+

on the vertices of C, such that if

is a simplicially affine map such that IA.'(v)-A.(v)i

each vertex v

f

< o(v) for

C and A.'= A. on ap, then there is a PL E-isotopy

ht: P --> P such that A. = A.'h 1 . Moreover ht is fixed on ap, and on every simplex of C on which A = A.', In addition A.' is PL nonsingular; and for each vertex v of C, Lip (A.' -A)< E(v) on ISt(v)l. Proof. Let v (J:

f

ap- P be a vertex of C. There is a local representation

ISt(v)l .... Rm- 1 x R of A. To see this, let U C ISt(v)l be a neighbor-

hood of v and ¢: U .... Rm- 1 x R a local representation of A., so that (1) rr 2 ¢=A. Choose 0 < k < 1 so that if x

f

St(v) then kx + (1-k)v

¢ 0 : ISt(v)l --> Rm- 1 x R by

¢ 0 (x) = ¢(x)- ¢(v) , so that (2) rr 2 ¢ 0 (x) = A.(x)- A.(v) and define (J :

ISt(v)l --> Rm- 1 x R

by

fJ(x) = k- 1 ¢ 0 (kx+ (1-k)v) + ¢(v) Then

since rr 2 : Rm- 1 x R--> R is linear, = k- 1 [A.(kx+(1-k)v)-A.(v)] + A.(v)

f

U. Define

44

MORRIS W. HIRSCH

by (1) and (2), = k- 1 [k.\(x)+ (1-k).\(v)-.\(v)] + .\(v) because A is affine on each simplex, = k- 1 [k.\(x)-k.\(v)] + .\(v) =

.\(x) .

Suppose then that 0: ISt(v)l --> Rm- 1 x R is a PL embedding such that

1T 2 0

= A. Let C' be a subdivision of C in which 0 is simplicially

affine. Given s > 0, let r > 0 be so small that if w

f

IStc(v)l-laStc(v,

is a vertex of C' and IY -A(w)l ::; r, there exists a point z ( int O(IStc(v)l)

n

(Rn- 1xy)

such that lz -O(w)l < s. Let .\': Stc(v)--> R be a simplicially affine map such that .\'=A on astc(v). Thus .\' and .\ agree on every vertex of Stc(v) except v. Suppose I.\' -AI < r. Let

be a PL map which is simplicially affine in the subdivision L of Stc(v) induced by C' such that for each vertex w (3) O'(w)

f

int OIStc(v)l

(4) IO'(w)- O(w)l

f

L we have

n Rm- 1 x.\'(y),

and

< s.

Observe that

since this is true for each vertex of L by (3), and 0' and .\' are simplicially affine in L, while

772

is a linear map. It is easy to see that

as s--> 0, (4) implies that 10'-01--> 0; also ILip(0'-0)1-+ 0 on IStc(v)l and therefore 0' will be an embedding if s is sufficiently small Moreover given q > c we can choose s so small that the PD homotopy, fixed on astc(v),

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS I

45

will be a PD isotopy, and can be approximated by a PL q-isotopy

such that

e0 = e,e1 = e'

and et is fixed on \aStc(v)\. Define a PL

e

isotopy ht of \Stc(v)\ by ht = et- 1 0 . Extend ht to be fixed on

e

P- \Stc(v)\. Then ht = 1 on \astc(v)\, and A'h 1 = A'e1- 1 0 =

e

77 2 e1 e1- 1 0 =A.

Observe that ,\' is PL nonsingular since h 1 exists,

anditiseasytoseethat Lip(A'-A)-.0 and \A'-A\-.0 as s-->0

· smce

77

2 1 · 1·1near. 2 : R n- x R --> R 1s

We have proved 8.1 in the special case where ,\'=A on every vertex of C except v. The general case follows by an obvious globalization passing from A to ,\' by changing A successively over an enumeration of the vertices of C. The reader can supply the details.

Isolating singularities by separating vertices

8.2 THEOREM. Let M be a smooth manifold, P a PL manifold and T : P --> M a PD homeomorphism. Let ,\ : P --> R be a PL nonsingular map which is constant on each component of

ap.

Let N C M be a closed

set such that ,\ T-l : M -. R is regular on bd N. Given

E:

P --> R+ there

zs a PL E-isotopy ht of P such that (a) the map Ah 1 T- 1 : M--> R has only isolated singular points in a neighborhood of N;

(b) Ah 1 T- 1 is regular on bd N (c) ht is fixed on (P-T- 1 N) U

aP.

Proof. Since the set of regular points of AT- 1 is open, bd N has a

neighborhood U C M on which ,\ T- 1 is regular.

46

MORRIS W. HIRSCH

Give P a triangulation C in which .\T- 1 is simplicially affine and so fine that every simplex meeting

1-- 1 (bd N)

lies in U. Let V be the

set of vertices of C lying in P- caPu lstcT- 1 bd Ni) For each vertex v of C choose .\'(v) v f, w and v

f

f

R such that .\'(v) f, .\'(w) if

V. This defines a simplicially affine map .\': C .... R. If

the numbers IA.'(v)-A.(v)l are sufficiently small, the theorem will be satisfied by 8.1 and 5.4. With a view toward the main theorem, we apply the last result to smoothings of M x I. 8.3 THEOREM. Let M be a PL manifold without boundary and a a

smoothing of M x I. Let N C M be a closed set such that a is a product in a neighborhood of (bd N) x I. Given

E:

rel[(M-N)xiUaMx!0,11] toasmoothing

M x I .... R+' a is E-isotopic

f3

suchthat

772

:(Mxl)f3-->I

has only isolated singular points in a neighborhood of N x I, and

77 2

is

smooth of rank 1 on a neighborhood of N x IO, 11. Proof. By uniqueness of collaring (13.10) we may assume that each point of N x 0 has a neighborhood U x [0, r] with r > 0 such that a\U x [O,r] is a product; similarly for N x 1. Therefore we may assume

772

smooth

of rank 1, and hence regular, in a neighborhood of N x !O, 11. The theorem is now a consequence of 8.2 replacing M of 8.2 by M x < 0, 1>, N of 8.2 by N x etc. §9. Removing isolated singularities In this section we prove 4.2, the simplified form of the main theorem. It is proved together with 9.1 by double induction, as follows:

4.2 (n-1)

;::. 9.1 (n+1)

;::. 4.2 (n)

starting with 4.2 (-1) which is vacuous, or 9.1 (1) which is trivial.

47

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS I

Removable singularities Throughout Section 9, K denotes a complex which is the star of the vertex 0 in a rectilinear triangulation of Rm. A simplicially smooth em-

bedding T: K--. Rm, T(O)

=

0 is given; this means that T: \K\--. Rm is

an embedding and T\a: a--. Rm is smooth of rank m for each m simplex

a < K. For example T may be part of a smooth triangulation. We shall denote by rr : K --. R a simplicially affine map which is PL nonsingular (Section 8), and such that rr T- 1 : T\ K\ --. R has an isolated singular point at 0

T\ K\. It follows that rr T- 1 is regular on T\ K\ -0.

f

We say that (rr, T, K) has a removable singularity if there exists a PD isotopy Tt: (1) Tt

=

\K\ --. Rm of T such that T0 in a neighborhood of \aK\ and

(2) rr T1- 1 : T\ K\ --. R is regular.

Note that rr Tt-l

is not required to be regular for 0

< t < 1 except im-

plicitly by (1) on a neighborhood of \aK\. We call Tt a regularizing

isotopy for (rr, T, K). 9.1 THEOREM. Let T: K--. Rm and rr: K--. R be as above.

Then

(rr, T, K) has a removable singularity.

Some lemmas We prove some lemmas whose purpose is to allow T in 9.1 to be replaced by isotopic embeddings and K to be replaced by "substars." The difficulty is that if T 1 : K --. R m is prismatically isotopic to T, rr T1 1 need not be regular on T 1 IK\ - 0. One convenient sufficient condition for this is that rr(v) T1

=

I=

rr(O) for boundary vertices of K. Another is that

f 1 T where ft is a diffeotopy; the inductive hypothesis of the main

theorem is used in 9.12 and 9.13 to construct such a diffeotopy. Still another method of making rr T1 1 regular on T 1 \ K\ - 0 is to make sure that T 1 is sufficiently Lipschitz close to T.

48

MORRIS W. HIRSCH

9.2 LEMMA. Let L be the star of 0 in a triangulation of IK!. If (rriL, TIL, L) has a removable singularity at 0, so has (rr, T, K). Proof. Let st: ILl--> Rm be a regularizing isotopy for (rriL, TIL, L). De·

fine Tt: IKI __, Rm to equal St on ILl and T on IKI- ILl. Then Tt is a regularizing isotopy for (rr, T, K).

9.3 LEMMA. Let L be the star of 0 in a subdivision of K. Let rr': L--> R be a simplicially affine map such that (a) there is a PL isotopy ht of IKl, fixed on IKl - ILl such that rr = rr'h 1

(b) (rr', TIL, L) has a removable singularity. Then (rr, T, K) has a removable singularity. Proof. Let st: ILl --> Rm be a regularizing isotopy for (rr', Tl L, L). Ex-

tend st toequal Ton IKI-ILI. Put Tt=Stht:IKI-->Rm. Then -1 , -1 rr T 1-1 = 77 h-1 1 8 1 = rr 8 1 , which is regular. Therefore T t is a regularizing isotopy for (rr, T, K). 9.4 LEMMA. Given 0 < s 1 < s 2 < 1, there exists E > 0 with the following property. Suppose there exists T 1 :I Kl --> Rm, a PO embedding such that

(a) T 1 is PO isotopic to T rel a neighborhood of IaKI (b) rrT! 1 is regular on int ls 1 KI- 0 (c) Lip(T 1 -T) 0 and Lip(TT! 1 -1) .... 0 as

IT-T 1 1

__,

0 and Lip(T-T 1 ) __, 0.

49

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS I 9.5 LEMMA. Suppose rr(v)

-f rr(O) for every vertex v f dK. Let Tt: K-->Rm

be a prismatic isotopy of T. Then rr T1 1 is regular on int T 1 Kl - 0. 1

Proof. Follows from 6.9. Some special isotopies Let A C B C IKl be any subsets. A map f: B .... Rm is called radial on A if for each interval [tx, x] C A with t some s

f

I. If in addition s

=

f

I, f[tx, x]

t, that is if f(tx)

=

[sf(x), f(x)] for

=

tf(x) whenever t

f

I

and [tx, x] C A, we call f affinely radial on A. 9.6 LEMMA. Suppose TIKI= om

c

Rm, the unit ball. Assume T is

affinelyradialinanneighborhood U of ldKI in IKI. Let O

on+ 1 . This is a prismatic isotopy of T fixed in

a neighborhood of laKI such that T 1 lasKI topy, 77T1 1

is regular on on+ 1 -

=

sD~. Since ¢t is a diffeo-

0. Also T 1 is affinely radial in a

neighborhood of la(sK)I. Observe that T 1 ((asK) words 77T1 1 (0)

n

n 77- 1 (0)) =

a(son+ 1 )

sSn = ssn- 1 = 77n+ 1- 1 (0)

n 77 n+ 1-

n sSn.

1 (0).

In other

Moreover 77T1 1 is

smooth of rank 1 in a neighborhood of ssn- 1 in sSn. By uniqueness of smooth collaring, there is a diffeotopy ht of sSn such that (1) 17 n+ 1 h 1 = 17 T; 1

on a neighborhood of ssn- 1 in ssn. We extend ht to a diffeotopy of on+ 1 in a way similar to that of the extension of ¢t above, and compose ht with T 1 to find a prismatic -->

on+ 1 of T 1 with the following properties (0

=

T 1 in a neighborhood of laKI

isotopy T 1 +t: K (2) T 1 +t

(3) T 2 1sKI

=

:S t :S

1):

son+ 1

(4) T 2 . is affinely radial in a neighborhood of la(sK)I

(5) 77T2 1 : on+ 1 --. R is regular on on+ 1 - 0 (6) 77T2 1

=

77n+ 1 on a neighborhood of ssn- 1 in sSn.

This last follows from (1) since

From (6) and (4) we get (7) 77T2 1 in

=

17n+ 1 on a neighborhood of ssn- 1

on+ 1 . By 9.3 it suffices to prove 9.1 for (771sK, T 2 1sK, sK). This proves 9.13.

54

MORRIS W. HIRSCH

PL approximation and the Alexander trick We now complete the proof of 9.1 (n+1). Let T: K -> Dn+ 1 be as in 9.13. Explicitly, we assume 77(0)

=

0 and:

(1) There exists 0 < q < 1 such that T is affinely radial in a neighborhood of IKI - int lqKI. (2) There exists b > 0 such that 77 of 77- 1 [-b, b]

n (IKI

- int lqKI

77 n+ 1 T on a neighborhood

=

P.

=

Let q < r < s < 1. Given 0 > 0, there exists by a PD homeomorphism T 1 : IKl .... on+ 1 such that: (3) T 1

=

T on IKI - int lsKI

(4) T 1 : lrKI .... Rm+ 1 is PL (5) IT 1 -TI < o and Lip IT 1 -TI < o (6) T 1 is PD isotopic to T rel IKl - int IsKI (7) 77n+ 1 T 1 If

o

=

77n+ 1

=

77 on P.

is sufficiently small, then from (1) we can easily obtain (8) T 1 lrKI is strictly convex, and 0

f

int T 1 lrKI.

We now define a new PD homeomorphism

completely determined by T 1 and r, by declaring (9) T 2

=

T on IKI- int lrKI

and (10) T 2 : lrKI

->

Rn+l is affinely radial.

In other words, T 2 II rKI is the cone from 0 on T 2 : I(a(rK)I From (4) and (8) we have (11) T 2 II rKI is a PL embedding.

->

Rn+l.

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS I

55

Put Pn laKI

=

Q = lxtlaKI: 77(x):S: bl.

Let C(Q)=Itxt separately on C(L+) and C(L_), where L+

=

Ia( L\rr(a) c

L_

=

{afL\rr(a)C R_l.

R+ l

The existence of ¢t\L+ follows from 6.11, putting C L(v)

=

L+' f

=

rr, g

=

=

C(L+), v

=

0,

"m+ 1 . Similarly for L_. This completes the proof

of 9.14. We have proved the implication: Theorem 4.2 (n-1)

'> Theorem 9.1 (n+1)

Proof of simplified form of main theorem

We now prove 9.1 (n+1)

l>

4.2 (n), which will complete the proofs of

9.1 and 4.2. In 4.2 (n) we have a PL n-manifold M without boundary, a closed set N C M of diameter d and a smoothing a of M x I which is a product in a neighborhood of (bd N) x I. Given E > 0 we are required to find f3 f S(M xI) such that a :::: d+E f3 rel M x 0 U (M- N) x I and which is a product in a neighborhood of N x I. By 8.3 and an E/3-isotopy we

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS I

may assume that

57

: (M x 1){3 --> I is smooth of rank 1 in a neighborhoo:I

TT 2

of (bd N xI) U (N x bd I) and has only isolated singular points in a neighborhood W of N x I. By applying 9.1 (n+1) to a small neighborhood of each singular points of

TT 2

: (W -aw)a--> I,

c:/3-isotopy that

TT 2

noting that dim W = n+1, we can assume after an

: (M x I)a --> I is regular in a neighborhood of N x I.

By 6.6 there is an E/3 + d isotopy rel M x 0 U (M- N) x I which fulfills the requirements of 4.2 (n). §10. Strongly relative form of the simplified main theorem 10.1 THEOREM. Let M be a PL manifold without boundary, N C M a compact set, and K C N a closed set. Let a be a smoothing of M x I which is a product in a neighborhood

uX

I of (K n bd N)

X

I. Given

E > 0 there exists a PD isotopy Ht: M x I .... (Mx I)a such that: (a) Ht

=

1 on M x 0 U (M- N) x I;

(b) H;a is a product smoothing on a neighborhood of K; (c) \TT 2 Ht-TT 2 \


Mw is a PD isotopy rel M- N, where

w = a\M x 0. Put

58

MORRIS W. HIRSCH

Then Gt is fixed on M x 0 U (M- N) x I, and there are diffeomorphisms G1 : w x

f1 1 x 1

t -

()

x

t

F1 ---->a in a neighborhood of N x I. Hence G:a

is a product in a neighborhood of N x I. The trouble is that Gt need not be a PD isotopy. By 13.9 however, we can approximate Gt by a PD isotopy Ht: M x I .... (Mx I)a such that Ht = Gt on M x 0 U (M- N) x I, for all t < I, and also H 0 = G0 , H 1 = G1 . This will prove 10.1. We proceed to fill in some of the details of this argument. Let

N0 C int N be a compact PL submanifold of K - U.

10.2 LEMMA. Given

E:

> 0 there is a PD isotopy F t: M x I .... (M x I)a

such that:

(a) F1 ((aN 0 ) x I is a smooth submanifold; (b) the induced smoothing on (aN 0 ) x I is a product (c) Ft is a product isotopy ft x 1 in a closed neighborhood

u0

X

I of K

n aN 0 in int N.

(d) Ft is fixed on a neighborhood of (M-int N) x I (e) lrr 2 F -rr 2 1 < E:.

Proof. By uniqueness of collaring we may assume a is a product in a neighborhood of N 0 xI. By 7.4 there is a small PD isotopy of (Mx O,a 0 ) taking

aN 0

onto a smooth manifold. Extend this isotopy to a small

isotopy ut of (M x I)a in such a way that ut is a product in a neighborhood of No (M-int N)

X

X

0 u (K n aNo)

I. Let

f3

I, and ut is fixed in a neighborhood of

X

= ul a

f

S(M X I). Then (aNo X 0)

u

[(K n aNo)x I]= X

has a neighborhood in (aN 0 ) x I which is a smooth submanifold of {3; and the smoothing is a product in a neighborhood of (K n aNo)

X

I. By

7.4 there is a small PD isotopy v of (Mxl)f3 taking (aN 0 ) xI onto a smooth submanifold of

f3

which is fixed in a neighborhood of X and on

a neighborhood of (M- int N) x I; in particular it is fixed on M x 0. Let

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS I

y=

v;f3;

59

then (oN 0 ) xI is a smooth submanifold of y, and the smooth-

ing is a product in a neighborhood of (K n oNo) isotopy w t of (M x I) a'

wt

=

X

I. Observe that the PD

{u2 u0:St:S~ u1 v2t-1,

~ :S t :S

is a product in a neighborhood of (K n oNo)

X

1

I. Hence, replacing a by

w;a, it suffices to prove 10.2 under the extra assumption: (oN 0 ) xI is a smooth submanifold of (Mx I)a and the induced smoothing w of (oNo)

X

I is a product in a neighborhood of (K n oNo)

Given

o > 0,

X

I.

by 4.2 there is a PD isotopy gt of [(oN 0 )x I]w such

that g;w is a product smoothing, gt is fixed in a neighborhood of (KnoN 0 ) xI, and lrr 2 gt-rr 2 l < o. This isotopy extends to the required isotopy Ft of (M x I)a if

o

is sufficiently small, proving 10.2.

To prove 10.1, define a PD homeomorphism g: (Mx I)-. (Mx I)a x I by g(p, t) = (Ft(p), t) where F is as in 10.2. Set Ft(x, 0) = (ut(x), 0), let a 0 = aIM x 0, and define a PD homeomorphism ut x 1 x 1 = f: M x I x I-> Ma x I x I where

=aiM x 0. 1 2 We may assume Ft = F0 for 0 :S t :S 3 and Ft = F1 for 3 :S t :S 1. Then gf- 1 : Ma x I x I-. (Mx I)a x I is a diffeomorphism on a neighbor0

a0

0

hood of A = M x I x 0 U [M x 0 U (M-N)x I] x I U (N xI) x 1. Moreover rr 3 g = rr 3 f = 7T 3 : M x I x I-> I. By 13.9 there is a PD homeomorphism h : M x I x I -> (M x I)a x I agreeing with gf- 1 in a neighborhood of A, preserving rr 3 , and approximating gf- 1 as closely as desired. Put h(x, s, t) = (Ht(x, s), t). ·Then Ht: M x I-. (Mx I)a satisfies 10.1.

§11. Proof of the main theorem Let K, N, M, a, Let

o: M x

E:

be as in 4.1. Put m =dim M.

I-. R+ be such that if u 0 , ... ,urn are 8-maps of M x I

then the composition urn o ••• o u1 is an €-map.

60

MORRIS W. HIRSCH

Give M a triangulation such that (1) diam

a RP by setting £cf(v) vertex v

f

=

f(v) for every

C. He proves that if f is nondegenerate, then for any

and any triangulation C 0 of P there exists a refinement C of

E

>0

c0

such that

Our second remark concerning Whitehead's theorems is that if ¢: RP-> Rs is an affine map such that ¢f: C 0 -> Rs is simplicially affine, then ¢a £cf

=

¢f for every subdivision C of C 0 • We have

proved: 13.1 THEOREM. Let PC Rq be a compact polyhedron and f: P .... RP a

nondegenerate PD map. Let ¢ : RP .... Rs be affine and suppose ¢f: P .... Rs is PL. Let P 0 C P be a compact polyhedron such that f IP 0

zs PL. Given (a)

E

> 0 there exists a PL map g: P -> RP such that

II f- gil Lip

P a, g: M ->

Qf3

f3

be smooth-

be PD homeo-

morphisms. Suppose A C P is a closed set such that gf- 1 : Pa->

Qf3

IS

a diffeomorphism in a neighborhood U of A, and P- U is compact. Given

E

> 0 there exists a PD homeomorphism h: P-> Qf3 such that

(a) h

=

gf- 1 in a neighborhood of A,

(b) ih-gC 1 i Rs are PL maps such that ¢ : P a .... Rs and (}: Qf3 ~ R are pseudo-affine, and ¢f

=

1/J, (Jg = 1/J,

then (c) (Jh = ¢.

Proof. Let n be the dimension of P, M and Q. Let B 1 C B C P and C C M be compact PL submanifolds of dimension n such that P-UC int B 1 C int f(C), f(C) C int B B cP-A.

Given

o > 0,

by 13.6 there exists a PD homeomorphism f 0 : M .... P a

such that:

f 0 1C is PL

f0

=

f on M- f- 1 B

llf-foiiLip < 0 · (For statements about norms we assume P, M, Q embedded in Rq as polyhedra and P a,

Qf3

embedded in RP as smooth submanifolds.)

Consider the PD map

72

MORRIS W. HIRSCH

as an approximation on CIB 1 to the PD map gC 1 \B- int B 1 : B- int

s1

-> Qf3, this last map being PD because B- int B 1 C U by (1). By 13.5 if

1\(gfi)l- gCl)\CIBli\Lip is small enough, there exists a PD map h 0 : B- int B 1 -> Qf3 such that h0

= gf 01 on CIB 1 , Oh0 = cp on B- int B 1 and l\h0 - gC 1 1\Lip < E.

Extend h 0 to h: P-> Q by making h

=

gf01 on B 1 . This completes

the proof of 13.9. Observe that we lose control of Lip (gf01 ) outside of U; for this reason the approximation (b) of 13.9 is only

c0 .

Uniqueness of collaring

13.10 PROPOSITION. Let M be a PL manifold, CIM

=

0 and a a

smoothing of M x R+. Let N C M x R+ and K x 0 C (Mx 0)

nN

be

closed sets. Let W C M x R be an open set containing M x 0 U N and let be a PD embedding. Suppose (a) f \M x 0 = 1 (b) f

1 in a neighborhood U C W of K x 0

=

n bd N.

Given E: W-> R+ there is a PD E-isotopy ft: W --> (Mx R)a of f

=

£0

such that (c) ft

=

f outside N and on M x 0, 0 'S t

'S

(d) f 1

=

1 in a neighborhood of K x 0 in W.

1

Proof. We may assume N C W n f(W). Since the proposition is strongly relative in form (see Section 2) we shall prove it only for the case K compact, leaving the easy globalization to the reader. In this case we may assume N

=

N 0 x (0, q], q > 0, where N0

taining K; and U of K

n bd N0 .

=

U0 x (0, q] where U0

C

C

M is a compact set con-

M is an open neighborhood

For simplicity of notation assume q

=

1.

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS I

73

By 13.7 we can make f PL in a neighborhood of K x 0, keeping f fixed on W- N. Therefore we assume f is PL. Henceforth we ignore smoothings and work entirely in the PL category. Choose 0 < b < c < 1 and a compact neighborhood N1 C int N0 of K0 = K- U0 such that f(N 1 x [0,0]) C N0 x [0,1 > f- 1 (N 1 x[O,O])c N0 x[0,1> N1 x [0,2b]

c

f(N 0 x [0,0>) .

Extend f to F:WU (MxMxR, setting FIMxt of M x R such that if (x,y)fN 1 xt= 1 on (M-No)

X

R+

u

No

X

The Isotopy ft =

[c, oo>.

cpt

1 Fcf>tiW:

W--> M x R+ has the required properties.

The Alexander trick The following beautiful idea, due to Alexander [1] is what makes the theory of PD isotopies apparently more powerful than that of prismatic isotopies. Alexander worked in what we would call the topological category. The proof below is essentially his proof given a geometrical interpretation. 13.11 THEOREM (Alexander). Let E be a PL n-cell and f: E-> E a PL

homeomorphism fixed on aE. Then f is PL isotopic to 1 rel aE. Proof. Take E as a convex polyhedron in Rn with 0 as an interior point. Define F to be the homeomorphism of X = E x 1 U (aE) x I which is f on E x 1 and the identity on (aE) x I. Consider Ex I C Rn x R to be the cone from (0, 0) on X. Let G: Ex I .... Ex I be the cone on F, i.e., the radial extension of F. Since G preserves rr 2 : E x I --> I, we can write G(x, t) = (ft(x), t). Then f 1 = f and f 0 = 1E; hence ft is a PL isotopy from 1E to f.

BIBLIOGRAPHY [1].

Alexander, J., On deformations of the n-cell. Proc. Nat. Acad. Sc. 9 (1923), pp. 406-407.

[2].

Cairns, S., Homeomorphisms between topological manifolds and analytic manifolds. Annals of Math. 41 (1940), pp. 796-808.

[3].

Hirsch, M. W., Obstructions to smoothing manifolds and maps. Bull. Amer. Math. Soc. 69 (1963), pp. 352-356.

[4].

83 (1966).

, On tangential equivalence of manifolds. Ann. of Math.

[5].

and Mazur, B., Smoothings of piecewise linear manifolds. Cambridge Univ. 1964 (mimeo).

[6].

and Milnor, Bull. Amer. Math. Soc.

[7].

and Pugh, C., Stable manifolds for hyperbolic sets, Proceedings of Symposia in Pure Mathematics XIV, Providence 1970.

[8].

Kervaire, M., A manifold which does not admit any differentiable structure. Comm. Math. Helv. 34 (1960), pp. 304-312.

[9].

Kuiper, N., On smoothings of triangulated and combinatorial manifolds. Differential and Combinatorial topology. A symposium in honor of Marston Morse. Princeton 1965, pp. 3-22.

J.,

Some curious involutions of spheres.

[10]. Lashof, R., and Rothenberg, M., Microbundles and smoothing. Topology 3 (1965), pp. 357-380.

[11]. Mazur, B., Seminaire de Topologie combinatoire et differentielle. lnst. Hautes Etudes Scien. 1962. [12]. Munkres, J ., Obstruction to the smoothing of piecewise differentiable homeomorphisms. Ann. of Math. 72 (1960), pp. 521-554. [13]. [14]. [15].

Ill.

J.

, Obstructions to imposing differentiable structures. Math. 8 (1964), pp. 361-376.

, Concordance is equivalent to smoothability. Topology 5 (1966), pp. 371-389. Ill.

J.

, Compatibility of imposed differentiable structures. Math. 12 (1968), pp. 610-615.

74

BIBLIOGRAPHY

[16]. [17].

75

, Elementary Differential Topology. Ann. of Math. Study no. 54, Princeton 1961. Mich. Math.

, Concordance of differential structures: two approaches. J. 14 (1967), pp. 183-191.

[18]. Rourke,C., and Sanderson, B., Block Bundles. Annals of Math. 87 (1968), pp. 1-28, 255-277, 431-483. [19]. Smale, S., Generalized Poincare's Conjecture in dimensions greater than 4. Annals Math. 74 (1961), pp. 391-406. [20]. Sullivan, D., On the Hauptvermutung for manifolds. Bull. Amer. Math. Soc. 63 (1967), pp. 598-600. [21]. Thorn, R., Des varietes triangulees aux varietes differentiables. Proc. Int. Cong. Math. Edinburgh 1958, pp. 248-255. [22]. Whitehead, J. H. C., On pp. 809-824. [23].

d

complexes. Ann. Math. 41 (1940),

, Manifolds with transverse fields in Euclidean space. Ann. Math. 73 (1961), pp. 154-212.

[24]. Zeeman, E. C., and Hudson, J. F. P., On regular neighborhoods. Proc. London Math. Soc. 14 (1964), pp. 714-745.

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II: CLASSIFICATION Morris W. Hirsch and Barry Mazur

Introduction Let S(M) denote the set of concordance classes of smoothings of a PL manifold M. The object of this paper is to reduce the study of S(M) to tractable, or at least standard, topological problems. This is done in several related ways. The first approach is based on the fact that if M is smoothable, then the diagonal M~ C M x M has a vector bundle neighborhood. It turns out that S(M) is naturally isomorphic to the set of stable equivalence classes of such neighborhoods. The proof of this is purely geometrical. Block bundle theory identifies the set of stable equivalence classes of vector bundle neighborhoods of

M~

in M x M with the set of homotopy

classes of sections of a fibration &(M) over M with fibre PL/0. This fibration is induced from the fibration BO

->

BPL by the map M--> BPL

that classifies the stable normal block bundle of

M~

in M x M (or

equivalently, the stable tangent microbundle of M). In this way the study of S(M) is reduced to the study of sections of the fibre space &(M). For any a

f

S(M) one can establish, by geometric means, an abelian

group structure on S(M) with a equal to the origin. Forming f' = PL/0, the fibre of the fibration (e) BPL ... BO, we may also establish a natural isomorphism [M, f']

~ S(M)

taking 0 to a.

77

78

MORRIS W. HIRSCH AND BARRY MAZUR

By such considerations, the functor [-,r] is endowed with an

r

abelian group structure. Consequently

is endowed with a homotopy

abelian and homotopy associative h-space structure. The fibration (E) may then be shown to be

r -principal,

and S(M)

corresponds to homotopy classes of its sections. Going full circle, S(M) is then a principle homogeneous space over the abelian group [M, r]. Moreover' choosing

a (

S(M), the natural iso-

morphism(*) above is given by the rule: [M,ll ~ S(M)

y

I->

y·a

If M is a smooth manifold, let S(M) denote S(M') where M' is

some PL manifold isomorphic to a smooth triangulation of M. Then there is then a natural isomorphism S(M) ~ [M, r]. In other words' the functor M

1->

S(M), from smooth manifolds to sets, is equivalent to the homotopy

functor M

t-->

[M, r], from smooth manifolds to abelian groups. Historical remarks

At the International Co~gress of Mathematicians in 1958, Rene Thorn [29] suggested that smoothings ought to be classified by sections of a fibration. Subsequently

J.

Munkres [17; 18] found obstruction theories

for certain smoothing problems, and A. Gleason (1959, unpublished) proved that every contractible open PL manifold could be smoothed. These results were further evidence for Thorn's idea. Milnor conjectured that .\S(Sn) is isomorphic to

11

n(PL/0), which suggested that the fibre

should be PL/0. Moreover, he invented microbundles [15; 16] and showed that a PL manifold M is smoothable if and only if its tangent microbundle has a vector bundle structure. In the meantime the stability theorem S(M) ~ S(M x g{) was proved by Hirsch [6] and Mazur-Poenaru [14] and was used to verify Milnor's conjecture.t Another obstruction theory was obtained by Hirsch [6].

t

We denote the real number by ffi and Euclidean n-space by ffin.

79

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II

The classification of S(M) by liftings of M --> BPL

o~er

BO .... BPL

was proved in Mazur-Poenaru [14]. The subject was much discussed at the topology conference held in Seattle in 1963, where the general outline emerged of a classification of concordance classes of smoothings of M by maps M --> PL/0. Lashof and Rothenberg [13] published a partial classification based on these ideas. In 1966 Haefliger [5] and others obtained results on "ambient" smoothings of a PL submanifold of a smooth manifold, in connection with smooth knots. Recently D. White [30] has classified ambient smoothings using semisimplicial complexes and block bundles. The present paper was begun in 1963 at Cambridge University, following the Seattle conference. A set of mimeographed notes was produced but political turmoil at the University of California interrupted completion of the paper. In the meantime new results in topology were discovered; some of these have been used to simplify the theory of smoothings. In particular the theory of block bundles of Rourke-Sanderson [23], Morlet [17] and Kato [11] makes it possible to avoid the use of semisimplicial complexes (except in Section 6) and the stable PL tubular neighborhood theorem of Lashof-Rothenberg. (This theorem is used implicitly in Section 6 in identifying the classifying spaces for stable PL microbundles and stable block bundles.) Outline of this paper

The concordance classification of smoothings of a PL manifold M is carried out in stages. First the problem is shown to be stable: the natural map S(M)--> S(Mx~n) is bijective, where S(M) is the set of concordance classes of smoothings of M. This was done in Part I. The next step is to extend this to nontrivial vector bundles; there is a bijective map where x

X! : S(M) --> S(E) =

(p, E, M) is a piecewise differentiable vector bundle. This is

carried out in Sections 1 and 2; the main results are 2.6 and its variations 2.7, 2.8 and 2.9.

MORRIS W. HIRSCH AND BARRY MAZUR

80

In Section 3 linearizations of a PL manifold pair (V, M) are studied. A linearization of (V, M) is a PD vector bundle (p, E, M) where E C V is a neighborhood of M and p: E --> M is a retraction; the set of equivalence classes of linearizations of (V, M) is called ~(V, M). The direct limit: is ~s(V, M), the set of stable linearization classes. The main results of Section 3 are as follows. Theorem 3.2 says: M is smoothable if and only if ~s(MxM,Mf1) I=¢, where ML'1 is the diago-

nal. This is a strengthening of Milnor's theorem that M is smoothable if and only if the tangent microbundle of M has a vector bundle st:ucture. Theorem 3.4 says: the map

is a complete pairing. Here S(V, M) is the set of concordance classes of

germs of smoothings of neighborhoods of M in V. The map assigns to a linearization x of (V x :R m, Mx 0) and a smoothing a of V x the smoothing f3 of M such that Mf3 x neighborhood of M x 0 in V x

:R m.

:R m

:R m,

is concordant to x !a in a

To say is a complete pairing

means that the maps a ~ (x,a) and x ~ (x,a) are bijective when x or a , respective Iy, is fixed. The final result of Section 3 is the classification Theorem 3.10. This says that the map Exp: S(M)--> ~s(Mx M, Mf1) is bijective, where Exp assigns to a smoothing a of M the stable linearization of a neighborhood of the diagonal, obtained from the exponential map of a Riemannian metric for a. The proof of 3.10 is based on Lemmas 3.8 and 3.9, which ultimately depend on the smooth tubular neighborhood theorem for ML'1 C M x M. Lemma 3.8 says that if a,w where (

f

S(M) then (Exp a)!w = (wxa)f1 = (axw)f1

)f1 means the ML'1 germ of a smoothing of a neighborhood of

the diagonal, and Exp: S(M)--> ~(Mx M, Mf1) is the unstabilized form of Exp. Lemma 3.9 interprets 3.8 to prove:

81

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II

(Exp a, w)

=

(Exp w, a) .

Theorem 3.10 is a formal consequence of this relation and 3.4. The classification 3.10 is an isomorphism between two sets of geometrical objects: smoothings of M, and stable linearizations of a neighborhood of the diagonal. In Section 4 the study of .fs(V, M) is reduced to a homotopy question by the theory of block bundles. Therefore: S(M) is isomorphic to homotopy classes of liftings of the map M .... BPL over t: BO --> BPL. Here M --> BPL classifies the stable tangent block bundle of M, or equivalently, the stable tangent microbundle of M; while t: BO .... BPL is defined by triangulating vector bundles. Pulling back the fibration t over M via M --> BPL gives an induced fibration &(M)--> M whose fibre is PL/0. Therefore S(M) ~ homotopy classes of sections of &(M). The fibre

r

= PL/0 turns out to be an h-space which is homotopy

associative and homotopy commutative. This is proved in Section 4 using the finiteness of S(sn) and a geometrically defined abelian group structure on S(M) (for smoothable M). The fibration &(M) is shown to be !'-principle and the isomorphism S(M) ~ [M, rJ obtained. Obstruction theories for smoothing a manifold, or a PD homeomorphism between smooth manifolds, are derived in Section 5. In Section 6 an alternative description of h-structure on PL/0 is given, relating it to the h-structures on BO and BPL. No homotopy theory is used. Results used from I In deriving the geometric classification 3.10, the main result needed from Part I is the M x ~n theorem: the natural map S(M)--> S(Mx~n)

zs

bijective. This is Theorem 7.8 of I. The stronger M x I theorem is needed whenever the fact that concordance implies isotopy is used; for the proof in Section 4 (but not Section 6) that PL/0 is an h-space, for example, and in developing some of the obstruction theories in Section 5.

82

MORRIS W. HIRSCH AND BARRY MAZUR

§1. Vector bundles (1.1) A vector bundle atlas on a map p: E .... B is a set ={(.,U.)!. A of homeomorphisms ¢·: p- 1 u 1..... 1

u.1 x

~n such that:

1

1

1(

(1) {UilifA is an indexed open cover of B; (2) for each i f A the diagram

commutes, rr 1 being the projection; (3) for each i, j fA there are maps g .. : U. n U ..... GL(n) such J1 1 J that the homeomorphism ¢. 0 ¢:- 1 of (U. n u.) X ~ n is J 1 1 J given by (x,y) t--> (x,gji(x)y). The elements (¢i, Ui) are called local trivializations; the gij are

transition functions. A maximal vector bundle atlas on (p, E, B) is a vector bundle struc-

ture. The quadruple (p, E, B, ) is a vector bundle. (1.2) Vector bundles with "additional structure" of various kinds are defined by imposing restrictions on the family of transition functions. If the base space B is a polyhedron, for example, it makes sense tore-

n Uj--> GL(n) be PD, i.e., smooth (= C"") on each simplex of some triangulation of U i n U j. A PD vector bundle quire that each map gji: Ui

.atlas is a vector bundle atlas having this property. A maximal PD vector bundle atlas 0 is called a PD vector bundle structure; the quadruple (p,E,B, 0 ) is a PD vector bundle.

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II

83

(1.3) Let (p, E, B, x 1 is a map f: E 0 -> E 1 which covers a map g: B 0 -> B 1 , such that for each (¢, U) (f*¢, g- 1 u)

E

E

~n is the projection. Now suppose that B 0 and B 1 are polyhedra. If (

1

(pk, Ek, Bk, k) is a PD vector

=

is a PD bundle map provided the in-

duced map g: B 0 --> B 1 is PL, and also that f* 1

=

0"

(1.4) THEOREM (Covering homotopy theorem for PD vector bundles).

Let B be a polyhedron, A C B a subpolyhedron and x a PD vector bundle over B x I. Let r: B x I --> B x 0 be the retraction r(x, t)

=

(x, 0).

If £0 : x \Ax I .... x \Ax 0 is a PD vector bundle retraction covering r

\Ax I, then f 0 extends to a PD vector bundle retraction f: x-->x!Bx 0

covering r.

Proof. The proof is almost identical to the usual simplex-by-simplex proof of the covering homotopy theorem for vector bundles over a simplicial complex, and is left to the reader. The next result implies existence and uniqueness (up to PD vector bundle isomorphism) of PD structures on vector bundles.

(1.5) THEOREM. (a) Let B be a polyhedron, x a vector bundle over B, and IJI0 a PD structure on

xi A.

Then there exists a PD structure

IJI on x which restricts to IJI0 over A. (b) Let x 0 , x 1 be PD vector bundles over polyhedra B 0 , B 1 and f: x 0 --> x 1 a vector bundle map covering a PL map g: B 0 --> B 1 . Let

A C B 0 be a subpolyhedron such that the restriction fA: x 0 \A--> x 1 _is a PO vector bundle map. Then there exists a homotopy ft: x 0 --> x 1 of

vector bundle maps such that for all 0 S t :S 1:

and

(i)

ft covers g;

(ii)

ftA = fA;

(iii) £1 : x 0 --> x 1 rs a PD vector bundle map.

85

SMOOTIIINGS OF PIECEWISE LINEAR MANIFOLDS II

Proof. We prove (b) first. By considering the commuting diagram

of bundle maps, we may assume B 0 = B 1 = B, and g = 1 8

.

The standard

induction on dim(B- A), and simplex-by-simplex proof of the inductive step, permit us to assume that B is a simplex

and A=

~

a~.

The

covering homotopy theorem implies that x 0 and x 1 are isomorphic to the trivial PD vector bundle e

=

(rr 1 , ~ x 9{ n, ~) (whose PD atlas con-

tains the identity map of ~ x 9{ n). Therefore we may assume x 0 = x 1 =e. Then the bundle map f : e .... e is determined by a map h : ~ .... GL(n) which is PD on a~. A homotopy ht: ~ .... GL(n) of h

=

h 0 such that

ht = h on a~ and h 1 is PD determines the desired homotopy ft. This proves (b). To prove (a) we may assume B

= ~,

A

= a~,

and x is a trivial

vector bundle. Let f: x .... e be a vector bundle isomorphism covering 1~. By (b) the vector bundle rna p fa~ :X Ia~

. . e Ia~

is homotopic'

through vector bundle maps, to a PD vector bundle map g: (xla~, 'I' 0 ) .... ela~. The homotopy from fa~ to g extends to a homotopy ft: x .... e of

vector bundle maps covering 1~. The PD structure 'I' on x induced by f 1 extends '1'0 . The proof of 1.4 is complete. (1.6) Triangulations of PD vector bundles To each PD vector bundle x

=

(p, E, B, ) we associate a class of

triangulations of the space E, as follows. A triangulation T of E is called compatible with x (or with ) if there exists a triangulation T0 of B and an atlas 0 C ci> adapted to T (see 1.3) such that for each simplex ~ C B of T0 , p- 1 ~ is a subcomplex of T, and in addition each simplex of p- 1 ~ is smoothly embedded in the smooth manifold

p- 1 ~. In other words T induces a smooth triangulation of p- 1 ~.

86

MORRIS W. HIRSCH AND BARRY MAZUR

Thismeansthatif (¢,U) il x 5{n is a smooth embedding. Such a triangulation of E always exists. This is proved by applying J.H.C. Whitehead's theory of smooth triangulations to il x 5{n, extending

ail x 5{n

a given smooth triangulation of

which corresponds via a local

trivialization to a compatible triangulation of x\ail. Compatible triangulations of x are unique up to PL isomorphism. If the full strength of Whitehead's theory is used, as developed for example in I, Theorem 13.4, then the compatible triangulation T of E can be chosen to have the following property. The map p: E -> B is PL in the PL structure determined by T, and (p, E, B) is a locally trivial bundle in the PL category, with fibre the PL manifold 5{n. If T' is another compatible triangulation of E having the same property, then there exists an isotopy ft: E-> E covering lB, having the following properties: f 0

= lE;

f 1 : (E, T') -> (E, T) is a PL isomorphism; and

ft\p- 1 11 is a PD isotopy from the PL structure induced by T' into the differential structure induced by x, for each simplex il of T0 . Moreover ft\p- 111 is the identity if T and T' induce the same PL structure on p- 1 11. (For the existence of compatible triangulations use 13.4 of I, with M = il x 5{n and ¢ = rr 2 :il x 5{n-> il, etc. For isotopy either I, 13.4 or the covering homotopy theorem for PL bundle maps can be used.) If B is a· PL manifold, the PL structure on E induced by a com-

patible triangulation makes E a PL manifold. (1.7) Whitney sums

Let x = (p, E, M, ) and y

=

product PD vector bundle x x y

(q, F, N, 'I') be PD vector bundles. The =

(p x q, Ex F, Mx N, ®) has the PD

vector bundle structure ® determined by the PD vector bundle atlas

! (¢ x l{f, Ux V)\ (¢, U)

(g(x), h(x)y) .

and f3 are the induced differential structures on E' and E, this

is equivalent to saying that f :

E'fJ' --.

Ef3 is a smooth map.

(1.8) THEOREM (Covering homotopy theorem for differentiable vector bundles). Let x

=

(p, E, Max I,~) be a differentiable vector bundle where

Ma is a smooth manifold. Let A C M be a closed set and U C M an open neighborhood of A. Let

be differentiable vector bundle retraction covering the retraction r\Ux I, where r: M x I --. M x 0 is defined by r(x 1 t)

=

(x 1 0). Then there exists a

differentiable vector bundle retraction f: x-> x\Ma x 0 which covers r and extends f 0 \V x I /or some neighborhood V C M of A.

89

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II

We leave the proof to the reader; for example, the proof of 1.9(b) below can be combined with the standard covering homotopy theorem for vector bundles. The next result is the uniqueness and existence theorem for differentiable vector bundle structures. (1. 9) THEOREM. (a) Let Ma be a smooth manifold and x

=

(p, E, M, )

a vector bundle. Let A C M be a closed set, U C M a neighborhood of

A, and '1'0 a differentiable vector bundle structure on

x\ U

over Ua.

Then there is a differentiable vector bundle structure 'I' on x over Ma such that '1'\V

=

'1'0 \V for some neighborhood V of A in U.

(b) Let xi= (pi,Ei,Mi,) be differentiable vector bundles over smooth manifolds Mi for i = 0, 1. Let f: x 0 -. x 1 be a vector bundle map covering a smooth map g: M0 --. M1 . Suppose U C M0 is a neighborhood of a closed set A C M0 such that fu: x 0 \U--. x 1 is a differentiable vector bundle map. Then there exists a homotopy ft : x 0

->

x 1 of vector

bundle maps such that f 0 = f, ft = f over a neighborhood V of A in U, and f 1 : x 0 -> x 1 is a differentiable vector bundle map. Proof. A proof analogous to that of 1.5 is not difficult. Alternatively, to prove (b) for example, observe that vector bundle maps over g correspond to sections of a certain bundle y over M0 with fibre GL(n), and group GL(n) acting by conjugation. y has a natural differentiable bundle structure (in the sense of Steenrod [28]). A differentiable section of y corresponds to a differentiable bundle map, and now (b) follows from standard approximation theorems. To prove (a), let z = (q, E 0 , G) be a differentiable vector bundle over a Grassmann manifold such that x is induced from z by a map g: M-> G, covered by a bundle map f: x--> z. We may assume g differentiable. By (b) we may assume that fv :x\V -->z is a differentiable vector bundle map for some neighborhood V C U of A. Then the differentiable vector bundle structure 'I' induced by f from z satisfies (a) of 1. 9.

90

MORRIS W. HIRSCH AND BARRY MAZUR

(1.10)

Differentiable and PD vector bundles

Note that if M is a PL manifold and a a smoothing of M, then a differentiable vector bundle (P, E, M, ) over a is also a PD vector bundle. (More precisely, is contained in a unique PD vector bundle structure

0 .)

We describe this situation by saying that the smooth vector

bundle structure induces the PD vector bundle structure. If {3 is the differential structure on E induced by and a, we shall also say that

({3,a) is a smoothing of the PD vector bundle (P, E, M, 0 ). Similarly, a differentiable vector bundle map is also a PD vector bundle map if the map of base spaces is simultaneously smooth and PL. Examples of such maps include: identity maps; projections Max I-+ Ma; inclusions Max I Ol -+Max ~n, etc.

(1.11) LEMMA. (a) In Theorem 1.9(a), suppose M is a PL manifold and

a a (compatible) smoothing of M. Suppose also that vector bundle structure, and that chosen so that 'I' C

0

induces

1

\U.

1

C is a PD

Then 'I' can be

1 .

(b) In 1.9(b), suppose that M0 and M1 are PL manifolds with

smoothings, and that f is a PD vector bundle map. Then the homotopy ft can be chosen so that the map x 0 x I -+ x 1 , (x, t)

t->

ft(x), is a PD

vector bundle map. Proof. The proof of 1.9 also proves 1.11. Smoothings of total spaces For the next lemmas, let xi= (pi,Ei,M,i) be a PD vector bundle over a PL manifold M, i = 0, 1. Let f: x 0 -+ x 1 be an isomorphism of PD vector bundles covering 1M. Suppose E 0 has a PL structure compatible with 0 .

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II

91

(1.12) LEMMA. Let a be a differential structure on E 1 . The induced differential structure £*a on E 0 is compatible with the PL structure

on E 0 in each of the following cases: (a) E 1 has a PL structure compatible with a and f: E 0

....

E 1 zs PL;

(b) a is compatible with the PD vector bundle structure 1 .

Proof. Left to the reader. (1.13) LEMMA. Let a be a smoothing of M and f3i a differential struc-

ture on Ei(i=O, 1) such that (f3i,a) is a smoothing of the PD vector bundle xi (see 1.10). Let f: x 0 -> x 1 be a PD vector bundle isomorphism over 1M. Then f*f3 1 and {3 0 are concordant smoothings of E 0 . Proof. Let ft: x 0 -> x 1 be an isotopy of PD vector bundle maps as in 1.11, with £0 = f and f1 an isomorphism of the differentiable vector bundles over Ma corresponding to {3 0 and {3 1 • Let F: E 0 xI .... E 1 xI be the map F(x, t) = (ft(x), t). Let

t

be the standard smoothing of I;

then l.ll(b) implies that F*(/31 x t) is compatible with the product PL structure on E 0 x I. Since £1 : (E 0 , {3 0 ) .... (E 1 , {3 1 ) is a diffeomorphism, F*(f3 1 x t) is a concordance between {3 0 and £*{3 1 • §2. The map x!: S(M) .... S(E) If M is a PL manifold then

s• (M)

denotes the set of smoothings of

M, while S(M) denotes the set of concordance classes of smoothings. The concordance class of a is denoted by [a]. We recall from I, 7.8: (2.1) THEOREM (Product smoothing theorem). The natural map S(M)-> S(Mx9tn) is bijective for all n (

z+.

Theorem 2.6 below generalizes this to arbitrary vector bundles over M. Other versions of this isomorphism are 2.6 and 2.7.

92

MORRIS W. HIRSCH AND BARRY MAZUR

(2.2) LEMMA. Let E and M be PL manifolds and x = (p, E, M, Cll) a PD vector bundle such that ci> is compatible with the PL structure on E. Let a 0 , a 1 be smoothings of M and {3 0 , {3 1 smoothings of E such that both ({3 0 , a 0 ) and ({3 1 , a 1 ) are smoothings of x. If a 0 and a 1 are concordant, then {3 0 and {3 1 are concordant. Proof. Consider the PD vector bundle xx I= (pxl,Exi,Mxi,Cll*) in-

duced from x by rr 1 : M x I--> M. Give Ex I the product PL structure; this is compatible with ci>*. Let a

S•(Mx I) be a concordance from a 0 on M x 0 to a 1 on M x 1. By 1.4(a) and l.S(a) there exists a differenf

tiable vector bundle structure 'l' on x x I over (Mx I) which induces ci>*, and such that if f3 is the smoothing of Ex I corresponding to 'l', then (Ex i, f3 i) is a smooth submanifold of (Ex I, {3) for i

=

0, 1. Then

f3 is a concordance from {3 0 to {3 1 • (2.3) Let x

=

(p, E, M, Cll) be a PD vector bundle over a PL manifold M.

Suppose E has a PL structure compatible with ci>, making E a PL manifold. For every smoothing a of M, there exists by 1.9(a) and l.ll(a) a smoothing f3 of E such that (f3, a) is a smoothing of x. By 2.2 the concordance class of f3 is uniquely determined by that of a. The correspondence [a]

r>

[{3] defines a map denoted by

The truth of the following statement is obvious: (2.4) LEMMA. Let e = (p,Mx~n,M) be a trivial PD vector bundle over a PL manifold M. Give M x ~ n the product PL structure. Let pn be the natural smoothing of ~ n. Then the image of [a] under e! : S(M)--> S(Mx~n) is [axpn],

for all [a]

f

S(M).

SMOO'TIIINGS OF PIECEWISE LINEAR MANIFOLDS II

93

Now let x, y be PD vector bundles over M whose total spaces Ex, Ey have compatible PL structures. Let y 0 be the PD vector bundle over Ex induced from y by the projection p of x. We identify Ey 0 with E(xmy) as in 1.7. The same PL structure on Ey 0 is compatible with both y 0 and xey. (2.5) LEMMA. Let M be a PL manifold. Then the following diagram

commutes:

Proof. The diagram

commutes. Therefore if a, f3, y are smoothings of M, Ex and Ey 0 respectively, such that (f3, a) is a smoothing of x and (y, {3) is a smoothing of y 0 (see 1.9) then {y,a) is a smoothing of xey. This implies the lemma. (2.6) THEOREM (Bundle smoothing theorem). Let x = ·(p, E, M, ) be a PD vector bundle over a PL manifold M. Let E have a PL structure

compatible with . Then the map x! : S(M)

-+

S(E) is bijective.

Proof. Let y be vector bundle over M such that xey is isomorphic to the trivial vector bundle e = (p,Mx!Rn,M). By (1.5) y has a PD structure and the corresponding PD structure on xey is isomorphic to

94

MORRIS W. HIRSCH AND BARRY MAZUR

the canonical PD structure on e. Therefore we may assume y is a PD vector bundle, and let

f: xEily -+ e

be a PD vector bundle isomorphism over 1M. Give M x ~n the product PL structure. Then by 1.12 the map f*.: S(E(xEily)-+ S(Mx~n) is defined, and by 1.13 the diagram

commutes. By 2.5 the diagram I

(xEily)·

I

• S(E(xEily) = S(Ey 0 )

Yo

1

S(Ex)

commutes where Yo= Px *y. Therefore e! :S(M)-+ S(Mx~n) can be factored e! = f*

0

Yo!

0

x!. The product smoothing theorem states that e!

is bijective. Therefore x! is injective, and Yo! is surjective since f* is bijective. But x was arbitrary; hence y 0! is also injective, making x! bijective as well. Germs of smoothings Let V be a PL manifold and X C V a subset. The set s•(V, X) of X-germs of smoothings of V is defined as the inverse limit of the maps r ji s•cu i) -+

s•(U j)

as U i varies over open neighborhoods of X in V,

and rji is induced by restriction if Uj CUi. Two X-germs of smoothings are concordant if they have concordant representatives on some neighborhood of X. The set of concordance classes of X-germs is denoted by S(V,X). Let g : S(V) -+ Scv, X) be the natural map.

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS

n

95

(2. 7) LEMMA. Let x = (p, E, M) be a PD vector bundle over a PL manifold M, whose total space has a compatible PL structure. Identify M with the zero section of E. Then the map g: S(E)-+ S(E, M) is bijective. Proof. As discussed in 1.6, it may be assumed that the projection map p : E -+ M is PL locally trivial. Let A C M be a subpolyhedron and W C E a neighborhood of p- 1 (A)

U M. There is a PL isotopy (not necessarily surjective) ft: E-+E

such that:

ft(x) = x in a neighborhood of p- 1 (A) U M . The existence of such an isotopy is easily proved for the case where M is a simplex 1:1, E = 1:1

X

9t n,

and A = al:i; the general case reduces to

this by the PL local triviality of p : E -+ M and the usual simplex-bysimplex proof. To prove g surjective let WC E be a neighborhood of M and a a smoothing of W. Let ft be an isotopy as above with A =

rp;

then the

M germ of fia is a. To prove g injective, let a 0 ,a 1 be smoothings of E which are concordant on a neighborhood V C E of M. We may choose a concordance

f3' E s·(vx I) between aoiV and aliV which extends to a smoothing

f3

of a neighborhood W C Ex I of A= (Mx 0) U (Mx 1) U (Vx I). Let ft: Ex I-+ Ex I be a PL isotopy such that f 0 = 1, f 1 (Ex I) C W, and ft is the identity on a neighborhood of A. Then fif3 is a concordance between a 0 and a 1 • Note that we have actually proved that every M-germ of a smoothing of E extends to a smoothing of E; and the germ of a concordance between M-germs of smoothings of E extends to a concordance between the two smoothings of E.

96

MORRIS W. HIRSCH AND BARRY MAZUR Let x

=

(p, E,M,) be as in 2.7. The composite map S(M) £

I

(E) __£ S(E, M)

I

will also be denoted by x ·. (2.8) COROLLARY. The map x!: S(M) .... S(E, M) is bijective.

Proof. 2. 7 and 2.6. The map xx! : S(M, X) .... S(E, X) Let x = (p, E,M) be as in 2.7 and let XC M be any subset. A map xx!: S(M, X)-> S(E, X) is defined as follows. Let a be a smoothing of a neighborhood W C M of X, let [a] C S(M, X) be the concordance class of the X-germ of a. Let tt be the restriction of x over W. Then tt![a]

f

S(p- 1 W) is represented by a smoothing

is a neighborhood of X in E, I

Define xx ·[a]

=

f3

f3

of p- 1 w. Since p- 1 w

represents an element [y]

f

S(E, X).

[y]; it is easy to see that this definition is independent

of the choices made. Since tt! :S(W)-> S(p- 1 W) is bijective no matter which neighborhood W is chosen, we have proved: (2.9) LEMMA. The map xx!: S(M, X)-> S(E, X) is bijective.

§3. Linearization Let V be a PL manifold and M C V a PL submanifold. A lineari-

zation of (V, M) is a PD vector bundle x

=

(p, E, M) such that:

(1) E is a neighborhood in V of M, (2) p : E .... M is a retraction, and (3) the PL structure on E induced from V is compatible with x. The set of linearizations of (V, M) is denoted by Two linearizations

X·=

1

s:_•(V, M).

(p·, E., M) of (V, M), i 1 1

if there exists a linearization x

=

=

0, 1, are equivalent

(p, E, Mx I) of (Vx I, Mx I) such that

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SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II

for some neighborhood W C V of M, p(x, i) i

=

=

(pi(x), i) for x

f

(Wx i)

n E,

0, 1. The set of equivalence classes of linearizations of (V, M) is

denoted by ~(V, M). Let x

=

(P, E, M) be a PD vector bundle, and let x ~ e q be the

Whitney sum of x and the trivial PD vector bundle M x g{q .... M. We consider the total space of x ~ e q to be E x g{ q and the projection to be the composition. p

o

rr 1 :Ex g{q .... E .... M.

If x is a linearization of (V, M) then x ~ e q is a linearization of

(Vxg{q,MxO). In this way maps

are defined. The direct limit of this sequence of maps is the set of stable linearizations of (V,M), denoted by ~s(V,M). We denote by s :~(V,M) .... ~(V, M) the natural map. The next theorem is stated in a stronger form than is needed. It would suffice to prove the following in place of (b): if V and M are smoothable then for some integer q 2: 0, ~(Vx g{q, Mx 0) ;i (2). The proof of this uses embedding theorems which are considerably simpler than those used in the proof of (b). (3.1) LEMMA. Let (V, M) be a PL manifold pair such that V 1s

smoothable. (a) If ~(V, M) is not empty, then M is smoothable.

(b) If M is smoothable and dim V > ~(V, M)

Proof. (a) Let x

~ (dim M) + 1, then

;i (2).

=

(p, E, M) be a linearization of (V, M). Since E is

open in V, S(E) ;i (2). By 2.6, S(M) ;i (2). (b) We may assume the inclusion M .... V is a homotopy equivalence. Suppose V and M have smoothings

f3,

a respectively. By Haefliger

[ 4], the dimension restriction implies that the inclusion M .... V is

98

MORRIS W. HIRSCH AND BARRY MAZUR

homotopic to a smooth embedding f: Ma .... V{3· V{3 has a smooth triangulation T making f(M) a subcomplex of

r,

and such that f : M .... VT

is PL. (See I, Section 13.) By uniqueness of smooth triangulations there is a PD isotopy gt: VT .... V such that g0 = 1v, and g1 : VT .... V is a PL homeomorphism between the PL structure defined by T and the original PL structure on V. Then h = g1 of: M .... V is a PL embedding homotopic to the inclusion such that h(M) is a smooth submanifold of (V, (g1 1 )*{3). Therefore we may suppose the inclusion M .... V is homotopic to a smooth embedding f : Ma .... V{3 such that f : M .... V is a PL embedding. By Hudson [10] the dimension restriction implies that there is a PL isotopy ut: V .... V such that u 0 = 1v and u1 IM =f. Then

y

=

u~{3 is a smoothing of V in which Ma is a smooth submanifold.

Now Ma has a differentiable normal vector bundle x in Vj and by the tubular neighborhood theorem we may consider x a linearization of (V, M). This proves (3.1). The dimension restriction in 3.1(b) cannot be greatly relaxed. In [7] there is given an example of a locally flat PL submanifold M4 C S7 such that (S 7 , M4 ) has no linearization; but every four dimensional PL manifold can be smoothed (Cairns [2]). (3.2) THEOREM. Let M be a PL manifold, aM=

0.

Let ML\ C M x M

denote the diagonal. Then the foiiowing statements are pairwise equivalent:

(a) M is smoothable;

0; fs(Mx M, Mi\) ~ 0.

(b) f(Mx M, Mi\) ~ (c)

Proof. (a)

==;>

(b): If a is a smoothing of M then Mi\ is a smooth

submanifold of Ma x Ma. (b)

;:. (c): Obvious.

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II

99

(c)===> (a): Suppose fs(Mx M, M~) f. 0. Then for some s > 0, f(MxMx9{s, ~xO)f-0. Put V=MxRs. Then f(VxV,V~)f-0 because there is a PL homeomorphism

By 2.1 it suffices to prove V smoothable. Therefore, replacing M by V, it suffices to prove:

Let x =

(p,E,M~)

is a neighborhood of

be a linearization of and p : E

M~

-> M~

(MxM,M~),

where ECMxM

is a retraction. Stable unique-

ness of PL normal bundles [8; 13; 16] implies that for q 2:: 0 there is a PL embedding g: Ex 9{q ... M x M x 9{q making the following diagram commute:

M~

X0

M~

1

l

g

EX 9{q

I

X0

M X M X 9{q

l

M~

M~

where A.(x, y, z) = (x, x) and unlabeled maps are inclusions. (This is because the map M x M ...

(x, y)

M~,

1-+

(x, x), restricts to a PL bundle

in some neighborhood of M~ .) Since p11 1 :Ex 9{q ... M~ is a PD vector bundle, the map g induces a linearization of (Mx Mx 9{q, M~ x 0) whose projection is given by (x, y, z) like

cp

1-+

(x, x, 0). From a PL homeomorphism

above, we obtain a linearization of ((Mx9{q) x (Mx9{q), (Mx9{q)~)

whose projection is: (x, y)

1-+

(x, x). Since it suffices to prove M x 9{q

smooth, it suffices to prove: M is smoothable provided there exists a linearization x = (p, E, M) of (Mx M, M~) such that p(x, y) = (x, x). We assume without loss of generality that M C 9{S, and there is PL retraction r: N ... M of an open neighborhood N C 9{S of M. Let

100

MORRIS W. HIRSCH AND BARRY MAZUR

W={(x,v)tMx~slx+vtN, and (r(x+v),x)EE!

Let E(r*x) be the total space of the PD vector bundle over N induced by r: N _, M from x, identifying M with

in the natural way. Give

M~

E(r*x) its natural PL structure. By definition E(r*x) = {(y, z, w)

f

Nx Nx Mi(z, w)

E, r(y) = z!

f

Define a map f:W _, E(r*x),f(x,v) = (x+v,r(x+v),x). Then f is injective and PL. Since the manifolds W and E(r*x) have the same dimension f maps W onto an open subset of E(r*x) by a PL homeomorphism. The base space N of r*x is smoothable, being open in ~s. Hence E(r*x) is smoothable by 1.9(a). Therefore W is smoothable. Since W C M x ~k is an open set containing M x 0, it follows from 2.8 that M is smoothable. The proof of 3.2 is complete.

The map : 2.s(V, M) x S(M) _, S(V, M) Let V be a PL manifold and M a PL submanifold. A map •

•

ell : 2_ (V' M)

c•

c

cD (M) _, cD(V' M)

X

is defined by setting •

ell (x,a)

=

I

x·[a) ;

it is easy to see that ell• (x,a) depends only on the equivalence class of

•

x and the concordance class of a. Therefore ell

ell : (V' M)

X

S(M) _, S(V' M)

It is clear that the following diagram commutes:

(3.3)

induces a map

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II

101

where the vertical maps are the natural ones. The map S(V, M) .... S(Vx~q,rxO) is bijective by (2.9); hence cl> induces a map

In order to state the next theorem we define a map of sets f:AxB-+C to be a complete pairing if it satisfies the following three conditions: (1) if any two of the sets A, B, C are nonempty so is the third; (2) f is right complete:

for all a

A the map

f

fa : B .... C, b

1-->

f(a, b)

1-->

f(a, b)

is bijective; (3) f is left complete:

for all b

B the map

f

fb : A .... C, a is bijective.

(3.4) THEOREM. Let V be a PL manifold and M C V a PL submani-

fold such that

aM =

M

n av.

CJ> :

Then the map

fs(V, M) x S(M) .... S(V, M)

is a complete pairing. Proof. Condition (1) in the definition of complete pairing follows from 1.9(a), and 3.1, 2.1. Condition (2) follows immediately from 2.9. It remains to prove (3):

cl>[a] : fs (V, M) .... S(V, M) is bijective

102

MORRIS W. HIRSCH AND BARRY MAZUR

for every smoothing a of M. Referring to diagram 3.3, it suffices to show that for q sufficiently large, the map

is a complete pairing. This follows from: (3.5) LEMMA. II dim V ~ 2 dim M + 5, then

ci> : f(v, M) x S(M) _, S(v, M) is a complete pairing.

1

To prove this, let [x]

£

f(V, M) and [a]

£

S(M). The fact that Cl>[x]

=

x · : S(M)-> S(V, M) is bijective is proved in 2.8 (and needs no dimension restriction). It remains to prove that Cl>[a] : f(V, M) -> S(V, M) is bijective. To prove surjectivity let f3 be a smoothing of a neighborhood of M in V; we may replace V by this neighborhood and so let f3 be a smoothing of V. Since dim V > 2 dim M there is a smooth embedding f: Ma-> VfJ· Considered as a PO embedding M-> V{3• f is PO isotopic to a PL embedding g: M-> V, by the theory of smooth triangulation. (For there is a smooth triangulation of Vf3 making f(Ma) a subcomplex; and there is a PO

isotopy of this triangulation taking it to a PL isomorphism with the

original PL structure in V.) The PL embedding g: M-+ V is homotopic, and hence PL isotopic, to the inclusion, because dim V ~ 2 dim M + 3. By Hudson [9] this isotopy extends to a PL isotopy of V. (It is here that the assumption on

aM

and av is used.) It follows that there is a PO isotopy ht: V-> Vf3

such that h 0 = lv, h1 (M) = f(Ma). Give (V, M) the linearization x induced by f from the smooth tubular neighborhood of f(Ma) in VfJ· Then x ![a] = [{3], which means that Cl>[a][x] = [{3]. To prove injectivity of Cl>[a] let x 0 and x 1 be linearizations of (V, M) such that x 0 ![a] = x 1 ![a]. This means that there is a smoothing f3 of V x I with Ma x 0 and Ma x 1 smooth submanifolds, and xi is

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS

II

103

the tubular neighborhood of MaX i in acvx I)f3, i = 0, 1. Because of the dimension assumptions, there is a smooth embedding F: Ma xI .... (Vx I)f3 extending the inclusion of Ma

X

ai,

and F is PD isotopic to the inclu-

a

sion keeping M X I fixed, by a PD isotopy Gt : v

X

I .... (V X 1){3 keeping

V x a1 fixed. Thus G1 (Mx I)= F(Ma x 1). A tubular neighborhood of F(Ma xI), pulled back by G1 , provides a concordance between x 0 and

x 1 . This completes the proof of 3.4. By using more sophisticated embedding and isotopy theorems of Haefliger [4] and Hudson [10], 3.5 can be improved to (3.6) LEMMA. Let a be a smoothing of M. The map «l>[a] : f(V, M) .... S(V, M), sending [x] to x'[a], is surjective if dim V ~~(dim M+ 1); and injective if dim V

> ~ (dim M+ 1).

The map Exp: S(M) .... f(Mx M, M~)

Let M be a PL manifold without boundary. To every smoothing a of M we define a class of linearizations of (Mx M, M~) as follows. Let T(Ma) denote the total space of the tangent vector bundle of Ma. Pick a smooth Riemannian metric for a. There is a neighborhood

W C T(Ma) of the zero section so small that for each x

t:

M, the exponen-

tial map

is a diffeomorphism onto a convex neighborhood of x. Let r: T(Ma) .... W be a smooth fibre preserving embedding which is the identity on a neighborhood of the zero section. Define a smooth embedding by

Thus e maps the zero section onto the diagonal in the obvious way.

104

MORRIS W. HIRSCH AND BARRY MAZUR By means of e we obtain a linearization (q, E, ML'l) of (Mx M, ML'l):

put E=e(T(Ma)) and q=d 0 p 0 e- 1 where d:M-.MxM isthediagonal embedding and p: T(Ma)-.. M is the bundle projection. Give (q, E, ML'l) the unique smooth vector bundle structure making e - 1 a smooth vector • c• e• bundle isomorphism. In this way a map Exp 1 : cJ (M) ~ • .L (Mx M, M) is defined.

•

The linearization Exp 1(a) of (Mx M, ML'l) depends on the choice of the Riemannian metric, the neighborhood W, and the embedding r. Different choices, however, lead to an equivalent linearization, since any two smooth Riemannian metrics on Ma can be connected by an interval in the convex cone of such metrics, leading to an isotopy between their exponential maps on some neighborhood of the zero section; and any two embeddings such as r are smoothly isotopic rei a neighborhood of the zero section. Moreover, if

a0

and a 1 are isotopic smoothings of M,

the resulting linearizations are equivalent. Therefore we have defined a map

Let A: Mx M-> Mx M be the "flip" (x, y)

r->

(y, x). If we replace the

map e: T(Ma)-> M x M by A o e: T(Ma)-> M x M in the definition of Exp 1 , maps Exp;:S.(M)->~·(MxM,ML'l) and Exp 2 :S(M) .... ~(MxM,ML'l) are obtained. (3.7) LEMMA. Exp 2 = Exp 1 .

Proof. This is just the tubular neighborhood theorem applied to the "vertical" and "horizontal" tubular neighborhoods of ML'l in Max Ma, which define

Exp~a]

and

Exp~a]

respectively.

We denote by Exp the map Exp = Exp 1 = Exp 2 : S(M) .... ~(Mx M, ML'l) .

105

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II

Let a and w be smoothings of M. Then w x a is a smoothing of M x M; we denote by [w x a]~ the concordance class of the M~ -germ of w xa. Thus [wxa]~ fS(MxM,M~).

(3.8) LEMMA. (a) (Exp [a])![w] = [w x a] (b) [wxa]~

=

[axw]~.

Proof. (a) By definition, (Exp [a])![w] is the concordance class of the

M~-germ of a smoothing of the total space E of (Exp 1a)![w] which makes the vector bundle x 0

=

(p, E, M~) into a smooth vector bundle

over Mw. Such a smoothing can be obtained as follows. Let gt: M--> Ma be a PD homotopy such that g 0

=

1M' g1 : Mw

->

Ma is a smooth map,

and (x, gt(x)) f E for all x f M, t f I. Define Gt: M~

->

E C M x M by

Gt(x, x) = (x, gt(x)). Extend Gt to a neighborhood N of M~ in E as follows. If (x, y) < E then (x, y) belongs to the fibre over (x, x) in the vector bundle exp 1 (a)= (q, E, M~). Hence we may think of (x, y) as a vector; (x, x) is the origin. Then define Gt(x, y) vector addition. More precisely, Gt(x, y)

=

=

Gt(x, x) + (x, y),

e(e- 1 Gt(x, x) + e- 1 (x, y)),

where e: T(Ma)-> M x M defines the vector bundle structure on E. The homotopy Gt is well defined on a neighborhood N of is in fact a PD isotopy N G1

°e

->

M~

in E, and

Mw x Ma. It is clear that the map e and

induce equivalent linearizations x 0 , x 1 respectively of (N, M~),

hence x![w]

=

xi[w]. That induced by G1

°e

is a tubular neighborhood

of the smooth submanifold G1 (M) C Mw x Ma, since G1 (M) is the graph of the smooth map g1 : Mw

->

Ma. This means that xi [w] = (Gr[w x a])~.

Since G1 is PD isotopic to the identity map of N, (Gr[w x a])~

=

[w x a]~, which proves (a). To prove (b), we have from (a) and (3. 7): [w xa]~

(Exp 1 [a])![w]

= (Exp2 [a])![w] = A *((Exp1 [a])![w])

[A*(wxa)]~ = [axw]~.

106

MORRIS W. HIRSCH AND BARRY MAZUR Denote by Exp: S(M)--> f>s(Mx M, ML'l) the composition S(M)

Exp

(3.9) LEMMA. The maps

and

satisfy the relation ll>(Exp a,w)

for all a, w

E

=

ll>(Exp w,a)

S(M).

Proof. Note that a, w denote concordance classes of smoothings of M. It suffices to prove that ll>(Exp a,w)

=

ll>(Exp w,a)

By definition of It>, this is equivalent to I

(Exp a)·w

I

(Exp w)·a ,

=

which follows from 3.8.

(3.10) THEOREM. Let M be a PL manifold without boundary. Then the map Exp: S(M)--> f> 8 (Mx M, ML'l) is bijective. Proof. This is a purely formal consequence of 3.4, 3.9, and the following fact. (3.11) LEMMA. Let A, B, Z be sets and It>: Ax B--> Z, e: B--> A maps.

Suppose that It> is a complete pairing (see 3.4), and that ll>(e(a),w)

for all a, w

E

=

ll>(e(w),a)

B. Then e is bijective.

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II

Proof. Fix a

f

B. The map w

1-->

107

(e(a),w) is a bijection B-> Z, since

is right complete. Therefore the map w

1-->

(e(w),a) is a bijection

B -> Z. Because is left complete, e must be bijective. §4. Classifications Theorem 3.10 says that the classification of smoothings of M is the same as the classification of stable linearizations of (Mx M, Mf1). This last classification is translated to a homotopy problem via the theory of block bundles [23]. We do not repeat the definition of block bundle, but

merely note the following facts. If M is a locally flat PL submanifold of V, then a regular neighbor-

hood of M in V determines an isomorphism class of block bundles over M; this class is independent of the choice of regular neighborhood. The

functor which to each polyhedron P assigns the set of isomorphism classes of block bundles over P of fibre dimension n, is a homotopy

---

functor and has a classifying space (or semi-simplicial complex) B PLn. ~

(There is a semisimplicial group PLn and B PLn may be taken to be its classifying space.) The operation of triangulating vector PD bundles induces a map

----

tn: B O(n) --. B PL(n), which we may take to be a fibration with the follow-

---

ing property. Let

f : M --. B PL(n) be the classifying map of the normal block bundle of M in V. Then equivalence classes of linearizations of (M, V) correspond bijectively to homotopy classes of liftings

---

---

g: M -> B O(n) of f over tn.

---

There are natural inclusions B PL(n)-> B PL(n+ 1). Passing to the direct limit, we denote by B PL the classifying space for stable equivalence classes of block bundles. Equivalence classes of stable linearizations of (V, M) correspond bijectively to homotopy classes of liftings

----

g: M --. BO of fs : M -> B PL over t: B PL -> BO, where fs is the comf B ---position M--+ PLn -> B ---PL, and t is the direct limit of I tnl.

108

MORRIS W. HIRSCH AND BARRY MAZUR The fibre of tn: BO(n) .... B PLn is denoted by PLn/On, and the

fibre of t by PL/0. (Later we shall put PL/0 =

r .)

For a PL manifold M without boundary, let fM: M .... B

PL

classify

the stable normal block bundle of M~ in M x M. Let &(M) .... M be the fibration induced by fM from t: BO .... B PL. Then 3.10 has the following interpretation. (4.1) THEOREM. Concordance classes of smoothings of M correspond

bijectively to homotopy classes of sections of the bundle &(M) .... M ,..__

whose fibre is PL/0. The correspondence is natural for PL homeomorphisms. The last sentence means that if g: N .... M is a PL homeomorphism, we may take fN = fM 0 g, and &(N)

=

g*&(M). Therefore there is a natu-

ral correspondence glt between sections of &(M) and &(N), preserving homotopies. If a section a of &(M) corresponds to a smoothing a of

M, then gIt a corresponds to g *a.

Another classification is obtained as follows. Let M be a PL submanifold of V; assume V is smoothable and let w be a fixed smooth-

ing of V. By 3.4, for every smoothing a of M there is a unique equiva1

lence class of stable linearizations x of (V, M) such that x·[a] = [w]. This correspondence a

1->

x defines a bijection S(M) .... ~s (V, M), which

depends on [w]. Consider the special case V

=

M. Then ~s(M, M) is the set of

equivalence classes of stable linearizations of the trivial stable block bundle over M, and is canonically identified with homotopy classes of liftings of the constant map M .... B PL over t : BO .... B PL. In other words

homotopy classes of maps M .... PL/0. This proves the following classification theorem.

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II

109

(4.2) THEOREM. Let M be a smoothable PL manifold. For each con-

cordance class w

f

S(M) there is a bijection

~w

: S(M) .... [M,PL/0] ,

with ~w (w) being the homotopy class of the constant map. If g: Mw .... Ne is both a diffeomorphism and a PL homeomorphism, then the foliowing diagram commutes:

~e S(N) - - - [N,PL/0] g*

l

S(M)

l

g#

~w

,.._.

[M, PL/0]

From the point of view of smooth manifolds, 4.2 is unsatisfactory in that it refers to a specific compatible PL structure on the smooth manifold Mw. In order to deal with smooth manifolds rather than smoothed PL manifolds we make the following definitions. Let M be a topological manifold and w a smoothing of M. Are-

smoothing of Mw is a smoothing of M which has a smooth triangulation in common with w. If a 0 and a 1 are resmoothings of Mw, they are called w-concordant if there is a smoothing a of M x I which induces

ai on M x i (i= 0, 1), and which has a smooth triangulation t: P x I-> (Mx I)a where P is a PL manifold and P x I has the product PL structure. It is not assumed that t is level-preserving. It is clear that w-concordance is a symmetric and reflexive relation on the set of differential structures on M; transitivity is easily proved using the theory of smooth triangulations. The set of w-concordance classes of resmoothings of Mw is denoted by S(Mw). Let K be a PL structure on M compatible with w; denote by MK the corresponding PL manifold. A map tK: S(MK)--> S(Mw) is defined by tK[a] = [a]w = the w-concordance class of the smoothing a. Let L be another PL structure on M compatible with w. From the theory of smooth triangulations we obtain a PL isomorphism f : MK --> ML

110

MORRIS W. HIRSCH AND BARRY MAZUR

which is PD isotopic to 1M as a PD map MK--> Mw. There is induced a bijective map q~L : S(ML)--> S(MK) which does not depend on the choice of f. The proof of the following lemma is left to the reader. (4.3) LEMMA. (a) The diagram

commutes. (b) tK and tL are bijective; hence q~L is bijective.

Let D be the category of smooth manifolds and diffeomorphisms, and

S the category of pointed sets and maps. Define a contravariant functor S: D--> S by assigning to Mw the set S(Mw) with base point [wlw; and to a diffeomorphism g: Mw --> Ne, assign the map S(g): S(Ne)--> S(Mw),

[a]e __, [g*a]w. Let j" : D --> T be the natural functor from D to the category T of topological spaces and continuous maps. Let 1

=

PL/0

be the fibre of BO --> B

PL;

let it also denote the

functor 1: T --> S which assigns to a continuous map h: X --> Y the map htt: [Y, 1] __,[X, 1] of homotopy sets. (4.4) THEOREM. The functors S and 1oj": D--> S are naturally equivalent. In other words, the homotopy functor [M, 1], pulled back to D, is naturally equivalent to

S.

Proof. For each smooth manifold Mw, let K structure on M. Define a map

=

Kw be a compatible PL

111

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II

to be the composition

[M,l]. By 4.2 and 4.3 H is well defined independent of the choice of K, and is bijective. The naturality of H also follows easily from 4.2 and 4.3. Thus H is the required natural equivalence. Let M,N be PL manifolds and

0,;

respective smoothings of M,N.

According to 4.4, if f: M -> N is a continuous map, there is defined a map S(f): S(N;)-> S(M0

We proceed to explain S(f) directly. In doing so we

).

shall identify S(M0

)

with S(M) via a map tK as in 4.3.

First consider the zero section j: N-> Nx:Rq. It is not hard to see that the map

is the map [{3] r> [{3xpq], where pq is the natural smoothing on 9\q. By naturality, for [{3]

f

S(N;),

Moreover, since S is known to be a homotopy functor, we may replace j

o

f by any homotopic map. In particular, if q is sufficiently large we

may find a smooth embedding g:Ma-> N;x:Rq, homotopic to j

0

f; then

Now g(Ma) has a smooth normal bundle in N; x :Rq, which defines a linearization of (N x :R q, g(M)) (after a smooth triangulation of N; x :R q with g(M) a subcomplex). Denote this linearization by n = (p, E, g(M)), where E C N x :Rq is an open set containing g(M). Let {3 0 be the restriction of {3 x pn to E. By 2.8 there is a unique concordance class of smoothings y of g(M) such that n ![y]

=

[{3 0 J.

112

MORRIS W. HIRSCH AND BARRY MAZUR

(4.5) THEOREM. S(f)[f3]

=

[g*y].

Proof. Left to the reader. As another exercise the reader is invited to prove:

(4.6) THEOREM. Let x

=

(p, E, M) be a PD vector bundle, where E

and M are PL manifolds whose PL structures are compatible with x. Let [a] < S(M); put [e] = x ![a] < S(E). Then the map

is given by [{3]

f->

x ![{3].

As a corollary of 4.4 we have: (4.7) THEOREM. The functor

S: D ....

S from the category of smooth

manifolds and diffeomorphisms, extends to a homotopy functor on D. Remark. Let PLn denote the semisimplicial group of PL homeomorphisms of (Rn, 0); set PL = limn-.oo PLn. There is a natural homotopy equivalence B PL --. B

PL.

Therefore I' can be identified with the

fibre of BO --. B PL, sometimes denoted by PL/0. See also Section 6.

The abelian group structure on S(Mw) Let M be a PL manifold and w a fixed smoothing of M. Let d: M--. Mx M be the diagonal embedding. We define a binary operation Pw:S(Mw)xS(Mw)-.S(Mw) by Pw([a],[{3])=S(d)[ax{3] where S isthe functor of 4.7. Explicitly, Pw([a], [{3]) is represented by a smoothing y of M such that (Exp w)! [y] = [axf3]L1 , the ML1 germ of ax {3. Equivalently, by 3.8, y is determined by the equation

113

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II

[w x y ]1"1 = [ax {3]1"1 .

(4.8) LEMMA. Pw is commutative and is a complete pairing. Proof. Follows from 3.8 and 3.10. (4.9) THEOREM. The operation Pw m:1kes S(Mw) into an abelian group

whose identity element is [w]. If f: Mw

->

Ne is a continuous map, then

is a homomorphism of groups. Proof. We have already proved commutativity, and clearly [w] is a twosided identity. Inverses exist since the pairing is complete. It remains to prove associativity. Let M0 C Mx Mx M = M( 3 ) be the diagonal: M0 = l(x,y,z)

Ne, then S(f) (af3) = (S(f)a) (S(f) {3). It

is sufficient to observe that the diagram f X f MxM-_;_;..;....;;-NxN

f M-----N

MORRIS W. HIRSCH AND BARRY MAZUR

114

commutes, where dM and dN are the diagonal maps, and S(dNf)(ax/3)= S(f) (a/3) while S((fx f) dM) (ax {3) = (S(f)a)(S(f) {3). We can now improve the homotopy functor S: D--> S. Let A be the category of abelian groups and homomorphisms and A --> S the forgetful functor.

(4.10) THEOREM. The homotopy functor S: D--> S lifts to a homotopy functor D--> A.

Proof.

Follows from 4. 9.

We shall denote by S the functor D --> A determined by the group structure on S(Mw) specified by 4.9. We emphasize that the group structure on S(Mw) depends on the concordance class w

E

S(M). Any two such structures are isomorphic, how-

ever. Abelian group structure on

Recall that 1 ...._,

[X,i]

denotes the fibre of the map BO --> B PL (also de-

noted by PL/0). Let P denote the category of spaces having the homotopy type of polyhedra, and homotopy classes of maps. An object X of P is a topological space homotopy equivalent to a subcomplex of a rectilinear triangulation of a Euclidean space; equivalently, X is a homotopy equivalent to a finite dimensional, locally finite CW complex having countably many components. Let 1: P --> S be the functor X

f->

[X, lJ.

(4.11) THEOREM. The functor 1: P--> S lifts to a functor P--> A. Proof.

Let P be an object of P. Choose a homotopy equivalence

¢: P--> Q from P to a polyhedron Q C ~n. Let N C ~n be an open set admitting Q as a deformation retract. Let v be the smoothing of N

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II induced by the inclusion N -->

:R n.

115

The composition P __1>_. Q C N is a

homotopy equivalence and defines a bijection [P,i] ~ [N,l] From 4.4 there is a bijection

By 4.10, S(Nc) is an abelian group; we denote it by A(P). Let P' be another object of P and let ¢': P'--> Q' be a homotopy equivalence to a polyhedron Q' C :R n'; let N' C :R n' be an open set having ¢'(P') as a deformation retract; and let v' be the induced smoothing of

N'. Given a map g: P --> P', there is a unique homotopy class of maps G: N --> N' such that the diagram p _ _:g:.......,.. P'

¢1

1¢'

G N _....:.__ N'

commutes up to homotopy. Define A(g): A(P')--> A(P) by S(G): S(N:U-)--> S(Nu)· Then A(g) is a homomorphism of groups; it is easy to see that A is a functor P--> A which lifts 1. This proves 4.11. (4.12) THEOREM. There is a multiplication making [' into an h-space which is homotopy abelian and homotopy associative and which has a homotopy inversion. The resulting abelian group structure on [X,['] coincides with that of Theorems 4.4 and 4.10 if X is homotopy equivalent to a polyhedron. An elementary proof of 4.12 is outlined in Section 6. Here we follow Lashof and Rothenberg [12] in using the following deep result. (4.13). THEOREM. [Sn,l] is finite for all n _:: : 0.

116

MORRIS W. HIRSCH AND BARRY MAZUR

Proof of 4.13. By 4.2 [Sn,r'] is isomorphic to the set of concordance classes of smoothings of the standard PL n-sphere sn. From I, concordance implies orientation preserving diffeomorphism, while it is well known that the converse is true for sn. For n ~ 5 and the generalized Poincare conjecture of Smale [25], orientation preserving diffeomorphism is the same relation on smoothings of sn as h-cobordism. Finiteness of the h-cobordism groups for n ~ 5 is proved by Kervaire and Milnor [12]. For n ~ 3 Munkres [22] and Smale [26] show that all smoothings of sn are concordant, while Cerf [3] does the same for S4 • Tl1us 4.13 is proved.

(4.14) LEMMA. Let X be a countable complex and Y a space with finite homotopy groups. Then the natural map A : [X, r] -> inv lim [Xi, r]

is bijective, where Xi denotes the i-skeleton of X. Proof. A is always surjective. To prove A injective, let f: X -> r be such that for each i there exists a map gi: cxi-> y such that gil Xi= fjXi. Here CXi is the cone on Xi; it is assumed all the gi take the vertex of the cone to Yo. For each 0-cell aj, j = 1, 2, ... , of X, the maps gijCaj fall into finitely many homotopy classes of paths with fixed end points. Starting with j = 1, we find a subsequence

l gi l,

n = 1, ... , of the gi such n that g. jca., n = 1, 2, ... , are all homotopic with fixed end points. ln J Passing to j = 2, 3, ... , we choose further subsequences until we find a subsequence, also denoted by

l gin I, such that for each j, ginjCaj""" gi 1 1Caj,

with fixed end points. Therefore g. jCX 0 """ gi jCX 0 rei Xo, all n. 1n

Since in~ n, fi to g·

~

n

1

is defined in CXn. Let hn = CXn .... Y be homotopic

rel x 0 , such that hn = gi

1

on CX 0 . Thus we have found a

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II

117

single null homotopy of f IX 0 which extends to a null homotopy of fIX·

1

for each xi. Proceeding inductively, suppose we have a map g: CXn __, Y which

is a null homotopy of f IXn, and which extends for each i > n to a null homotopy ui: cxi

-->

y of f Ixi. An argument similar to the above shows

I, m = 1, 2, ... , of the u-, such that m 1 uimiCXn+l"" ui 1 tel Xn+l" Define gn+l = ui 1 1CXn+l" Then ~+ 1 1CXn gn, and gn+l extends to a null homotopy of f IXj for each j 2 n + 2. that there is a subsequence lui

=

By induction we have found a sequence gn: CXn-> Y such that ~+ 1 1CXn =

gn, g = 1, 2, ... ; and gn\X =f. Define g: CX-> Y by g\CXn = gn.

Then g is a null homotopy of f.

(4.15) LEMMA. Let F be the category of spaces having the homotopy type of countable CW complexes. There is a lifting of the functor f' : F -> S, f'(X) = (X,['], into the category A of abelian groups, which

extends the lifting of f': P-> S of 4.10. Moreover if X ( F, then the natural map A: [X, f'] -> inv lim [Xi, X] is an isomorphism of groups. Proof. From 4.14 we know that A is an isomorphism of sets. The sequence [X 0 , f'] ..- (X 1 , f'] ..- · · · is an inductive sequence of homomorphisms of abelian groups. Hence the desired group structure on [X,['] is ind Fm (Xi, l->00

n.

Proof of 4.12. Let F be as in 4.15. By 4.13 f' x f' is in F. Therefore there is an abelian group structure on [[' x f', [']. Define fl. ( [f'xf' ,f'] to be (rr 1 ] + [rr 2 ], where rr i: f' x f'

->

f' is the projection on the ith factor.

Then it is easy to see that p. makes f' into an h-space, and hence [X, f'] has a natural abelian group structure for all spaces X, completing the proof of 4.12.

(4.16) THEOREM. The fibration E

..._.,

=

(P, BO, B PL) is f'-principal. If

f: X -> B PL is such that the fibration f*E induced by f has a section,

118

MORRIS W. HIRSCH AND BARRY MAZUR

then f*E is fibre homotopically trivial. Choosing a homotopy class of sections of f*E induces a bijection between [X, f'] and homotopy class-

es of sections of f*E. To say that E is f' -principal means the following. There is a map /1- :BOX

r

-->

BO such that the diagram

BT~ BOx['

/1-

1

L

BO

commutes; where p. 0 is the natural map; and

BO ---'P:..-- B PL such that the two maps

cso x n x r __. so x r __. so so x cr x n __. so x r __. so are homotopic, the homotopy preserving the projections into B PL. Moreover, it is required that on the fibre f' of E over the base point, p.\f'xf' is the h-space structure of I~, and that p.\BO v f' is the identity on BO and on

r.

We first prove:

(4.17) LEMMA. Let X and Y be countable CW complexes. Let

I

f

l,

XvY - - - - B O

X --~g--B PL

be a commuting diagram. Then there is a commuting diagram

119

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II

X

X

F y _ __.;:;._BO

X

--=-g-- B PL

I

I

where F extends f. Proof. We are assuming X v Y is the union of X and Y with only

x 0 =Yo in common; that p- 1 g(x 0 ) = f'; and f(x 0 ) =identity of f'. First suppose X and Y are finite polyhedrons. Consider the composition u : X x Y ____, X C X v Y as a lift of p

o

J'...

BO

u : X x Y -> B PL. Homotopy classes of lifts of p

o

u

correspond bijectively to the group [X x Y, f'], with u corresponding to

0 by 4.12. Let F: X x Y-> BO be the lift of p

o

u corresponding to the

homotopy class of the composition w:XxY-->YCXvY.i..f' Notethat w\X""'O, while w\Y=f\Y. ltfollowsthat F\XvY ishomotopic through lifts of p

o

f to f. Therefore we may assume F\ X v Y = f,

proving 4.17 for finite complexes. For countable complexes X, Y, let Fi: Xi x Yi -> BO be the map just constructed. Then Fi+ 1 \Xi x Y i and Fi are homotopic lifts rel Xi v Y i• by naturality of the correspondence between lifts and maps into f'. Therefore each Fi extends over Xi+l x Y i+l; hence over xi+2

X

yi+2; etc. This proves 4.17. (Here xl

c

x2

c ...

is an increas-

ing sequence of finite complexes whose union is X; similarly for Y .) In order to prove 4.16, observe that BO and f' have the homotopy type of countable CW complexes. Therefore the map (BO) v f' -> BO

120

MORRIS W. HIRSCH AND BARRY MAZUR

that is the identity on BO and on [', extends to a map f1: BOx[' .... BO making the diagram BOxf'-> BO commute. Moreover, the construction ~

BO of f1 shows that

flir X r

~

->BPL is the h-operation of r. For the last state-

ment of 4.16, it follows from the construction that the two maps (BOxrxr)-> BO are homotopic, as lifts of the compositions BOX r

X['

-> BO-> B PL, on every skeleton. Therefore, the map BOX r X r -> r,. measuring the difference of lifts, is null homotopic on every skeleton. Therefore it is globally null homotopic by 4.13 and 4.14. This completes the proof of 4.16. (4.18) COROLLARY. Lemma 4.17 is true for arbitrary spaces X, Y.

Proof. Let f: X vY .... BO be as in 4.17. Since f(Y) C r, we can factor

f thus: XvY

BO v [' _____. BO ;

and f 0 extends to f 1 . X x Y-> BO. Then f1

o

f 1 :X x Y -> BO is the re-

quired extension f1 where f1: BOX r -> BO is from 4.16. (4.19)COROLLARY. Let E->B beafibrationinducedfrom p:BO->BPL

by a map f: B .... B PL. If E has a section then it is fibre homotopically trivial. Proof. The bundle E inherits a map 11': EX r -> E from f1: BOX r -> BO.

If s: B .... E is a section, define f: Bx ['-> E to be the composition Bxr ~ Exf' ____!!:___.E. Then f is a fibre homotopy equivalence.

121

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II

§5. Obstruction theories The classification theorems 4.1 and 4.2 lead immediately to obstruction theories for concordance classes of smoothings. In order to obtain obstruction theories for individual smoothings the following concordance extension theorem is needed. (5.1) THEOREM. Let M be a PL manifold, A C M a closed subset and

U C M an open neighborhood of A. Let a 0 be a smoothing of M and a 1 a smoothing of U x I which agrees with a 0

on U x 0. Then there

exists a smoothing (3 of M x I which agrees with a 0 on M x 0 and with a 1 in a neighborhood of Ax I. Proof. The first step is to find a neighborhood W C MxI of Mx 0 U Ax I =

X and a smoothing y of W agreeing with a 0 and a 1 in a neighbor-

hood V C Mx I of X. To do this, apply uniqueness of PD collarings (1, 13.10) to conclude that the germ of any PD collaring of part of the

boundary of a smooth manifold can be extended to a collaring of the whole boundary. In the present situation this implies the existence of a PD embedding f: N --> (U x I)a

where N C U x [0, 1 > is an open neighbor-

l

hood of U x 0, having the following properties. There are closed sets B,C in M suchthat ACintB, BCintC, CCU, and f

=

identity on N

n [(U-C)x I]

and f :N

n

(int B X I)a

0

Xl ....

(U X I)a

1

is a smooth embedding. Let W = ((M-C)x I) U (int Bx I) U N. The smoothing y of W is defined to be the product

ao

X

L on (M- C)

X

I; al on N

n

(int Bx I);

and f*a 1 on N. The properties of f show that these three smoothings agree where they overlap; clearly y has the desired properties. The next step is to find a PL embedding G: Mx I--> W which is the identity on M x 0 and on a neighborhood D x I of A x I; this is left to the reader.

122

MORRIS W. HIRSCH AND BARRY MAZUR

Next use (I, 7 .4) to push G(Mx 1) onto a smooth submanifold of W

Y'

by a PD ell!bedding H: G(Mx I) .... W which fixes M x 0 and a neighborhood of A x I. (The smoothing y can be extended to a smoothing of (M x

if the reader objects to the boundary

of W in applying (I, 7.4).) Thus H

0

G: Mx I .... Wy is a PD embedding

onto a smooth submanifold, leaving fixed M x 0 and a neighborhood of M x I. Therefore (H o G)*y is the desired smoothing of M x I.

An immediate consequence of the concordance extension theorem is:

(5.2) COROLLARY. Let A be a closed subset of a PL manifold M. Let a be an A-germ of a smoothing of M (see Section 2). Then a extends to a smoothing of M if and only if there is a smoothing of M whose A-germ is concordant to a. Combining the classification theorem 4.2 and Corollary 5.2 we obtain: (5.3) THEOREM. Let M be a PL manifold and &(M) the fibration over M with fibre

PLIO

induced from B 0 .... BP"L by a map M .... BPI: classifying the tangent block bundle of M. Let A C M be a closed set and a

an A-germ of a smoothing of M. Let s: A .... &!A be a section whose homotopy class corresponds to a. Then a extends to a smoothing of M if and only if s extends to a section of &. From 5.3 we can interpret the obstructions to smoothing M as obstructions to a section of &. These lie in groups

Conceivably the coefficients might be twisted, but actually this does .._....

not happen. To see why, observe that the fibre PL/0 is connected because 1-manifolds are uniquely smoothable up to concordance. (In fact

-

123

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II

PL/0 is 6-connected.) Therefore E has a cross-section over the

1-skeleton of B

PL.

On the other hand E is

r -principal (4.16),

hence

E is fibre homotopically trivial over the 1-skeleton. This implies simple coefficients. If A C M is a closed set and a is a smoothing of a neighborhood of

Mi-l U A in M, the obstruction cohomology class to extending the Mi_ 2 U A germ of a over a neighborhood of Mi U A lies in

It vanishes if and only if such an extension is possible. Since a concordance is a smoothing, obstructions to extending a concordance are readily obtained. Let a 0 , a 1 be smoothings of M and y a concordance between their Mi_ 2 U A germs; y is a smoothing of a neighborhood in M x I of (Mi_ 2 U A) x I. Assume y extends to a concordance between the Mi-l U A germs. The obstruction class to extending y to a concordance between the Mi U A germs of a 0 and a 1 lies in a group naturally isomorphic to

It vanishes if and only if such an extension of y is possible. Obstructions to smoothing a PD homeomorphism are obtained as follows. Let M be a triangulated PL manifold with smoothing a. Let

Nf3 be a smooth manifold and f: M-+ Nf3 a PD homeomorphism. Suppose f is a diffeomorphism in a neighborhood of Mi-l U A. The problem is to find a PD isotopy ft of f

=

f 0 so that f1 is a diffeomorphism in a

neighborhood of Mi U A; it is required that ft = f in a neighborhood of Mi_ 2 U A. The smoothings a and f*f3 agree in a neighborhood of Mi-l U A, and the desired isotopy exists if and only if a and f*f3 are concordant in a neighborhood of Mi U A, the concordance being a product in a neighborhood of (Mi_ 2 U A) x I. Therefore the obstruction to finding such an isotopy is defined as the obstruction to finding such a concordance.

124

MORRIS W. HIRSCH AND BARRY MAZUR

It lies in

and vanishes if and only if there is a PD isotopy ft: M -. N as above. Munkres [20, 21] shows that these obstructions are dual to his homology obstructions [18, 19]. Another interpretation of the problem of smoothing f is as follows.

£*{3 is concordant to £*a if and only if the two linearizations Exp (a) and Exp (£*{3) of (M x M, M~) are equivalent (3.10). Let t(a), t({J) denote respectively the tangent PD vector bundles of Ma, Np, considered as linearizations of (MxM,M~) and (NxN,N~). The map fxf:MxM-. Nx N restricts to what might be called a PD map of: t(a)-. t({J), covering f: M -> N. Then f is PD isotopic to a diffeomorphism Ma -> Np if and only if Of is PD isotopic to a vector bundle isomorphism t(a)-.t({J). Perhaps Of should be called the "combinatorial" of f, in analogy with the "differential" of a differentiable map. §6. h-structures on BO, B PL and [' In this section we give a new description of the h-structure on ['. Details are omitted since Boardman and Vogt [1] promise a thorough treatment of the whole subject. We shall work in the semisimplicial category. PL(n) is the semisimplicial group whose k-simplices are PL automorphisms (preserving 0-sections) of the trivial Rn bundle over ~k; PL

lim PL(n). B PL(n) is a classifying space for PL(n), and n->oo B PL = lim B PL(n) is then a classifying space for PL. n->oo The stable tubular neighborhood theorem for the PL category [8, 13] =

implies that every block bundle is stably isomorphic to a PL bundle, the latter being unique up to stable isomorphism. This means we can ~

identify B PL and B PL. Henceforth we ignore PL.

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II

(6.1)

125

The map tn : BO(n) --> B PL(n)

We follow Lashof and Rothenberg in defining tn. We replace the orthogonal group O(n) by the semisimplicial group complex O(n) of PD singular simplices of O(n). Lashof and Rothenberg define a choice of BO(n) and a map BO(n) --> B PL(n) as follows. Let p(n): E PL(n) --> B PL(n) be a universal principal bundle for PL(n). Thus PL(n) acts freely on E PL(n) on the right with orbit space B PL(n), and E PL(n) is contractible. Since the inclusion PD(n) --> PL(n) is a homotopy equivalence, the associated bundle EO(n)

=

E PL(n) xPL(n) PD(n)

is also contractible. The right action of O(n) on PD(n) induces a free action of O(n) on EO(n). The orbit space of this action is then BO(n) = E PL(n) x PL(n) (PD(n)/O(n)) . The projection E PL(n) x PD(n) -. E PL(n) induces a semi-simplicial fibration PD(n)/O(n)

C

t

BO(n) __!!___.. B PL(n) ,

and in the limit, another fibration PD/0 C BO

(1)

-L

B PL

The multiplications on BO and B PL induce the operation of Whitney sums on stable bundles. Let A ::R"" x :R""--> :R"" be an isometry(= norm preserving bijective linear map). Define homomorphisms A0 : 0

X

0 --> 0 '

APL : PL X PL --> PL

by (f, g)

f->

A(fx g)A - l

.

126

MORRIS W. HIRSCH AND BARRY MAZUR

The induced maps on classifying spaces are denoted llo: BOxBO-> BO,

llpL: B PLxB PL-> B PL

(6.2) THEOREM. p. 0 and llpL are homotopy abelian, homotopy associative h-space structures. Moreover p.0 can be chosen within its homotopy class so that the following diagram commutes: llo

BOx BO - - - - - B O

txt[

"PL

It

B PLxB PL - - - - B PL

where t is the fibration (1). Consequently the fibre h-space.

r

r

of t is an

is also homotopy associative and homotopy commutative.

Proof. For 0 < k < q let qO(k) C O(q) be the semisimplicial subgroup of PD singular simplices whose values leave fixed the first q-k coordinates. Put

O(q, n)

=

O(q)/qO(q-n) .

This is the same as the PD singular complex of the Stiefel manifold Vq,n of orthogonal n-frames in :Rq. Put PL(q, n)

=

PL(q)/qPL(q-n). Define O(oo, n)

=

dir limq->oo O(q, n)

O(oo, oo)

=

inv limn->oo O(oo,n)

Then

so O(oo, n) is contractible for each n. Since each map O(oo, n+1)-> O(oo, n) is a fibration, O(oo, oo) is contractible. Let Hom (O(n), O(q)) be the PD singular complex of Hom (O(n), O(q)), the space of homomorphisms O(n) -> O(q). Define a map

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II

127

"" Hom(O(n), 0). Define Hom(O(n), O) to be the directlimitofthemaps BO is an h-structure. We must show that the composite map fl,: BO ____, BOx

!x 0 l-->

BOx BO L

BO

is homotopic to the identity, where x 0 is the base point of BO. Now it is easy to see that fl'= B( 0 (a)). Since O(oo,oo) is contractible, a is

129

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II

connected to the identity. Clearly 0 (c) is the identity c

f

O(oo,oo),

where c is the inverse limit of the inclusions O(n)-> O(oo). Hence fJ-' = B( 0 (c)). Clearly 0 (c) is the identity homomorphism of 0; so fJ-'

is homotopic to the identity. Likewise, the composition fJ-': BO ___,

lx 0 l x

BO ___, BOx BO L

BO

is homotopic to the identity. In a similar way, using the connectedness of 0 00 oo' it is proved that t

fJ-

is homotopy associative and homotopy commutative. A similar argument can be used for B PL. Instead of proving

PL(oo·,oo) contractible, we construct a map O(oo,oo)-. PL(oo,oo) which extends the natural map of vertices. Such a map proves that PL(oo,oo) is connected, and then the same proof shows that B PL is a homotopy commutative and homotopy associative h-space. We must define maps O(q, n)-> PL(q, n) which extend the natural map of vertices; which commute with the inclusions O(q, n) -> O(q, n+l) and PL(q, n) -. PL(q, n+l) and which also commute with the fibrations O(q, n+l) -> O(q, n) and PL(q, n+l) -. PL(q, n) We start with the Lashof-Rothenberg map tq: BO(q)

->

B PL(q) of

6.1 above. A similar construction produces a map B(q O(q-n)) -> B(qPL(q-n)), and the following diagram commutes up to homotopy: B(qO(q-n)) - - - - B O ( q )

l

B(qPL(q-n))

l

B PL(q)

130

MORRIS W. HIRSCH AND BARRY MAZUR

Each diagram O(q, n)

C

O(q+l, n)

PL(q, n)

C

PL(q+l, n)

l

(6.3)

!

commutes up to homotopy, since with proper choices of universal bundles it commutes exactly. By recursion on q, using the homotopy extension property for the pairs (O(q+l, n), O(q, n)) we adjust each vertical map within its homotopy class so that (6.3) commutes for all (q, n). Therefore a map O(oo, n)--> PL(oo, n) is obtained. Each diagram O(oo, n) - - - - O(oo, n+l)

(6.4)

1

PL( , n)

1

PL( , n+l)

commutes up to homotopy, and the vertical maps are fibrations. By recursion on n we adjust each vertical map in succession until 6.4 commutes exactly for all n. The result is the desired map 0(oo, oo) .... PL( oo, oo ). This completes the ptoof that BO and B PL are homotopy abelian h-groups. It follows that the fibre 1 To know that 1

of t : BO -> B .PL is an h-group.

is homotopy-commutative and homotopy associative, we

need to show that t: BO -> B PL commutes with homotopies of associativity and commutativity. This is true and is a consequence of the constructions used. This completes the proof of 6.2. It is not hard to see that the multiplications on BO and B PL are compatible with Whitney sum maps: BO(n)x BO(m)-> BO(n+ m) and B PL(n)x B PL(m)--. B PL(n+ m). It is left to the reader to verify that for finite dimensional CW complexes, X, the group structure on [X, 1] is the same using either the multiplication on [' defined in this section or the preceding one.

SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II

131

Remarks:

1. PL(oo, n) and PL(oo, oo) are contractible. 2. It seems likely that by exploiting the contractibility of PL(oo, oo) and 0(oo, oo) (and not just their connectedness) one could show directly that the h-spaces r, BO and B PL satisfy the conditions A(n) of Stasheff [27] for all n, which would mean they have classifying spaces. There should also be maps Br --> B(BO)--> B(B PL). Boardman and Vogt claim that r

and B PL are in fact

infinite loop spaces (as is known for BO), and hence have classifying spaces. 3. It is well known that BO is a group. Perhaps the shortest construction is to observe that 0 is a normal subgroup of the contactible group 0 00 of isometries of group which can be used for BO.

5\

00

•

Hence 0 00 /0 is a

BIBLIOGRAPHY [1].

Boardman, J., and Vogt, R., Homotopy everything H-spaces. Bull. Amer. Math. Soc. 74 (1968), 1117-1122.

[2].

Cairns, S., Homeomorphisms between topological manifolds and analytic manifolds. Annals Math. 41 (1940), 796-808.

[3].

Cerf, J., Sur les diffeomorphismes de la sphere de dimension trois (f'4 = 0). Lecture notes in Mathematics. 53 (1968), Springer-Verlag.

[ 4].

Haefliger, A., Plongement differentiables de varietes dans varietes. Comm. Math. Helv. 36 (1961), 48-82.

[5].

, Knotted spheres and related geometrical problems. Proc. Int. Congress of Math., Moscow, 1966.

[6].

Hirsch, M., Obstructions to smoothing manifolds and maps. Bull. Amer. Math. Soc. 69 (1963), 352-356.

[7].

, On tubular neighborhoods of piecewise linear and topological manifolds, Proc. of Conference on Manifolds, Prindle, Weber and Schmidt 1968, 63-80.

[8].

, On normal microbundles. Topology 5 (1966), 229-240.

[9].

Hudson, J., Extending piecewise linear isotopies. Proc. London Math. Soc. (3), 16 (1966), 651-668.

[10].

, Piecewise linear embeddings and isotopies. Bull. Amer. Math. Soc. 72 (1966), 536-7.

[11]. Kato, M., Combinatorial prebundles I, II. Osaka

J.

Math. 4 (1968).

[12]. Kervaire, M., and Milnor, J., Groups of homotopy spheres, part I. Annals Math. 77 (1963), 504-577. [ 13]. Lash of, R., and Rothenberg, M., Microbundles and smoothing. Topology 3 (1965), 357-380.

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[14]. Mazur, B., and Poenaru, V., Seminaire de Topologie combinatoire et differentielle. Inst. Hautes Etudes Scient. 1962. [15]. Milnor, J., Microbundles, Part I. Topology 3, supplement 1 (1964), 53-80. [16].

, Topological manifolds and smooth manifolds. Proc. Int. Cong. Math., Stockholm 1962, 132-138.

[17]. Morlet, C., Voisinages tubulaires des varietes semilineaires. C. R. Acad. Sci. Paris, 262 (1966), 740-743. [18]. Munkres, J., Obstructions to the smoothing of piecewise differentiable homeomorphisms. Annals Math. 72 (1960), 521-554. [19]. [20]. [21]. [22].

, Obstruction to imposing differentiable structures. Ill. J. Math. 8 (1964), 361-376. , Concordance is equivalent to smoothability. Topology 5 (1966), 371-389. , Compatibility of imposed differentiable structures. Ill. J. Math. 12 (1968), 610-615.

J.

, Differentiable isotopies on the 2-sphere. Mich. Math. 7 (1960), 193-197.

[23]. Rourke, C., and Sanderson, B., Block bundles. Ann. Math. 87 (1968), 1-28; 256-278; 431-483. [24]. - - - - - - - - - , !".-sets. University of Warwick 1969 (mimeographed). [25]. Smale, S., Generalized Poincare's conjecture in dimensions greater than 4. Annals Math. 74 (1961), 391-406. [26].

, Diffeomorphisms of the 2-sphere. Proc. Amer. Math. Soc. 10 (1959), 621-626.

[27]. Stasheff, J., Homotopy associativity of H-spaces I. Trans. Amer. Math. Soc. 108 (1963), 275-292. [28]. Steenrod, N., The Topology of Fibre Bundles. Princeton, 1951. [29]. Thorn, R., Des variete::> triangulees aux varietes differentiables. Proc. Int. Cong. Math., Edinburgh 1958, 248-255. [30]. White, D., Smoothing of embeddings and classifying spaces. Thesis, U. Geneva, 1971.

Library of Congress Cataloging in Publication Data Hirsch, Morris, 1933Smoothings of piecewise linear manifolds. (Annals of mathematics studies, no. 80) Bibliography: p. 1. Piecewise linear topology. 2. Manifolds (Mathematics) Barry, joint author. II. Title. III. Series. QA613.4.H57 514'.224 74-2967 ISBN 0-691-08145-X

I.

Mazur,

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9 780691 081458