Flat bundles, von Neumann algebras and $K$-theory with $\mathbb{R}/\mathbb{Z}$-coefficients

Abstract

Let $M$ be a closed manifold and $\alpha :{\pi }_1(M)\to U_n$ a representation. We give a purely $K$-theoretic description of the associated element $[\alpha ]$ in the $K$-theory of $M$ with $\mathbb{R}/\mathbb{Z}$-coefficients. To that end, it is convenient to describe the $\mathbb{R}/\mathbb{Z}$-$K$-theory as a relative $K$-theory with respect to the inclusion of $\C$ in a finite von Neumann algebra $B$. We use the following fact: there is, associated with $\alpha$, a finite von Neumann algebra $B$ together with a flat bundle $\cE\to M$ with fibers $B$, such that $E_\a\otimes \cE$ is canonically isomorphic with $\C^n\otimes \cE$, where $E_{\alpha }$ denotes the flat bundle with fiber $\C^n$ associated with $\alpha$. We also discuss the spectral flow and rho type description of the pairing of the class $[\alpha ]$ with the $K$-homology class of an elliptic selfadjoint (pseudo)-differential operator $D$ of order 1.

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