The natural transformation form K(Z) to THH(Z)

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The natural transformation from

K(Z)

to

THH(Z)

Marcel Bokstedt

Fakultat fUr Mathematik Universitat Bielefeld 4800 Bielefeld, FRG

In [2 ], [3] we have studied the topological Hochschild homology. In particular we have shown that there is a map K(R) ------:;» THH (R) of rings up to homotopy. In this paper, we will show that this map for R

=~

is nontrivial on homotopy, groups in positive degrees. As an applica-

tion we will show that the map K(Zl)

[ 4] is not a 2-primary equivalence, in contradiction to a conjecture in [ 5 ].

I want to thank I.Madsen for pointing out an error in an earlier version of this paper.

Let

R be an FSP in the sense of [2]. This determines a ring up

to homotopy [8 ]. R is a functor from the category of finite, pointed, simplicial spaces, with a product l.l :

R(X) A R(Y) ,... R(XAY)

The product is assumed to be associative, with a unit. It is also assumed that the limit system

nnR(Sn)

stabilizes.

The corresponding ring is

Examples are the identity functor, and the functor which to a simplicial set associates the free abelian simplicial group generated by it. These examples correspond to the rings up to hombtopy For such an Hochschi ld homology

R, we defined the K-theory

QSO K(R)

resp.

Zl.

and the topological

THH (R) . There are maps [2.]:

(homotopy units of limn~(Sn» ... K(F) .... THH(F)

- Z -

By nuturality, we obtain a commutative diagram BG

B '!lIZ

~

1 A(*)

1 t

K(Z)

~

1 Qso = THH(QSO) Theorem 1.1.

THH(Zl) •

:>

The map TT

Zp - I (K(%.)

-->

TI

Zp - I (THH(:£)

R&

Zip

is surjective. The rest of this section is devoted to a proof of 1.1. In [11] it is shown that the boundary map TT Zp- 1(K(Zl)

)

is surjective onto the first nontrivial homotopy group. In addition zip

R&

TIZp_Z(B ZlIZ, BG) ---> TIZp_Z(K(Z),A(*»

is an isomorphism. It follows that 1.1 reduces to the statement Claim I.Z.

TT

Zp - Z (BSG)· ---~>

TIZP_Z(QSo,THH(Z»

is surjective onto the image of a'oTHH : TI Zp - 1 (THH(71)) Recall from [ 3] that

o

TIZp_Z(QS ,THH(71»).

--:>

THH(71)

is a product. of Eilcenberg-

MacLane spectra, so that

TI. THH(Z) =

Zll j

1

It is also known

that if

=

and

I~

I~/p l ~/p

i i

=0 = Zj

i

Zj-l

p

is odd

i = 0

0< i < Zp-3

i = Zp-3 Zp-3 < i < 4p-3

- 3 -

The map

QSo ----->

THH(Z)

factors over the inclusion Z --> THH(Z).

In particular, it is trivial on homotopy groups in positive dimensions. It follows that

={

o

IT

Let

_ (QS ,THH(Z» 2p 2

Z/2 ~ Z/2

p = 2

zip

p

odd.

R be a commutative FSP. We have the following commutative

diagram· SIx R*

1

A B R*

1.3

S1 x R* --->

free loop space on The map

R*

ABR*

1

THH(R)

;>

~ where

S1 x R

>

IC'/ BR*

is obtained by the inclusion of

R*

in the

BR* twisted by the circle action on ABR* • R is the inclusion of the homotopy units in

---;>

R,

and the map SIx R

--;>

is given by combining the inclusion on

THH(R) R

--;>

THH(R)

with the action of

THH (R). For detai Is, see [ 2 ] •

The relevance to us is that the map BR* - - - > K(R)

BR* ---> THH(R)

factors as

- - - > THH(R).

We can get a hold on this map by computing the map sIx R - - > THH(R) and then applying diagram 1.3. In [3] it is shown that the following map is nontrivial: S

Here

f

1

f 1 g /\ ~ --> S+ /\ ~ --> THH'(7l) .

is the map splitting up

choice of basepoint. The map

g

~o

homotopy the map given by a

corresponds to the map of spaces > THHC!l)

S1 x 7Z

It is proved in [3], lemma 3.2, that the composition of the latter map with the map detecting

~2p_l(THH(r»

~

lip

THH~ _ _> S2p-l /\ ~/p represents the Steenrod power

pI .

$

SI

- 4 -

Consider the diagram of spectra

S~

1\

QSo

S I I\Z

--->

j,

Z ->

S~

(X I GS 0)

1\

J;

i

QSO = THH(QSo) ---> The map

-->

THH(QSo)

--->

-->

THH(Z)

THH(Z)

THH(Z) I THH(QSO) •

factors over the inclusion

THH(Z).

We obtain a new diagram at spectra. SI

QSo

1\

t

is detected by

pI

Tt

2p - I (S

1\

iZ

:>

>

THH (iZ)

>

Tt

~n

_ (OSO) 2p 3

SI

(Z / GSo) ={1

1\

2p-l S

is detected by

of the p-torsion Tt _ (SI 1\ QSO) 2P 2 in the sense that pI

p

1\

1

'!lIp !

, [ Ia] , VI. S.2.

a

of a. It follows that 2p-I

Z =

1\

j,

~ The nontrivial p-torsion

Thus, the generator

SI

-->

Tt

- (S I 2p I

1\

is nontrivial on the cofibre

!/QSo)

maps nontrivially into

I p).

The corresponding map on the space lev·el is a map of cofibres SI +

1\

QSo _>

z

:>

1

SI

Z

:>

THH(Z)

:>

+

1\

C I

1

1

C2 . •

C and C are wedges indexed by Z. Translation by induces homeoI 2 morphisms of the cofibres permuting the wedge components. The wedge component

S~

corresponding to the a-component of

Z

1\

resp.

THH(Z)

are mapped

as the cofibres in the following diagram. SI

---:>

----:>

1

THH (Z)

Since the generator of the p-torsion of

C ~ SI

-:>

o

Tt

2p - 1 (S ~

1 1\

SI +

THH(Z) 1\

1

1\

(Qso) 0

o

(I: (QSo) 0)

can be

desuspended to a generator of Tt

2p - I ( C)

F::I

ZIp

this group maps isomorphically to the corresponding homotopy group of THH(Z) . We can translate this statement into the corresponding statement

- 5 -

~thnl1L

till! I-COl1lp(H1Pnl. lJ~ inH

I. '1 Wt1 c()fHo'l11Jt~ l haC tho cumpoN i Le

s~ is nontrivial on

se

It. S I A

-->

TI

- • The image of 2p 1

TI

- (8 2p 1

1

A

1

8

B SG) - >

-~

It. (A B 8G) -"

TI

2p - 1(8

1

THH(Z) 1

A B 8G )

It.

agrees with the image of TI

_ (S 1 2p 1

A BSG) •

A

Claim 1.2 follows.

§

2.

The contradiction. Let

denote the etale K-theory [ 4 ]

Ket(R)

or the

ring

R.

There isoa map

Let

p

be a prime. It is conjectured that if

lip E R, then the map above

is close to being an isomorphism after completing at For instance, in

[5]

p.

it is conjectured that

1 ])A K(7[--2 2

>

Ket(7[~])A 2

(2)

is a homotopy equivalence. In this paragraph, I will show that this cannot be the case. From now on, all spaces will be completed at 2. Recall that there is a fibre square of rings up to homotopy Ket (2: [.!.]) 2

K{IR)

--~

=

1l x BO

=

o.Z x BU

~

~

~

KOF ) 3

K(¢)

K(Z[.!.]) _> Ket(Z[.!.]) 2 2 We want to derive a contradiction from this. Assume that the map

is a homotopy equivalence.

We can reconstruct the homotopy type of K(Z) sequence K(Z I 2)

- - - > K(Z)

and Quillen's computation [ 7 ] K(Z I 2)

[1], [5]

_0

-~

(everything is completed at 21 ).

Z

--~

using the localization

- 6 -

Con.. i df1 r l h(l

l~o"mlU l: II t

i V,I d I. agram nf rinRs up to IlOmo l 0py -~._. ~_.-)t

(lS 0

t

~

K(Z)

,1

~

THH(Z)

K(~) as above is given by the assumption KC!l[}])=K et (7l[}r

We claim that if

we obtain a contradiction. This contradiction arises on comparing the infinite loop maps of the O-component and the I-components in the diagram. QSo -----~

The map

K(~)

---~

K

et

(Z) factors over the ring

J.

There are fibrations

-~

Zl x BSO

where

i = 0,1,

and

J

et

---~

K

(Z[J..]) 2 i

.th component at

d enotes t h e 1

X.

1

X.

It follows

that we have diagrams of infinite loop spaces whose rows are fibrations et (:~[J..] ) 2 1.

K

--~

J.

1

r

B(Zl

---~

r

r

K(Zl) •

X. 1

>

1

1

1

*

THH(71) . 1

THH(Zl) . 1

---:>

B SO)

x

Under our assumption, we would have factorizations by infinite loop maps f. 1

K(Z) .

1

g. 1

BBSO

:>

THH(Z) . 1

:>

There is a fibration, see for instance [ 6 ] , U

--~

B(Z

BSO)

x

---:>

.

§ 24.

BSpin

/ //

This is related to the description of Ket (zl{< The map occurring in the pullback computing Ket (z [..!..]) is the fibre of 2

Z x BU

----~~

BU

3

which on the O-component is given by 1.J; - id , and on the I-component by 1.J;3/ id •

It follows that

. h omotopy equ1va . 1en t to t h e f'b K et(7l~)i ~~ 1S 1 re of

the map BO

--~

and that under our assumption BSO

BU

~

K(Zl).

3

1.J; -1 ~

BU

is the fibre of the map

1

BSU.

- 7 -

In other words, we have commutative diagram Spin

--:>

1 J.

--:>

1.

In

§

SU

--:>

BBSO

K~). - - : >

BBSO

1

,

.,/"'./

II

1.

3 we prove the following

Lemma 2.2 a) b)

H7 (BBSO; Z/4) ~ Z/4e ~2 The map

7 H (BBSO; Z/4) ---:>

f

7 H (SU

Z/4)

is trivial

on 2-torsion c)

A generator

under d)

a

and

=g

Sq 1a

%/4) maps nontrivially

f*

The reduction of D. g

where

7

of 7l/4 c H (BBSO

g

g

1+

~

9

7 H (BBSO ; %./2)

to

has coproduct

1 1 g + a 8 Sga + Sga 8 a

generate

3 H (BBSO ; '!l/2) ';: Z/2

and 4 H (BBSO ; Z/2) ';: '!l/2 , respectively.

The maps 3 H (THH(LZ).

; LZ/2) - : >

1.

3 H (BBSO

'!l/2)

are injective by 1.1. Recall from [2], theorem 1.1. that the unique element 7

1.

7

which is reduced from H ( ~/2)

if

i

=

7

is primitive if

Q I + 1 8 x ~7

1. It follows that

7 + x3

~

St

1

x

3

i = 0, and has diagonal +

S~

1

x

7 H (THH(Z). ; 'lZ/4) - > 1.

7 H (BBSO ; 'lJ./4)

hits an element of order 4 if and only if From Lemma 2.2 it follows that the' K('lZ)

3

Q x

3

maps nontrivially if and only if

But then

SU ->

of

; Z/2)

H (THH(Z).

D. x = x

x

-->

Co (JWt

fO S iiI!.

BBSO ---->

induce$ a nontrivial map on 7 H (-

i = 1.

Z/ II)

THH (Z) .

1.

i

=

1.

- 8 -

if and only if

i = 1. Since the two maps

SU

THHOZ)

---;>

only differ

by a translation, this gives the contradiction. 3.

§

Homology of BBSO

There is a fibration f

SU

BBSO

---:>

B Spin .

----;>

The purpose of this section is to show the following statement. Lemma 1)

7

7

f :H (BBSO ; Z/4) ---> H (SU ; Z/4)

The map

is trivial on

2-torsion. H (BBSO ; X/ 4) ~ 'Il/4 7

2)

A generator of Z/4

$

'll/2.

maps nontrivially under

f*.

The main reference is [9 J. In the notation of Stasheff ~

'Il/2[w.

i

=1=

2

H*(BBSO ; 1./2)

~

E[c.

j

>

3].

w.

Further, the image of decomposable in

j

H*(B Spin ; Z/2)

is

e.

~

for

i

is generated by

e

~

J

~

2

=1=

j

+ I]

+ 1, and

maps to an in-

H*(SU ; '!l/2).

7

H (BBSO ; Z/2) are in the image of

3

e

and

4

H*(B Spin), it follows that

e • Since f*

H (BBSO ; Z/2). We now have to examine the higher torsion of is given on

o

n

2

H*(BBSO ; 1.";2) 4

3

6

o

o

e

7 --;>

6

e

the reduction of a free generator of

a

e

7

i

where (t 2p + I ) suspends to w(2p+l)2 i +I given on H*(Spin ; Z/2) as follows:

o

n

. H s p1.n n

2

o The clasHes

R* (Spin ; :1:).

4

3

5

3 4

and

e 4e j ·

--;>

e

a;

e

S

6

3

in

7

t

s t 7 ... ] H*(B Spin; Z/2).

is

8

o

0 t

3

and

is

HS(BBSO; Z).

H*(Spin ; Z/2) ~Z/2 [t 2

e e

a

e 3e S

3 4

connecting

BBSO.

8

e e

There is a higher Bochstein of order

and

n < 8)

as follows (for

5

4

is trivial on

7

Sql

e

7

are reductions of

[r~e g~neratorY

of

- 9 -

Z/4- cohomology of

Now consider the spectral sequence associated to the fibration Spin 7 6

---~

SU

---~

BBSO

Z/4

-;;;-L

S

'L/2

i'--

4 3

Z/2

Z/4

-:;;;-I~ Z/4

Z/4

Z/2

Z/2

Z/4

Z/2

Z/2 $Z/4

Z/2 $Z/4

3

4

S

6

7

8

\

2 1

°

I---

2

1

°

2 E = H*(BBSO, H*(Spin, Z/4))

The spectral sequence converges to

Z/4

$

E(a 3

as

H*(SU ; Z/4) ~Z/4

$

H*(SU

Z) ~

a 7 •.. ).

Lemma 2) follows from part 1 and the fact that a class in

H7 (BBSO ; HO(Spin Z/4)

survives. There are not enough classes in the spectral sequence to hit all of it.

=>2.

To prove part of the lemma, notice that the unique element of 7 H (BBSO ; Z/4) which is divisible by two is hit by a differential (for instance, since it is in the image of

To finish the argument, we need to find a different element of order 2 7 in H (BBSO ; %/4) which is hit by a differential. 7 There is an element x of order 2 in H (B Spin; Z/4) which reduces to w E H7 (B Spin; Z/2). 7 (Since Sql w = w ). The image of x in H7 (BBSO ; Z/4) is a class y 6 7 7 which reduces to e E H (BBSO ; Z/2). Since the 2-divisible element 7 reduces to 0, x is a different one.

10

References 1.

Bokstedt, M.: The rational homotopy type of

nWhDiff (*)

• Lecture

Notes in Math. Vol 1051, 25 - 37. Springer, Berlin-HeidelbergNew York (1984) 2.

Bokstedt, M.: Topological Hochschild homology. To appear

3.

Bokstedt, M.: The topological Hochschild homology

4.

Dwyer, W.G.; Friedlander, E.M.: Algebraic and Etale K-theory. Trans-

ot

Z

and

zip.

actions of A.M.S. Vol 292, 247 - 280 (1986) 5.

Dwyer, W.G.; Friedlander, E.M., Conjectural calculations of the ,enay~1 linear group. Contemporary Math., AMS, Vol 55, I.

135 - 147

(1986) 6.

Milnor, J.: Morse theory. Annals of Math. studies, Princeton University Press (1963)

7.

Quillen, D.: On the cohomology and K-theory of the general linear group over finite fields, Annals of Math. 96, 552 - 586 (1972)

8.

Schwanz 1 , R. ; Vogt, R.: Homotopy invariance of

Aro

and

~

ring

spaces. 9. 10.

Stasheff, J.D.: Torsion in

BBSO, Pacific J. of Math. Vol 28 (1969)

Steenrod, N.E.; Epstein, D.B.A.: Cohomology Operations. Annals of Math. Studies Vol 50. Princeton University Press (1962)

II.

Waldhausen, F.: Algebraic K-theory of spaces, a manifold approach, Canad. Math. Soc. Conf. Proc. Vol 2. I, 141 - 184, AMS (1982)