Let M be a closed manifold and α: π 1 (M) → Un a representation. We give a purely K-theoretic description of the associated element in the K-theory group of M with ℝ/ℤ-coefficients ([α] ∈ K 1 (M;ℝ/ℤ)). To that end, it is convenient to describe the ℝ/ℤ-K-theory as a relative K-theory of the unital inclusion of ℂ into a finite von Neumann algebra B. We use the following fact: there is, associated with α, a finite von Neumann algebra B together with a flat bundle ε → M with fibers B, such that Eα ⊗ ε is canonically isomorphic with ℂn ⊗ ε, where Eα denotes the flat bundle with fiber ℂn associated with α. We also discuss the spectral flow and rho type description of the pairing of the class [α] with the K-homology class of an elliptic selfadjoint (pseudo)-differential operator D of order 1. Copyright © ISOPP 2014.

Flat bundles, von Neumann algebras and K-theory with R/Z-coefficients

ANTONINI, Paolo;
2014

Abstract

Let M be a closed manifold and α: π 1 (M) → Un a representation. We give a purely K-theoretic description of the associated element in the K-theory group of M with ℝ/ℤ-coefficients ([α] ∈ K 1 (M;ℝ/ℤ)). To that end, it is convenient to describe the ℝ/ℤ-K-theory as a relative K-theory of the unital inclusion of ℂ into a finite von Neumann algebra B. We use the following fact: there is, associated with α, a finite von Neumann algebra B together with a flat bundle ε → M with fibers B, such that Eα ⊗ ε is canonically isomorphic with ℂn ⊗ ε, where Eα denotes the flat bundle with fiber ℂn associated with α. We also discuss the spectral flow and rho type description of the pairing of the class [α] with the K-homology class of an elliptic selfadjoint (pseudo)-differential operator D of order 1. Copyright © ISOPP 2014.
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https://arxiv.org/abs/1308.0218
Antonini, Paolo; Azzali, S.; Skandalis, G.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/32467
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