Spin Cob ordism Determines

Real KTheory

Michael J Hopkins Mark A Hovey

MIT and Yale University

Cambridge MA New Haven CT

July

Intro duction

It is a classic theorem of Conner and Floyd CF that complex b ordism de

termines complex Khomology That is the map

K K X MU X

   MU



induced by the To dd genus MU K is an isomorphism of K mo dules for



all sp ectra X See St for descriptions of all the b ordism theories used in

this pap er The ConnerFloyd theorem was later generalized by Landweb er

in his exact functor theorem Lan Conner and Floyd also prove symplectic

b ordism determines real Ktheory

MSp X KO KO X

MSp  





is an isomorphism of KO mo dules for all X



The maps MU K and MSp KO used by Conner and Floyd were

c

extended to maps MSpin K and MSpin KO by Atiyah Bott and

Shapiro in ABS Recall that Spinn is the connected cover of SOn and

c

Spin n is another Lie group derived from Spinn and S The relevant

prop erty for our purp ose is that a bundle is orientable with resp ect to KO

c

resp Ktheory if and only if it is a Spin resp Spin bundle Thus this is

the natural place to lo ok for an isomorphism of ConnerFloyd typ e

1

Partially supp orte d by the NSF

We should note however that Ochanine Och has proven that

MSU X KO KO X

 MSU  



is not an isomorphism The problem is that the lefthand side is not a homol

ogy theory There is no known exact functor theorem for any other b ordism

theory except for those derived from MU

The purp ose of the present pap er is to prove

Theorem The maps

MSpin X KO KO X

MSpin  





and

c

c

K K X X MSpin

  MSpin





induced by the Atiyah BottShapiro orientations are natural isomorphisms

of KO resp K modules for al l spectra X

 

SpanierWhitehead duality then shows that Theorem is also true in

cohomology if we assume X is nite

The real case of Theorem ts in with the general philosophy that at the

prime one should use covers of the orthogonal group to replace the unitary

group ie MSO MSpin MO etc instead of MU At o dd primes

KO and K are essentially equivalent as are MSO and MU But at the prime

KO is a more subtle theory than K and MU cannot detect this though

Theorem says MSpin can

There is another cohomology theory exciting much current interest el

liptic cohomology Lan This theory is v p erio dic unlike KO which is

v p erio dic See Rav for a discussion of v p erio dic cohomology theories

n

However elliptic cohomology is currently only dened after inverting One

then applies the Landweb er exact functor theorem to get elliptic cohomology

from MU One might hop e in line with the philosophy ab ove that one could

dene elliptic cohomology at by using a cover of the orthogonal group most

likely MO This idea is due to Ochanine Och It has b een carried

out by Kreck and Stolz in KS but using MSpin The resulting theory is

not v p erio dic at however Unfortunately there is no known analog for

MO of the AndersonBrownPeterson splitting see b elow for MSpin

which is crucial in our pro of of Theorem

Our pro of of Theorem is very algebraic in nature A hint that Theorem

might b e true is provided by a result of Baum and Douglas BD They give

c

a geometric denition of K resp KOhomology using MSpin resp Spin

manifolds One might hop e that their work could lead to a more geometric

pro of of Theorem but we are currently unable to construct such a pro of

Instead we prove Theorem by proving it after lo calizing at each prime

Essentially all of the work is at p where we have available the splitting of

c

MSpin and MSpin due to Anderson Brown and Peterson ABP We recall

this splitting in section In section we prove that Theorem is equivalent

c

to showing that L MSpin resp L MSpin determines KO resp K Here

L is Ktheory lo calization See Rav or B for a discussion of L Section

contains a lemma which is at the heart of our pro of giving conditions under

which a ring sp ectrum R determines a mo dule sp ectrum M so that

M M X R X

   R



is a natural isomorphism for all X The essential idea is to resolve M as an

Rmo dule sp ectrum We apply this lemma in section completing the pro of

in the complex case In the real case however the lemma only tells us that

n n n n

MSpinM determines KOM where M is the mo d Mo ore

sp ectrum and n Section is devoted to showing that this is actually

enough to prove Theorem Finally the last section discusses o dd primes

In the real case Theorem is obvious at o dd primes The complex case is a

little more dicult but the classical metho d of Conner and Floyd works

The authors would like to thank Haynes Miller Mark Mahowald and Hal

Sadofsky for many helpful discussions

The AndersonBrownPeterson Splitting

Throughout this section all sp ectra will b e lo calized at unless otherwise

stated We recall the work of Anderson Brown and Peterson ABP on

c

MSpin and MSpin An exp osition of their work containing the details of

c

the MSpin case can b e found in St In addition an intensive study of

the homological asp ects of the AndersonBrownPeterson splitting has b een

carried out by Giambalvo and Pengelley GP

Let k o resp k u denote the connective cover of the real resp complex

Ktheory sp ectrum KO resp K Also let k o denote the connective

cover of k o Let HZZ denote the Z ZEilenb ergMacLane sp ectrum An

derson Brown and Peterson construct an additive splitting of MSpin resp

c

MSpin into a wedge of susp ensions of k o k o and HZZ resp k u

and HZZ

We will describ e their results in the real case and later indicate the simpli

cations that o ccur in the complex case Let J b e a partition that is a p ossi

P

j bly empty multiset of p ositive integers such as f g Let nJ

j 2J

so that n and nf g Anderson Brown and Peterson con

J J n

struct maps MSpin KO If J fng we denote by and we

;

denote which is the AtiyahBottShapiro orientation by

These maps interact with the multiplication on MSpin and KO by an

analogue of the Cartan formula for the Steenro d squares One way to describ e

this is as follows Let P b e the set of partitions and consider the set of

formal linear combinations ZP We can make this into a ring by dening

a multiplication on the set of partitions by set union and then into a Hopf

P P

n

0 00

algebra by dening fng fn k g fk g Supp ose J J J

k

Let

0

MSpin MSpin MSpin and KO KO KO

denote the ring sp ectrum multiplications Then the Cartan formula says that

P

00 0

J J 0 J

J

Theorem ABP Suppose that J Then if nJ is even

nJ

lifts to the nJ fold connective cover of KO k o If nJ is odd

J nJ

lifts to the nJ fold connective cover of KO k o



There exist a countable col lection z H MSpin ZZ such that

k

Y Y Y

J nJ

z MSpin k o

k

62J k

62JnJ even

Y Y

nJ deg z

k

k o HZZ

k

62JnJ odd

is a local homotopy equivalence

Note that though we have used the pro duct symb ol ab ove the pro duct

and the copro duct ie the wedge are the same in this case Note as well

J J

that we use the same notation for and the lift of to a connective cover

J

of KO Let us denote the left inverse of arising from the ab ove theorem by

J J I J

Note that is only dened if J and if I we have

I J

J

The fact that the with J do not app ear in this splitting whereas they

do app ear in the Cartan formula makes the real case more dicult than the

complex case

There is a similar splitting in the complex case except partitions con

J

taining are included and every lifts to the nJ fold connnective cover

nJ

of K k u The analogue of the Cartan formula remains true

Note that the ab ove splitting is not a ring sp ectrum splitting nor is

it even a k omo dule sp ectrum splitting as a lowdimensional calculation

veries Mahowald has asked if the ab ove splitting can b e mo died to make

it into a k omo dule sp ectrum splitting Mah If so the pro of of Theorem

could b e made much simpler However Stolz has recently proven that there

is in fact no way to do this KS Perhaps this helps explain why our pro of

of Theorem is so complicated

KTheory Lo calization

c

In this section we study the Ktheory lo calizations of MSpin and MSpin Let

p k o b e the p erio dicity element and let v p MSpin We also use

c

v for the complex analogue in MSpin Our main ob jective in this section

c

is to show that v MSpin and v MSpin are Klo cal which will imply that

it suces to prove Theorem after applying L We will stick to the real

case leaving the obvious mo dications needed in the complex case to the

reader

To prove this we will need to understand multiplication by v The fol

lowing two lemmas provide some of this understanding

J

Lemma Let J be a partition Then al l v except for v p

and possibly v and v These latter two are both even as elements

of the appropriate homotopy group Further v maps to under the forgetful

homomorphism to MO

J

Pro of If J and J the splitting shows v Stong shows

J

in St that if J lifts to the nJ connective cover of KO Thus for

dimensional reasons the comp osition

0

J

S k o MSpin KO

where the rst map is the unit is null for J Recall that p denotes the

p erio dicity element and denote the image of the Hopf map by The image

n n

of the unit on p ositive dimensional homotopy groups is fp p jn g

Indeed it is clear that the image must b e contained in this set and suitable

J n

elements from the image of J hit these elements So we have p

J n

for all J But is the image of under the unit so p

J n

This can only happ en if p is even for all J

J

In particular v is even for all J Anderson Brown and Peterson



J

ABP show that ch x C p x higher terms for x MSpin X

J

Here p x is the Pontrjagin class of x corresp onding to J Thus the p v are

J J

determine oriented cob ordism in also even for J Since p and p p

dimension v go es to an even element in MSO and thus go es to zero in

MO QE D

Our plan for proving that v MSpin is Klo cal is based on Bouselds

identication of Klo cal sp ectra B Let A M M b e the Adams

map on the mo d Mo ore sp ectrum We will show that

A MSpin M MSpin M

b ecomes a homotopy equivalence after inverting v To do this we need the

following lemma

J

Lemma Let I MSpin M consist of those with



for al l J with J Then I is precisely the subMSpin module of v torsion



elements

J

Pro of First note that if for all J with J then in

J

fact for all J This follows from the AndersonBrownPeterson

splitting Indeed such an must factor through a wedge of HZZ s Since

KO M is Klo cal any map from HZZ to KO M factors through

L HZZ which is contractible by the results of AH This proves that

J J

b efore lifting to a cover of KO But the map from a cover of

KO smashed with M to KO smashed with M is injective on homotopy

J J

so in fact after lifting the as well

0 0

Now supp ose I Find a partition J with J such that the p ower

0

J

of dividing is minimal among J with J Then using the

splitting we can write

X

n J

J

62J

0

is not divisible by and I It is then easy to see using the where

J

previous lemma and the Cartan formula that

0

n J n

0

mo d v p

J

i

Continuing in this way we see that v is never

Conversely supp ose I Consider the exact sequence of MSpin mo d



ules arising from the dening cobration for M

f g

MSpin MSpin M MSpin



 

J J

These maps commute with the so g for all J Thus g is in

the analog of I for MSpin Anderson Brown and Peterson call this set I



J

ie I is the subset of MSpin consisting of those classes with for





all J Now I is an ideal mapp ed monomorphically to MO ABP and v

 

maps to there so g v v g Hence there is a with f v

J J

Then f v for all J Since the kernal of f consists of

J

the even elements we must have b eing even for all J Thus

where I Thus v f v f v QE D



Now we can prove

c

Theorem v MSpin and v MSpin are Klocal

Pro of According to Bouseld B it suces to show that

A v MSpin M v MSpin M

 

is an isomorphism Since v MSpin M v MSpin M we

 

must prove the following two facts for MSpin M



If A is v torsion so is

n

There is a and an n such that A v is v torsion

Now A resp ects the AndersonBrownPeterson splitting and on the

homotopy of a cover of KO smashed with M A is multiplication by

p Thus we have

J J J

A x Ax px

Using the splitting write

X

J

mo d I

J

62J

Then

X

J

A p mo d I

J

62J

By the preceding lemma this cannot b e v torsion unless each p But

J

multiplication by p is injective so each and I so is v torsion

J

as well

We now prove the second fact ab ove Again write

X

J

mo d I

J

J

Consider the exact sequence of MSpin mo dules



^f ^g

MSpin ZZ MSpin M Tor MSpin



 

This sequence arises from the dening cob er sequence for M so f and

g come from maps of sp ectra which we also denote f and g Note that

f I I and g I I Thus

 

X

J

g g mo d I

J 

J

Since we are in a ZZ vector space Lemma and the Cartan formula imply

that

X

J

g v g p mo d I

J 

J

P

J

This is equivalent mo d I to g A We saw in the

 J

J

pro of of Lemma that v annihilates I Thus



X

J

g v g Av

J

J

P

0 J

If we let we nd that there is a such that

J

J

0

f v Av

Write

X

J

mo d I

J 

J

Again we are working mo d so

X

J

f v f p A f mo d I

J 

J

Thus f v A f is v torsion Hence we get that

0

v Av f

is v torsion completing the pro of QE D

Note that if the AndersonBrownPeterson splitting could b e made into a

k omo dule sp ectrum splitting which it can not b e KS Theorem would

W

b e obvious since then we would have v MSpin KO It can b e shown

62J

using St that the homotopy groups of v MSpin are what they should b e

for such an equivalence to hold

Corollary For any ring spectrum R

KO R MSpin R X

MSpin ^ R

 



KO R L MSpin R X

L MSpin ^ R

1

 



Also

c c

c

c

X X K L MSpin MSpin K

L MSpin MSpin  

1  

 

Pro of Since p is a unit in KO R



KO R L MSpin R X

L MSpin ^ R

1

 



1

KO R v L MSpin R X

v L MSpin ^ R

 

1



Similarly

KO R MSpin R X

MSpin ^ R

 



1

KO R v MSpin R X

v MSpin ^ R

 



Applying Theorem and recalling that direct limits commute with smashing

and applying L we get

v L MSpin R v L MSpin R L v MSpin R

v MSpin R v MSpin RQE D

Presentations of Mo dule Sp ectra

In this section we prove a general lemma that is at the heart of our argument

Supp ose R is a ring sp ectrum M an Rmo dule sp ectrum There is a natural

transformation

F R X M M X

M  R  



m

obtained as follows Let f S X R b e an element of R X and let



n

g S M an element of M Then F f g is the comp osite

 M

mn m n

S S S X R M X M

where the last map is the structure map of M There is a similar map in

cohomology We want to know under what conditions F is an isomorphism

M

The following lemma partially answers this question The idea is to nd a

presentation of M as an Rmo dule sp ectrum

Lemma Suppose M N and P are Rmodule spectra with F always sur

P

jective and F a natural isomorphism Suppose there are Rmodule maps

N

d P N and d N M with d d Suppose in addition there are

maps s M N and s N P with d s and s d d s homotopy

equivalences Then F is an isomorphism

M

Pro of The map d induces maps

N P R X R X

   R  R

 

and P X N X which we also denote by d The same is true for d Also

 

s induces a map N X P X which we also denote s and similarly for

 

s Note that s do es not induce a map R X N R X P as

 R   R 

 

it is not a map of Rmo dule sp ectra So we have the following commutative

diagram

F

P

R X P P X

R  





d d

2 2

F

N

N N X R X

  R





d d

1 1

F

M

R X M M X

 R  



First we prove surjectivity Since d s M X M X is an isomor

 

phism d N X M X is surjective Since F is an isomorphism

  N

F d d F is also surjective Therefore F is surjective

M N M

Now we prove injectivity As we p ointed out ab ove d N X M X

 

is surjective In particular d N M is surjective Since the tensor

 

pro duct is right exact d R X N R X M is also surjective

 R   R 

 

Now supp ose x R X M has F x By the ab ove remark

 R  M



there is a y R X N with d y x Then d F y F d y

 R  N M



Since s d d s N X N X is an isomorphism there is a z N X

  

such that s d z d s z F y Then we have

N

d F y d s d z d d s z d s d z

N

since d d Since d s is an isomorphism we have d z Thus

F y d s z

N

P with F w s z Since F is surjective there is a w R X

 P P  R



Then

F d w d F w d s z F y

N P N

Since F is an isomorphism we have d w y Thus

N

x d y d d w

and F is injective QE D

M

n n

MSpin ^ M (2 ) Determines KO ^ M (2 )

In this section we apply Lemma to obtain the result in the title of this

section We also complete the pro of of the complex case of Theorem All

sp ectra are lo calized at

c

We start with the complex case Take R N MSpin and let M k u

J

Let d b e the AtiyahBottShapiro orientation Recall that denotes

J

the right inverse to coming from the AndersonBrownPeterson splitting

Let s Then d is an Rmo dule map d s is a homotopy equivalence

and certainly F is an isomorphism

N

W

c

nJ

S Then F is an isomorphism Dene Now let P MSpin

P

J 6;

s to b e on the HZZ summands of R and on the b ottom k u summand

corresp onding to J On the other summands dene s to b e the com

p osite

s ^

'

1

nI nI c nI c nJ

k u k u S MSpin S MSpin S

J 6;

J nJ nJ

Let S k u denote the inclusion of the b ottom cell Then

dene d as the comp osite

W

J

^

c nJ c nJ

MSpin S MSpin k u

J 6; J 6;

W

J

^

c c c

MSpin MSpin MSpin

It is easy to see that d is a mo dule map and a diagram chase using the

J J

Cartan formula shows that for J d s is a homotopy equivalence

I J

and d s is for I J

Thus we get that the comp osite

W W

J J

d s s d

2 2 1 1

c c nJ nJ

MSpin MSpin k u k u

J J

is a homotopy equivalence The HZZ summands prevent us from applying

the main lemma right away But L HZZ AH so if we apply L

W W

J J

each of the maps and b ecome homotopy equivalences

Thus we have proved the following

Theorem

c

c

L MSpin L ku L ku X X

L MSpin

 



1



is a natural isomorphism of L ku modules



Invert the p erio dicity element on b oth sides of this isomorphism Then

since p L k u L p k u L K K we nd that

c

c

K K X X L MSpin

  L MSpin

1 



is a natural isomorphism Combined with Corollary this completes the

pro of of the complex case of Theorem

The idea ab ove was to shift a k u summand down to a b ottom summand

c

of a shifted copy of MSpin and then multiply by the b ottom cell to shift

it back up In the real case this works ne for the k o summands but not

for the k o summands Even after applying L there is no b ottom cell

n

L k o is QZ However if we smash with a Mo ore space M the

natural map k o k o induces a homotopy equivalence

n n

L k o M L k o M

n

So let N R L MSpin M for some xed n We need n

n n

so that M and hence R is a ring sp ectrum Let M L k o M

n

Note that M KO M Indeed the map b etween adjacent connected

covers of KO has b er a shifted copy of HZ or HZZ L kills HZZ and

converts HZ to HQAH Smashing with the Mo ore space then kills HQ Let

n

d L M which we will just call and let s using the same

convention Then d is an Rmo dule map d s is a homotopy equivalence

and F is an isomorphism

N

W W

0 00

nJ nJ 0

Let P R S S where J runs through partitions

0 00

J J

0 0 00 00

with J and nJ even and J runs through partitions with J and

nJ o dd Then F is an isomorphism

P

0 0 0

nJ nJ n J

S L k o M to b e the inclusion of the Dene

0

nJ n

b ottom cell of k o smashed with the units of L S and M Similarly

00 00 00

J nJ nJ n

dene S L k o M as ab ove followed by

the inverse of the homotopy equivalence

00 00

nJ n nJ n

L k o M L k o M

We can then dene the Rmo dule map d just as we did in the complex case

We dene s in the same way as in the complex case for the k o summands

That is it is the comp osite

0 0

s ^

1

nJ n nJ

L k o M R S P

Of course it is on the b ottom k o summand On the k o summands we

just precede the ab ove comp osite with the homotopy equivalence

00 00

nJ n nJ n

L k o M L k o M

Now we must check that d s s d induces an isomorphism on homotopy

groups Lo ok at a particular summand say a k o summand Moving

the susp ension co ordinate outside has the eect of dividing by an appropriate

p ower of the p erio dicity element Then s includes this in by d then

multiplies this by the b ottom class of the k o summand We would like

this to simply multiply by the same p ower of the p erio dicity element so that

d s would just include the k o summand Unfortunately the Cartan

J n

formula is a nontrivial sum b ecause may not b e on L k o M



if J However the following lemma shows that it is always even

n J

Lemma Suppose L k o M Then is divisible by unless



J

Pro of The pro of is mo deled on the pro of of Lemma in section An

Adams sp ectral sequence calculation shows that

n n

L k o M Z Z q p p L



where L is an ideal of relations generated by

n

q q q andq p

The degree of is that of is that of q is and that of p is The

comp osition

0

J

^

^

n n n n

L S M L k o M L MSpin M KO M

where the rst map is the unit is null for J Now the image of the unit

n n n n n n

map on homotopy contains the p p p p and p q Therefore

J

all these classes have We deduce from this that the other classes

J

must have even Indeed denote the Hopf map by as well Then

J n J n J n

p p so p must b e even A similar argument

J n

using instead of shows that p q is even QE D

By the lemma if we lo ok at the matrix for d s s d in a particular

dimension it will have ones on the diagonal and even numb ers everywhere

n

else This matrix is innite dimensional b ecause L MSpin M is not

lo cally nite However we are working over a b ounded torsion group and

in this case a map with such a matrix must b e an isomorphism Indeed

iterating such a map eventually pro deces the identity Thus we have proved

Theorem The map

n n

n

L MSpin M X KO M

L MSpin ^ M

1

 



n

KO M X



is an isomorphism for any n

The results of section then enable us to deduce

Corollary The map

n n

n

MSpin M X KO M

MSpin ^ M

 



n

KO M X



is an isomorphism for any n

n

MSpin and MSpin ^ M (2 )

In this section we nish the pro of of Theorem in the real case We need

the following two lemmas b oth of which are presumably wellknown We do

not assume our sp ectra are lo calized at for these lemmas

Lemma The identity map of a nite torsion spectrum T has nite order

Pro of Let DT denote the SpanierWhitehead dual of T Then selfmaps

of T are in one to one corresp ondance with homotopy classes of T DT

Now T DT is clearly nite and a homology calculation shows that it is

torsion Thus all of its homotopy groups are nite torsion QE D

Lemma If E is a homology theory such that E has bounded torsion then



for al l nite X E X also has bounded torsion



Pro of Induction on cells shows that the torsionfree part of X is



nitely generated Taking a minimal set of such generators we obtain a

cob er sequence

l

f g

h

n

k

T S X T

k

where T is a nite torsion sp ectrum By the previous lemma m T T is

null for some m Consider the exact sequence in Ehomology induced by the

W

n

k

ab ove cob er sequence By hyp othesis there is an n so that if w E S



is torsion then nw Supp ose x E X is torsion so N x for some



W

n

k

N Then h mx so there is a y E S such that g y mx We

  

claim that y is torsion Indeed N x so g N y Thus N y f z for

 

some z E T But then mz so mN y and y is torsion Thus



ny and therefore nmx QE D

We will also need the fact that MSpin X after lo calizing at has no



nonzero innitely divisible elements for nite X This is true since MSpin



consists of nitely generated ab elian groups so by induction on cells so do es

MSpin X



Finally we need the following lemma

Lemma If X is nite there are no nonzero innitely divisible elements

in MSpin X KO after localizing at

MSpin 





Pro of As ab ove consider the cob er sequence

g f

h

n n

k k

X T S S

where T is a nite torsion sp ectrum Lo calize at Then for some n

n

T T is null Applying MSpin we get the exact sequence



g

h

n

k

MSpin S MSpin X Ker f

 

Tensoring this with KO we get the exact sequence



g

h

n

k

KO S MSpin X KO Ker f KO

 MSpin  MSpin 



 

Let us rewrite this more economically in a xed degree as

g

h

M N P

n

Here P is all torsion and M is the lo calization of a nitely generated

0

group Thus M M Ker g is also the lo calization of a nitely generated

ab elian group and we have the short exact sequence

g

h

0

N P M

Now supp ose x N is innitely divisible so that there is a sequence of

elements x N with x x and x x Then hx is also innitely

m m m m

divisible but P is b ounded torsion so hx Thus x g y for

m m m

0

some y M As g is injective we have y y so each y is innitely

m m m m

0

divisible But M is the lo calization of a nitely generated ab elian group

so we must have y and so x QE D

m

We can now complete the pro of of Theorem at the prime It is easy

to see that

MSpin X KO KO X

MSpin  





is surjective indeed the isomorphism

MSp X KO KO X

MSp  





factors through MSpin X KO since the Atiyah BottShapiro ori

MSpin 





entation is an extension of MSp KO So it suces to prove injectivity

Recall that p denotes the p erio dicity element in k o v p Then since

MSpin KO is surjective in nonnegative degrees and p is an iso





morphism on b oth sides it suces to prove injectivity for elements of the

form x where x MSpin X We can also assume that X is the



lo calization of a nite sp ectrum since b oth homology functors and tensor

pro ducts commute with direct limits

By the isomorphism of the pro ceeding section it suces to prove that

the image of x in

n n

n

KO M MSpin M X

MSpin ^ M

 



is nonzero for some n By the preceeding lemmas we can cho ose q so large

q

that do es not divide x MSpin X KO nor x MSpin X

MSpin 

 



q

and such that kills the torsion in MSpin X



q

Supp ose that x b ecomes after smashing with the mo d Mo ore

sp ectrum Then there is an i such that

i q

v x y MSpin M X



q q

where MSpinM satises and y MSpinM X

 

q q

Let M M b e the map which is an isomorphism on the b ottom

q

cell and on the top cell Then we can cho ose ring structures on our

Mo ore sp ectra such that b ecomes a ring sp ectrum map Oka Thus in

q i

MSpin M X v x y and



Now consider the universal co ecient sequence

n n

MSpin X Z Z MSpin M X





n

TorMSpin X Z Z



By choice of q for n a multiple of q the righthand group is just

TorMSpin X Now induces a map b etween these sequences which is



q

hence on the righthand groups Thus

q

y MSpin X Z Z



0

We can therefore lift y to y MSpin X Similarly we can lift



0

to MSpin Then in MSpin X we have

 

i 0 0 q

v x y z MSpin X



0 q i q 0 q q

and Thus v x z y Therefore divides

i

v x and hence x This is a contradiction and concludes the pro of of

Theorem at the prime

Odd Primes

The purp ose of this section is to prove Theorem at o dd primes In this

section all sp ectra are assumed to b e lo calized at an o dd prime p There is

nothing to prove in the real case since at an o dd prime MSpin MSp

We follow the metho d of pro of used by Conner and Floyd CF to prove

K K X MU X

   MU



is an isomorphism Their argument requires us to work in cohomology and



to consider K as a ZZ graded cohomology theory After we are done

we can convert back to homology by using SpanierWhitehead duality and a

limit argument

Their pro of has three steps as follows

Prove that the map

  



G MU X K K X

MU

is surjective by nding a section K MU as in Lemma

Prove that G is an isomorphism for all X if and only if G is an isomor

phism for the universal example X MU

Prove that G is an isomorphism for X MU

c

In the interest of economical notation let us denote MSpin by E We

want to follow this program for

  



G E X K K X

E

The rst two steps are the same as in Conner and Floyd Since the Atiyah

BottShapiro orientation extends the To dd genus the comp osite of their

section K MU with the natural map MU E will b e a section for

E K Step of their argument go es through verbatim from CF except

the universal example is now X E

For Step we must do a little more work Recall from St that E





is evenly graded and torsionfree as a Z mo dule as is H E Z Let

p p

Er denote the r skeleton of E in a minimal CWdecomp osition Then



H Er Z is also evenly graded and torsionfree Therefore the Atiyah

p



1

Hirzebruch sp ectral sequence for E Er collapses The E term is

  

H Er Z E This is a free E mo dule so there is no ro om for mo dule

p

extensions and we have

  

E Er H E r Z E

p

 

The sequence H Er Z E satises the MittagLeer condition so

p

there is no lim term and we have

   

E E lim E Er lim H Er Z E

p

   

lim H Er Z E H E Z E

p p

 

as E mo dules The third equality ab ove holds b ecause E is a free Z

p

mo dule A similar argument shows

  

K E H E Z K

p



as K mo dules Thus G is an isomorphism for X E and the pro of of

Theorem is complete

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