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

Published online by Cambridge University Press:  24 February 2014

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Abstract

Let M be a closed manifold and α: π1 (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] ∈ K1 (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 M with fibers B, such that Eα is canonically isomorphic with ℂn, 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.

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Research Article
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Copyright © ISOPP 2014 

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