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Twisted K-Theory with Coefficients in a C -Algebra and Obstructions Against Twisted K-theory with coefficients in a C∗-algebra and obstructions against positive scalar curvature metrics Dissertation zur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades \Doctor rerum naturalium" der Georg-August-Universit¨atG¨ottingen vorgelegt von Ulrich Pennig aus Cloppenburg G¨ottingen2009 Referenten der Dissertation: Prof. Dr. Thomas Schick Prof. Dr. Andreas Thom Mitglieder der Pr¨ufungskommission: Prof. Dr. Thomas Schick Prof. Dr. Andreas Thom Prof. Dr. Victor Pidstrygach Prof. Dr. Ulrich Stuhler Prof. Dr. Thorsten Hohage Prof. Chenchang Zhu, Ph.D. Tag der m¨undlichen Pr¨ufung: 31. August 2009 To my parents Contents 1 Introduction 7 2 Preliminaries 15 2.1 Bundle gerbes and their modules . 15 2.1.1 The Dixmier-Douady class and stable isomorphism . 18 2.1.2 Bundle gerbes in higher gauge theory . 20 2.1.3 Twisted K-theory and bundle gerbes . 20 3 Twisted Hilbert A-module bundles 23 3.1 The projective unitary group of a C∗-algebra . 25 3.2 Morphisms of twisted Hilbert A-module bundles . 26 0 3.3 Twisted K-theory of locally compact spaces and KA(X; Y ) . 32 3.4 Stable isomorphism and Morita equivalence . 37 3.4.1 The frame bundle gerbe . 41 3.4.2 Bundle gerbe modules and twisted Hilbert Mn(C)-bundles 43 3.4.3 Twisted product . 44 3.4.4 K¨unneththeorem . 44 4 Twisted K-homology in the torsion case 49 4.1 Connections on bundle gerbe modules . 49 4.1.1 Flat extensions of Lie groups . 49 4.1.2 Connections . 51 4.1.3 Flat connections on general S1-bundle gerbes . 55 4.1.4 Trivial bundle gerbes . 59 4.1.5 Connections on tensor products . 61 4.1.6 Parallel transport in twisted Hilbert A-module bundles . 61 4.1.7 Curvature and Chern character of twisted Hilbert A-module bundles . 63 4.2 Index theory on twisted Hilbert A-module bundles . 73 4.2.1 Sobolev spaces . 73 4.2.2 Pseudodifferential operators on twisted Hilbert A-module bundles . 77 4.2.3 Some remarks about the symbol class . 91 4.3 Generalized projective Dirac operators . 93 4.3.1 Trivial bundle gerbes and countertwisting . 95 4.3.2 Spinor countertwisting . 99 4.3.3 Flat countertwisting . 103 4.3.4 Covering bundle gerbes and holonomy . 110 6 CONTENTS 4.4 Obstructions to positive scalar curvature . 112 4.4.1 The twisted Mishchenko-Fomenko line bundle . 118 4.5 KK-theory and projective Dirac operators . 119 5 An application: enlargeability and positive scalar curvature 129 5.1 Assembling almost flat twisted bundles . 131 5.2 Index theory . 137 5.2.1 Projective holonomy representations . 137 6 Perspectives 141 6.1 Central extensions . 141 6.2 Enlargeability and infinite covers . 142 6.3 Non-torsion twists . 143 Chapter 1 Introduction Twisted K-theory first appeared in the early 1970s in a paper by Donovan and Karoubi [20] as \K-theory with local coefficients". As the name suggests, it is an elaboration of topological K-theory, invented by Atiyah and Hirze- bruch in [4], that supports a notion of Poinare´ duality even in the case of non K-orientable (i.e. non spin(c)-) manifolds [72]. Like the local coefficients for ordinary cohomology are described by a class in H1(M; Z=2Z) for a paracom- pact Hausdorff space M, the extra data needed in case of K-theory is a twist represented by a certain (torsion) element in H3(M; Z). The latter group can also be seen to classify isomorphism classes of bundles AK ! M with fibers the compact operators on a separable Hilbert space, which was the starting point of a paper by Rosenberg [57], who showed that twisted K-theory as defined ∗ in [20] is isomorphic to K∗(C0(M; AK)) { the K-theory of the C -algebra of continuous sections of AK tending to zero at infinity. This description needs no restriction to torsion elements in H3(M; Z), but in case the twist is of finite order Bouwknegt, Carey, Mathai, Murray and Stevenson developed a geometric description of K0(C0(M; AK)) in terms of modules over bundle gerbes [12, 19], which provide a replacement for vector bundles in the twisted case. Recently, the rise of string theory gave some fresh impetus to the devel- opment of twisted K-theory [64, 13, 41, 77]: In string theory, space-time is modelled in such a way that its classical limit, in which quantum effects are neglected, is not just a Riemannian manifold M, but also carries the extra data of a B-field β on M, which supports topologically non-trivial dynamical struc- tures of the stringy space-time, called D-branes. Now, β coincides with the data needed to define a twisted K-theory group on M and the elements of this group can be identified with D-brane charges. Furthermore, there also is a du- ality principle, called T -duality, relating two different types of string theories, which amounts on the mathematical side to an interesting involution on twisted K-theory [18, 15]. Another application of twisted K-theory emanates from the representation theory of loop groups, in particular from a beautiful theorem proven by Freed, Hopkins and Teleman [22]: Let G be a simple, simply connected, compact Lie group and k 2 Z ' H3(G; Z) be a level. For each such k there is a canonical k central extension ΩbkG of the loop group ΩG. Let R (ΩbG) be the free abelian group generated by isomorphism classes of positive energy representations of k ΩbkG. There is a level preserving product, known as fusion, on R (ΩbG) turning 8 1. Introduction it into the Verlinde ring. The theorem says that it is isomorphic to the twisted _ d;[k+h ] _ equivariant K-theory KG (G), where h is the dual Coxeter number, [k + h_] 2 H3(G; Z) represents the twist, d is the dimension of G and G acts on itself by conjugation. Fusion turns into the Pontrjagin product on the K- theoretic side. Another reference is [16], where the formulation of this theorem as a commutative diagram is considered. As will be seen in the next chapter, twisted K-theory is closely related to higher algebraic structures, like bundle gerbes [12]. Therefore the development of the theory for singular spaces described by stacks or orbifolds comes in very natural. Results in this direction can be found in [1, 73]. Let A be a unital C∗-algebra, M be a compact Hausdorff space, then the conti- nuous functions with values in A, denoted by C(M; A) ' C(M) ⊗ A, will again be a C∗-algebra when equipped with the obvious supremum norm. There is a geometric description of K0(C(M; A)) as the Grothendieck group of iso- morphism classes of bundles with fibers finitely generated, projective Hilbert A-modules [61]. The present work takes this geometric picture as a starting point and develops a twisted version of it: Let A be a locally trivial bundle of C∗-algebras with structure group PU(A) = U(A)=U(1) and denote by C(M; A) the C∗-algebra of continuous sections. Replacing the above Hilbert module bun- dles by modules over a suitable bundle gerbe and thereby merging the above picture with the one developed in [12, 19], we arrive at a geometric interpre- tation of K0(C(M; A)) in terms of twisted Hilbert A-module bundles (see defi- nition 3.0.13). In particular, we prove a twisted Serre-Swan-theorem (3.2.8) saying that the category of twisted Hilbert A-module bundles is naturally equiv- alent to the one of finitely generated, projective C(M; A)-modules. A similar proposition holds in the case of locally compact spaces M, if some care is taken about the extendability of A to compactifications. A twisted Hilbert A-module bundle E lives over an auxiliary principal bundle P with structure group PU(A). From a geometric point of view, P might seem to be a somewhat bulky, even though not unmanageable object. Luckily, in all geometric applications we have in mind, P reduces to a principal Γ- bundle P , such that Γ is either a compact Lie group or a discrete group. Even without a reduction, but with restriction to torsion twists, the algebra C(M; A) is Morita equivalent to C(M; K) ⊗ A for a bundle of matrix algebras K! M (see theorem 3.4.1). Using the isomorphism induced on K-theory, each twisted K-cycle may be represented by a twisted Hilbert A-module bundle over some principal PU(n)-bundle P . We exploit this fact by developing a theory of connections on modules over certain lifting bundle gerbes that arise from flat central S1-extensions of Lie groups: 1 −! S1 −! Γb −! Γ −! 1 : In this case there is a notion of Chern character in the sense of Chern-Weil- theory, which takes values in the ordinary (untwisted) cohomology. It extends like in the setup of Mishchenko-Fomenko-index theory [47] to a Chern char- even acter with values in H (M; K0(A) ⊗ R). The usual approach to index theory via pseudodifferential operators does not transfer directly to the twisted case due to a crucial analytic difficulty: 0 ∗ Symbols, representing K-cycles in the twisted K-group Kπ∗A(T M), just yield transversally elliptic pseudodifferential operators on the corresponding twisted 9 Hilbert A-module bundles (with respect to the covertical subbundle H∗ ⊂ T ∗P ). Nevertheless, the machinery developed by Atiyah in [7] is extendable to this case if the algebra in question is supposed to come equipped with a trace.
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