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