Homotopical and operator algebraic twisted K -theory
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Mathematische Annalen
Homotopical and operator algebraic twisted K -theory Fabian Hebestreit1 · Steffen Sagave2 Received: 15 April 2020 / Revised: 5 August 2020 © The Author(s) 2020
Abstract Using the framework for multiplicative parametrized homotopy theory introduced in joint work with C. Schlichtkrull, we produce a multiplicative comparison between the homotopical and operator algebraic constructions of twisted K -theory, both in the real and complex case. We also improve several comparison results about twisted K -theory of C ∗ -algebras to include multiplicative structures. Our results can also be interpreted in the ∞-categorical setup for parametrized spectra. Mathematics Subject Classification 55P43 · 55P42 · 19L50
1 Introduction Twisted homology and cohomology, or (co)homology with local coefficients, was originally invented by Steenrod [27]. It was designed to capture cohomological invariants that can only be made sense of in ordinary (co)homology under orientation assumptions. Examples include obstruction classes against sections in fiber bundles, fundamental classes of manifolds and Thom classes of vector bundles. Donovan and Karoubi [11] generalized this to obtain a twisted version of topological K -theory. In integral (co)homology, a twisting datum is given by a line bundle and thus determined by its first Stiefel–Whitney class or, equivalently, by a map to K (Z/2, 1) τ≤1 BO B(O/SO). Donovan and Karoubi described twists in K -theory using principal PU-bundles, where PU denotes the projective unitary group of some (infinite dimensional) Hilbert space. By Kuiper’s theorem U is contractible, and hence prin-
Communicated by Thomas Schick.
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Steffen Sagave [email protected] Fabian Hebestreit [email protected]
1
Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
2
IMAPP, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, The Netherlands
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F. Hebestreit, S. Sagave
cipal PU-bundles are classified by their Dixmier–Douady classes in the third integral cohomology. Equivalently, they are encoded by maps to K (Z, 3) B(SO Spinc ), where denotes homotopy quotients. The theory can easily be extended to graded Hilbert spaces and to maps into B(O Spinc ). Moreover, there is an analogous theory in the case of real K -theory for principal PO-bundles and maps into K (Z/2, 2) τ≤2 BSO B(SO Spin) and more generally τ≤2 BO B(O Spin) for graded bundles. While the construction of Donovan and Karoubi relies on the operator algebraic framework for K -theory, there are also purely homotopical constructions of twisted K -theory. As a motivating example for the relevance of the interplay between these approaches we take the following conjecture of Stolz: By design, any closed dmanifold M determines a fundamental class [M] ∈ kod (M, or(M)) in its twisted, connective KO-homology, where or(M) : M → τ≤2 BO records the first two (normal) Stiefel–Whitney classes of M. If the universal cover of M admits a spin structure, then the twist or(M) factor
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