DOCSLIB.ORG

Spin Cobordism Determines Real K-Theory 1 Introduction

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Spin Cobordism Determines Real K-Theory 1 Introduction 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
Load more

Recommended publications
  • The Relation of Cobordism to K-Theories
    The Relation of Cobordism to K-Theories PhD student seminar winter term 2006/07 November 7, 2006 In the current winter term 2006/07 we want to learn something about the different flavours of cobordism theory - about its geometric constructions and highly algebraic calculations using spectral sequences. Further we want to learn the modern language of Thom spectra and survey some levels in the tower of the following Thom spectra MO MSO MSpin : : : Mfeg: After that there should be time for various special topics: we could calculate the homology or homotopy of Thom spectra, have a look at the theorem of Hartori-Stong, care about d-,e- and f-invariants, look at K-, KO- and MU-orientations and last but not least prove some theorems of Conner-Floyd type: ∼ MU∗X ⊗MU∗ K∗ = K∗X: Unfortunately, there is not a single book that covers all of this stuff or at least a main part of it. But on the other hand it is not a big problem to use several books at a time. As a motivating introduction I highly recommend the slides of Neil Strickland. The 9 slides contain a lot of pictures, it is fun reading them! • November 2nd, 2006: Talk 1 by Marc Siegmund: This should be an introduction to bordism, tell us the G geometric constructions and mention the ring structure of Ω∗ . You might take the books of Switzer (chapter 12) and Br¨oker/tom Dieck (chapter 2). Talk 2 by Julia Singer: The Pontrjagin-Thom construction should be given in detail. Have a look at the books of Hirsch, Switzer (chapter 12) or Br¨oker/tom Dieck (chapter 3).
    [Show full text]
  • Algebraic Cobordism
    Algebraic Cobordism Marc Levine January 22, 2009 Marc Levine Algebraic Cobordism Outline I Describe \oriented cohomology of smooth algebraic varieties" I Recall the fundamental properties of complex cobordism I Describe the fundamental properties of algebraic cobordism I Sketch the construction of algebraic cobordism I Give an application to Donaldson-Thomas invariants Marc Levine Algebraic Cobordism Algebraic topology and algebraic geometry Marc Levine Algebraic Cobordism Algebraic topology and algebraic geometry Naive algebraic analogs: Algebraic topology Algebraic geometry ∗ ∗ Singular homology H (X ; Z) $ Chow ring CH (X ) ∗ alg Topological K-theory Ktop(X ) $ Grothendieck group K0 (X ) Complex cobordism MU∗(X ) $ Algebraic cobordism Ω∗(X ) Marc Levine Algebraic Cobordism Algebraic topology and algebraic geometry Refined algebraic analogs: Algebraic topology Algebraic geometry The stable homotopy $ The motivic stable homotopy category SH category over k, SH(k) ∗ ∗;∗ Singular homology H (X ; Z) $ Motivic cohomology H (X ; Z) ∗ alg Topological K-theory Ktop(X ) $ Algebraic K-theory K∗ (X ) Complex cobordism MU∗(X ) $ Algebraic cobordism MGL∗;∗(X ) Marc Levine Algebraic Cobordism Cobordism and oriented cohomology Marc Levine Algebraic Cobordism Cobordism and oriented cohomology Complex cobordism is special Complex cobordism MU∗ is distinguished as the universal C-oriented cohomology theory on differentiable manifolds. We approach algebraic cobordism by defining oriented cohomology of smooth algebraic varieties, and constructing algebraic cobordism as the universal oriented cohomology theory. Marc Levine Algebraic Cobordism Cobordism and oriented cohomology Oriented cohomology What should \oriented cohomology of smooth varieties" be? Follow complex cobordism MU∗ as a model: k: a field. Sm=k: smooth quasi-projective varieties over k. An oriented cohomology theory A on Sm=k consists of: D1.
    [Show full text]
  • New Ideas in Algebraic Topology (K-Theory and Its Applications)
    NEW IDEAS IN ALGEBRAIC TOPOLOGY (K-THEORY AND ITS APPLICATIONS) S.P. NOVIKOV Contents Introduction 1 Chapter I. CLASSICAL CONCEPTS AND RESULTS 2 § 1. The concept of a fibre bundle 2 § 2. A general description of fibre bundles 4 § 3. Operations on fibre bundles 5 Chapter II. CHARACTERISTIC CLASSES AND COBORDISMS 5 § 4. The cohomological invariants of a fibre bundle. The characteristic classes of Stiefel–Whitney, Pontryagin and Chern 5 § 5. The characteristic numbers of Pontryagin, Chern and Stiefel. Cobordisms 7 § 6. The Hirzebruch genera. Theorems of Riemann–Roch type 8 § 7. Bott periodicity 9 § 8. Thom complexes 10 § 9. Notes on the invariance of the classes 10 Chapter III. GENERALIZED COHOMOLOGIES. THE K-FUNCTOR AND THE THEORY OF BORDISMS. MICROBUNDLES. 11 § 10. Generalized cohomologies. Examples. 11 Chapter IV. SOME APPLICATIONS OF THE K- AND J-FUNCTORS AND BORDISM THEORIES 16 § 11. Strict application of K-theory 16 § 12. Simultaneous applications of the K- and J-functors. Cohomology operation in K-theory 17 § 13. Bordism theory 19 APPENDIX 21 The Hirzebruch formula and coverings 21 Some pointers to the literature 22 References 22 Introduction In recent years there has been a widespread development in topology of the so-called generalized homology theories. Of these perhaps the most striking are K-theory and the bordism and cobordism theories. The term homology theory is used here, because these objects, often very different in their geometric meaning, Russian Math. Surveys. Volume 20, Number 3, May–June 1965. Translated by I.R. Porteous. 1 2 S.P. NOVIKOV share many of the properties of ordinary homology and cohomology, the analogy being extremely useful in solving concrete problems.
    [Show full text]
  • Floer Homology, Gauge Theory, and Low-Dimensional Topology
    Floer Homology, Gauge Theory, and Low-Dimensional Topology Clay Mathematics Proceedings Volume 5 Floer Homology, Gauge Theory, and Low-Dimensional Topology Proceedings of the Clay Mathematics Institute 2004 Summer School Alfréd Rényi Institute of Mathematics Budapest, Hungary June 5–26, 2004 David A. Ellwood Peter S. Ozsváth András I. Stipsicz Zoltán Szabó Editors American Mathematical Society Clay Mathematics Institute 2000 Mathematics Subject Classification. Primary 57R17, 57R55, 57R57, 57R58, 53D05, 53D40, 57M27, 14J26. The cover illustrates a Kinoshita-Terasaka knot (a knot with trivial Alexander polyno- mial), and two Kauffman states. These states represent the two generators of the Heegaard Floer homology of the knot in its topmost filtration level. The fact that these elements are homologically non-trivial can be used to show that the Seifert genus of this knot is two, a result first proved by David Gabai. Library of Congress Cataloging-in-Publication Data Clay Mathematics Institute. Summer School (2004 : Budapest, Hungary) Floer homology, gauge theory, and low-dimensional topology : proceedings of the Clay Mathe- matics Institute 2004 Summer School, Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary, June 5–26, 2004 / David A. Ellwood ...[et al.], editors. p. cm. — (Clay mathematics proceedings, ISSN 1534-6455 ; v. 5) ISBN 0-8218-3845-8 (alk. paper) 1. Low-dimensional topology—Congresses. 2. Symplectic geometry—Congresses. 3. Homol- ogy theory—Congresses. 4. Gauge fields (Physics)—Congresses. I. Ellwood, D. (David), 1966– II. Title. III. Series. QA612.14.C55 2004 514.22—dc22 2006042815 Copying and reprinting. Material in this book may be reproduced by any means for educa- tional and scientific purposes without fee or permission with the exception of reproduction by ser- vices that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given.
    [Show full text]
  • On the Bordism Ring of Complex Projective Space Claude Schochet
    proceedings of the american mathematical society Volume 37, Number 1, January 1973 ON THE BORDISM RING OF COMPLEX PROJECTIVE SPACE CLAUDE SCHOCHET Abstract. The bordism ring MUt{CPa>) is central to the theory of formal groups as applied by D. Quillen, J. F. Adams, and others recently to complex cobordism. In the present paper, rings Et(CPm) are considered, where E is an oriented ring spectrum, R=7rt(£), andpR=0 for a prime/». It is known that Et(CPcc) is freely generated as an .R-module by elements {ßT\r^0}. The ring structure, however, is not known. It is shown that the elements {/VI^O} form a simple system of generators for £t(CP°°) and that ßlr=s"rß„r mod(/?j, • • • , ßvr-i) for an element s e R (which corre- sponds to [CP"-1] when E=MUZV). This may lead to information concerning Et(K(Z, n)). 1. Introduction. Let E he an associative, commutative ring spectrum with unit, with R=n*E. Then E determines a generalized homology theory £* and a generalized cohomology theory E* (as in G. W. White- head [5]). Following J. F. Adams [1] (and using his notation throughout), assume that E is oriented in the following sense : There is given an element x e £*(CPX) such that £*(S2) is a free R- module on i*(x), where i:S2=CP1-+CP00 is the inclusion. (The Thom-Milnor spectrum MU which yields complex bordism theory and cobordism theory satisfies these hypotheses and is of seminal interest.) By a spectral sequence argument and general nonsense, Adams shows: (1.1) E*(CP ) is the graded ring of formal power series /?[[*]].
    [Show full text]
  • Bordism and Cobordism
    [ 200 ] BORDISM AND COBORDISM BY M. F. ATIYAH Received 28 March 1960 Introduction. In (10), (11) Wall determined the structure of the cobordism ring introduced by Thorn in (9). Among Wall's results is a certain exact sequence relating the oriented and unoriented cobordism groups. There is also another exact sequence, due to Rohlin(5), (6) and Dold(3) which is closely connected with that of Wall. These exact sequences are established by ad hoc methods. The purpose of this paper is to show that both these sequences are ' cohomology-type' exact sequences arising in the well-known way from mappings into a universal space. The appropriate ' cohomology' theory is constructed by taking as universal space the Thom complex MS0(n), for n large. This gives rise to (oriented) cobordism groups MS0*(X) of a space X. We consider also a 'singular homology' theory based on differentiable manifolds, and we define in this way the (oriented) bordism groups MS0*{X) of a space X. The justification for these definitions is that the main theorem of Thom (9) then gives a 'Poincare duality' isomorphism MS0*(X) ~ MSO^(X) for a compact oriented differentiable manifold X. The usual generalizations to manifolds which are open and not necessarily orientable also hold (Theorem (3-6)). The unoriented corbordism group jV'k of Thom enters the picture via the iso- k morphism (4-1) jrk ~ MS0*n- (P2n) (n large), where P2n is real projective 2%-space. The exact sequence of Wall is then just the cobordism sequence for the triad P2n, P2»-i> ^2n-2> while the exact sequence of Rohlin- Dold is essentially the corbordism sequence of the pair P2n, P2n_2.
    [Show full text]
  • On the Motivic Spectra Representing Algebraic Cobordism and Algebraic K-Theory
    Documenta Math. 359 On the Motivic Spectra Representing Algebraic Cobordism and Algebraic K-Theory David Gepner and Victor Snaith Received: September 9, 2008 Communicated by Lars Hesselholt Abstract. We show that the motivic spectrum representing alge- braic K-theory is a localization of the suspension spectrum of P∞, and similarly that the motivic spectrum representing periodic alge- braic cobordism is a localization of the suspension spectrum of BGL. In particular, working over C and passing to spaces of C-valued points, we obtain new proofs of the topological versions of these theorems, originally due to the second author. We conclude with a couple of applications: first, we give a short proof of the motivic Conner-Floyd theorem, and second, we show that algebraic K-theory and periodic algebraic cobordism are E∞ motivic spectra. 2000 Mathematics Subject Classification: 55N15; 55N22 1. Introduction 1.1. Background and motivation. Let (X, µ) be an E∞ monoid in the ∞ category of pointed spaces and let β ∈ πn(Σ X) be an element in the stable ∞ homotopy of X. Then Σ X is an E∞ ring spectrum, and we may invert the “multiplication by β” map − − ∞ Σ nβ∧1 Σ nΣ µ µ(β):Σ∞X ≃ Σ∞S0 ∧ Σ∞X −→ Σ−nΣ∞X ∧ Σ∞X −→ Σ−nΣ∞X. to obtain an E∞ ring spectrum −n β∗ Σ β∗ Σ∞X[1/β] := colim{Σ∞X −→ Σ−nΣ∞X −→ Σ−2nΣ∞X −→···} with the property that µ(β):Σ∞X[1/β] → Σ−nΣ∞X[1/β] is an equivalence. ∞ ∞ In fact, as is well-known, Σ X[1/β] is universal among E∞ Σ X-algebras A in which β becomes a unit.
    [Show full text]
  • Math 535B Final Project: Lecture and Paper
    Math 535b Final Project: Lecture and Paper 1. Overview and timeline The goal of the Math 535b final project is to explore some advanced aspect of Riemannian, symplectic, and/or K¨ahlergeometry (most likely chosen from a list below, but you are welcome to select and plan out a different topic, with instructor approval). You will then give two forms of exposition of this topic: • A 50 minute lecture, delivered to the class. • A 5+ page paper exposition of the topic. Your paper must be typeset (and I strongly recommend LATEX) The timeline for these assignments is: • Your lectures will be scheduled during the last two weeks of class, April 15-19 and 22- 26 - both during regular class time and during our make-up lecture slot, Wednesday between 4:30 and 6:30pm. We will schedule precise times later this week. If you have any scheduling constraints, please let me know ASAP. • Your final assignment will be due the last day of class, April 26. If it is necessary, this can be extended somewhat, but you will need to e-mail me in advance to schedule a final (hard) deadline. Some general requirements and/or suggestions: (1) In your paper, you are required to make reference to, in some non-trivial fashion, at least one original research paper. (you could also mention any insights you might have learned in your lecture though this is not necessary. Although many of the topics will be \textbook topics," i.e., they are now covered as advanced chapters in textbooks, there will be some component of the topic for which one or more original research papers are still the \best" or \most definitive" references.
    [Show full text]
  • Toric Polynomial Generators of Complex Cobordism
    TORIC POLYNOMIAL GENERATORS OF COMPLEX COBORDISM ANDREW WILFONG Abstract. Although it is well-known that the complex cobordism ring is a polynomial U ∼ ring Ω∗ = Z [α1; α2;:::], an explicit description for convenient generators α1; α2;::: has proven to be quite elusive. The focus of the following is to construct complex cobordism polynomial generators in many dimensions using smooth projective toric varieties. These generators are very convenient objects since they are smooth connected algebraic varieties with an underlying combinatorial structure that aids in various computations. By applying certain torus-equivariant blow-ups to a special class of smooth projective toric varieties, such generators can be constructed in every complex dimension that is odd or one less than a prime power. A large amount of evidence suggests that smooth projective toric varieties can serve as polynomial generators in the remaining dimensions as well. 1. Introduction U In 1960, Milnor and Novikov independently showed that the complex cobordism ring Ω∗ is isomorphic to the polynomial ring Z [α1; α2;:::], where αn has complex dimension n [14, 16]. The standard method for choosing generators αn involves taking products and disjoint unions i j of complex projective spaces and Milnor hypersurfaces Hi;j ⊂ CP × CP . This method provides a smooth algebraic not necessarily connected variety in each even real dimension U whose cobordism class can be chosen as a polynomial generator of Ω∗ . Replacing the disjoint unions with connected sums give other choices for polynomial generators. However, the operation of connected sum does not preserve algebraicity, so this operation results in a smooth connected not necessarily algebraic manifold as a complex cobordism generator in each dimension.
    [Show full text]
  • Vector Bundles. Characteristic Classes. Cobordism. Applications
    Math 754 Chapter IV: Vector Bundles. Characteristic classes. Cobordism. Applications Laurenţiu Maxim Department of Mathematics University of Wisconsin [email protected] May 3, 2018 Contents 1 Chern classes of complex vector bundles 2 2 Chern classes of complex vector bundles 2 3 Stiefel-Whitney classes of real vector bundles 5 4 Stiefel-Whitney classes of manifolds and applications 5 4.1 The embedding problem . .6 4.2 Boundary Problem. .9 5 Pontrjagin classes 11 5.1 Applications to the embedding problem . 14 6 Oriented cobordism and Pontrjagin numbers 15 7 Signature as an oriented cobordism invariant 17 8 Exotic 7-spheres 19 9 Exercises 20 1 1 Chern classes of complex vector bundles 2 Chern classes of complex vector bundles We begin with the following Proposition 2.1. ∗ ∼ H (BU(n); Z) = Z [c1; ··· ; cn] ; with deg ci = 2i ∗ Proof. Recall that H (U(n); Z) is a free Z-algebra on odd degree generators x1; ··· ; x2n−1, with deg(xi) = i, i.e., ∗ ∼ H (U(n); Z) = ΛZ[x1; ··· ; x2n−1]: Then using the Leray-Serre spectral sequence for the universal U(n)-bundle, and using the fact that EU(n) is contractible, yields the desired result. Alternatively, the functoriality of the universal bundle construction yields that for any subgroup H < G of a topological group G, there is a fibration G=H ,! BH ! BG. In our A 0 case, consider U(n − 1) as a subgroup of U(n) via the identification A 7! . Hence, 0 1 there exists fibration U(n)=U(n − 1) ∼= S2n−1 ,! BU(n − 1) ! BU(n): Then the Leray-Serre spectral sequence and induction on n gives the desired result, where 1 ∗ 1 ∼ we use the fact that BU(1) ' CP and H (CP ; Z) = Z[c] with deg c = 2.
    [Show full text]
  • What Is a Monotone Lagrangian Cobordism? Volume 31 (2012-2014), P
    Institut Fourier — Université de Grenoble I Actes du séminaire de Théorie spectrale et géométrie François CHARETTE What is a monotone Lagrangian cobordism? Volume 31 (2012-2014), p. 43-53. <http://tsg.cedram.org/item?id=TSG_2012-2014__31__43_0> © Institut Fourier, 2012-2014, tous droits réservés. L’accès aux articles du Séminaire de théorie spectrale et géométrie (http://tsg.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://tsg.cedram.org/legal/). cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Séminaire de théorie spectrale et géométrie Grenoble Volume 31 (2012-2014) 43-53 WHAT IS A MONOTONE LAGRANGIAN COBORDISM? François Charette Abstract. — We explain the notion of Lagrangian cobordism. A flexibil- 2 ity/rigidity dichotomy is illustrated by considering Lagrangian tori in C . Towards the end, we present a recent construction by Cornea and the author [8], of monotone cobordisms that are not trivial in a suitable sense. 1. Introduction In this note we explain the notion of Lagrangian cobordism that goes back to Arnol’d [1,2]. We pay special attention to a so–called flexibil- ity/rigidity dichotomy which is illustrated when studying Lagrangian tori in C2. The relevant definitions will be introduced in §2. Lagrangian submanifolds play an essential role in the understanding of symplectic manifolds and it is therefore natural to try to classify them, up to Hamiltonian isotopy for example. Even for Lagrangians in Cn the complete classification is unknown. A more attainable goal is to study them up to Lagrangian cobordism.
    [Show full text]
  • Geometric Equivariant Bordism and K-Theory
    CO40-9383.86 S3.00 - 00 C 1986 Pergamoo Press Ltd. GEOMETRIC EQUIVARIANT BORDISM AND K-THEORY IS MADSEN (Received I June 1985) THIS paper studies the relationship between equivariant bordism and equivariant K-theory, generalizing a celebrated result of Conner and Floyd to the equivariant setting. Let G be a finite group. Write Q:(X) for the bordism group of pairs (ML, f) of a closed k-dimensional oriented G-manifold and a G-map f: M’ 4 X, and let Q?(X) be the corresponding Z/Z-graded functor where we only distinguish the dimensions modulo 2. If G has odd order equivariant vector bundles are Kc( -; Z[i])-oriented, and there is a transformation 6: P(X) + KF(X; Z[=j]) which reduces to the G-signature when X is a single point. We show that 6 @ Z [j] induces an isomorphism between Q!(X)@ncRG@ Z[i] and K.G(X; Z[i]), where RG denotes the complex representation ring. This is combined with the universal coefficient theorem for Kc-theory to give: THEOREM. For G of odd order there is a functorial exact sequence 0 + Ext,,(KF(X), RG) @ Z[i] -+ Kb(X; Z[t]) -+ Hom,c(RI;(X), RG) @ .?I[&] -+ 0. This theorem has a number of applications to equivariant surgery theory and equivariant transversality, see e.g. [ 131. The Conner-Floyd isomorphism between equivariant cobordism and equivariant K- theory has previously been considered in [lo] and [17] for the stable unitary cobordism groups U F (X), where the Thorn isomorphism is built into the definition.
    [Show full text]