Positive Scalar Curvature and Callias-Type Index Theorems for Proper Actions Hao Guo August 8, 2018 Thesis submitted for the degree of Doctor of Philosophy in Pure Mathematics at the University of Adelaide Faculty of Engineering, Computer and Mathematical Sciences School of Mathematical Sciences Contents Signed Statement vi Acknowledgements viii Dedication x Abstract xii 1 Dirac Operators 1 1.1 Clifford Algebras . .1 1.2 The Spin and Spinc Groups . .3 1.3 Clifford Module Bundles . .4 1.4 Dirac Operators . .5 1.5 Proper Actions . .9 1.6 Almost-Connected Groups . 10 1.7 Equivariant Dirac Operators . 11 1.8 Index and Positive Scalar Curvature . 14 2 Fredholmness and G-Index 19 2.1 Group C∗-Algebras . 20 2.2 K-Theory of C∗-Algebras . 22 2.3 Hilbert Modules . 26 2.4 A G-Sobolev Module . 28 2.5 Boundedness and Fredholmness . 30 2.6 Preview of Results . 32 Bibliography 37 ii A The Cocompact Setting 39 A.1 Introduction . 42 A.2 Equivalence of Analytic and Geometric K-homologies . 46 A.2.1 Overview of the Proof . 50 A.2.2 Induction on Geometric K-homology . 52 A.2.3 Induction on Analytic K-homology . 57 A.2.4 Commutativity of the Diagram . 61 A.3 Poincar´eDuality . 64 A.3.1 Almost-connected Groups . 64 A.3.2 Discrete Groups . 72 A.4 Index Theory and K-homology Classes . 75 A.4.1 Higher Index . 75 A.4.2 Dirac Induction . 77 A.4.3 Higher Index Commutes with Induction . 80 A.5 A Trace Theorem for Equivariant Index . 83 A.6 Positive Scalar Curvature for Proper Cocompact Actions . 89 A.6.1 Obstructions . 89 A.6.2 Existence . 92 A.7 Two Applications of the Induction Principle . 96 A.7.1 Hattori's Vanishing Theorem . 96 A.7.2 An Analogue of Petrie's Conjecture . 101 Appendix: Equality of MZ Index and Integration Trace . 103 References . 105 B The Non-Cocompact Setting 113 B.1 Introduction . 116 B.2 Notation . 120 B.3 Sobolev Modules . 121 B.3.1 Boundedness and Adjointability . 123 B.3.2 Rellich's Lemma . 126 B.4 G-invertibility at Infinity . 127 B.4.1 A Simplified Definition of F .............. 135 B.4.2 G-invertible Operators as KK-elements . 137 B.4.3 A Simple Example . 140 B.5 G-Callias-type Operators . 142 iii B.5.1 Positivity of B2 + f ................... 142 B.5.2 Essential Self-adjointness and Regularity of B .... 145 B.5.3 Non-complete Cocompact Manifolds . 146 B.5.4 G-invertibility at Infinity of B ............. 150 B.6 The Endomorphism Φ . 152 B.6.1 The Construction . 152 B.6.2 Interpretation as a KK-product . 154 B.6.3 The G-corona . 156 B.7 Invariant Metrics of Positive Scalar Curvature . 162 B.7.1 An Existence Result . 162 B.7.2 A Non-cocompact Lichnerowicz-type Obstruction . 163 References . 167 iv Signed Statement I certify that this work contains no material which has been accepted for the award of any other degree or diploma in my name, in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. In addition, I certify that no part of this work will, in the future, be used in a submission in my name, for any other degree or diploma in any university or other tertiary institution without the prior approval of the University of Adelaide and where applicable, any partner institution responsible for the joint-award of this degree. I acknowledge that copyright of published works contained within this thesis resides with the copyright holder(s) of those works. I also give permission for the digital version of my thesis to be made available on the web, via the University's digital research repository, the Library Search and also through web search engines, unless permission has been granted by the University to restrict access for a period of time. I acknowledge the support I have received for my research through the provi- sion of an Australian Government Research Training Program Scholarship. Signed: June 30, 2018 vi Acknowledgements Firstly, I would like to acknowledge my principal supervisor, Elder Professor Mathai Varghese, who has been a continual source of inspiration and invalu- able insight throughout my candidature and my first encounter with index theory. It has been a privilege and an education to work with Mathai, and I am grateful in equal measure for his encouragement, advice and criticisms. Secondly, I would like to acknowledge Dr. Hang Wang, my associate supervisor. This journey has been so much more pleasant because of her patience, generosity and ability to tolerate my endless silly questions. I hope to be able to pursue mathematics with as much fervor and passion as she does. I would also like to gratefully acknowledge my fellow graduate students at the University of Adelaide, with whom I have had many fruitful discus- sions and attended the Basic Notions Seminar. Finally, I would like to thank my partner-in-crime Christine for always being there for me, and my family for their support through all these years. viii Dedication To all my family. x Abstract This thesis by publication is a study of the equivariant index theory of Dirac operators and Callias-type operators in two distinct settings, namely on cocompact and non-cocompact manifolds with a Lie group action. The first two chapters are a short resum´eof Dirac operators and index theory and form a common introduction to the papers in the appendices. Appendix A is joint work with my supervisors, Elder Professor Mathai Varghese and Dr. Hang Wang. For G an almost-connected Lie group acting properly and cocompactly on a manifold M, we study G-index theory of G- invariant Dirac operators. By establishing Poincar´eduality for equivariant K-theory and K-homology, we are able to extend the scope of our results to include all elements of equivariant analytic K-homology, which we also show is isomorphic to equivariant geometric K-homology. Our results are applied to prove: a rigidity result for almost-complex manifolds, generalising a vanishing theorem of Hattori; an analogue of Petrie's conjecture; and Lichnerowicz-type obstructions to G-invariant Riemannian metrics on M. Appendix B studies the much more general situation when the quotient M=G is non-compact and G is an arbitrary Lie group. I define G-Callias- type operators and show that they are C∗(G)-Fredholm by adapting analy- sis of Kasparov to new Hilbert C∗(G)-module analogues of Sobolev spaces. Questions of adjointability, regularity and essential self-adjointness are ad- dressed in detail. The estimates on G-Callias-type operators are based on the work of Bunke [8] in the non-equivariant context. We construct explicit admissible endomorphisms for G-Callias-type operators from the K-theory of the Higson G-corona of M, a highly non-trivial group. The index theory developed here is applied to prove a general obstruction theorem for G- invariant metrics of positive scalar curvature in the non-cocompact setting. xii Chapter 1 Dirac Operators This chapter is an exposition of some well-known material on Dirac opera- tors in the context of group actions on manifolds. The interested reader can pursue the topics further in books such as [14], [4] and [11]. The material on almost-connected groups can be found in the paper [1]. We begin by defining the Clifford algebra and the Spin and Spinc groups, followed by the notions of Dirac and Dirac-type operators acting on sections of a Clifford module bundle. We will recall what it means for a Lie group G to act properly on a man- ifold, and point out the simplifications that arise in the special case that G has only finitely many connected components. We recall the definitions of G-invariant Dirac and Dirac-type operators. We will also discuss Lich- nerowicz's vanishing theorem, a prototype of index-theoretic obstructions to the existence of metrics of positive scalar curvature. 1.1 Clifford Algebras Let K be a field with characteristic = 2. Let V be a vector space over 6 K equipped with a non-degenerate bilinear form g. The Clifford algebra Cl(V; g) associated to V and g is a unital associative K-algebra defined as the quotient Cl(V; g) := T (V )=I(V; g); 1 where T (V ) is the tensor algebra of V and I(V; g) is the ideal of T (V ) generated by the set of elements v v + g(v; v): v V : f ⊗ 2 g Let Id : V V be the identity map. Then Id is an involutive isometry ! − on (V; g) that induces a decomposition Cl(V; g) = Cl0(V; g) Cl1(V; g); ⊕ where Cl0(V; g) and Cl1(V; g) are the +1 and 1-eigenspaces of Id. This − 0 −1 gives Cl(V; g) a Z2-grading, and we refer to Cl(V; g) and Cl(V; g) respec- tively as the even and odd parts of Cl(V; g). As a vector space, Cl(V; g) is isomorphic to Λ∗V , the real exterior algebra of V . Define ; be the usual inner product on Rn. We shall often use the h i shorthand notation n Cln := Cl(R ; ; ): h i 0 It can be shown that Cln is isomorphic to Cln+1. Define Cln := Cln C: ⊗R n One sees that, as complex vector spaces, Cln ∼= Λ∗(C ).