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Group quasi-representations and index theory. (English) Zbl 1258.46029

From the introduction: “Let \(M\) be a closed connected Riemannian manifold with fundamental group \(G\) and let \(D\) be an elliptic operator on \(M\). Connes, Gromov and Moscovici introduced the notion of almost flat bundle and proved an index theorem showing that the pushforward of the equivariant index of \(D\) under a quasi-representation of \(G\) coming from parallel transport in an almost flat bundle \(E\) on \(M\) is equal to the index of \(D\) twisted by \(E\). Using this result, Connes, Gromov and Moscovici also showed that the signature with coefficients in an almost flat bundle is a homotopy invariant.
In this paper, we take a dual approach where the input data involves a group quasi-representation rather than an almost flat bundle. The first main result of the paper is a generalization of the index theorem of Connes, Gromov and Moscovici that allows for almost flat bundles \(E\) with fibers projective Hilbert modules over a tracial \(C^*\)-algebra (see Theorem 3.6). The second main result of the paper is a generalization of the Exel-Loring formula to surface groups, by reinterpreting it as an index theorem (see Theorem 4.2).”

MSC:

46L80 \(K\)-theory and operator algebras (including cyclic theory)
46L35 Classifications of \(C^*\)-algebras
19K35 Kasparov theory (\(KK\)-theory)
19K56 Index theory
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