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

E-ISSN： 1865-5394|13|2|275-303

ISSN： 1865-2433

Source： Journal of K-Theory, Vol.13, Iss.2, 2014-04, pp. : 275-303

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Abstract

Let M be a closed manifold and α: π ₁ (M) → U_n 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 ℰ → M with fibers B, such that E_α ⊗ ℰ is canonically isomorphic with ℂⁿ ⊗ ℰ, where E_α 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.