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Bordism, rho-invariants and the Baum-Connes conjecture. (English) Zbl 1158.58012

The rho-invariant is the eta invariant of a Dirac-type operator twisted by a virtual representation of dimension \(0\). This paper provides sufficient conditions for this invariant to vanish (for the spin Dirac operator) or to be a homotopy invariant (for the signature operator). The results are then shown to hold also for two other types of rho-invariants: the \(L^2\) rho-invariant and the delocalized rho-invariant.
Let \(\Gamma\) be a torsion-free discrete group, \(u:M'\to M\) a normal \(\Gamma\)-cover and \(\lambda_1,\lambda_2\) two finite-dimensional unitary representations of equal dimension, inducing flat bundles on \(M\). Assume that the Baum-Connes assembly map is an isomorphism from \(K_*(B\Gamma)\) to \(K^*(C^*\Gamma)\) where \(C^*\Gamma\) is the maximal \(C^*\)-algebra of \(\Gamma\). Then the \(\rho\) invariant of the signature operator depends on the covering \(u\) only up to \(\Gamma\)-homotopy equivalence. If \(M\) is spin with positive scalar curvature, then the rho invariant of the Dirac operator with respect to \(\lambda_1,\lambda_2\) vanishes.
The same results hold for the \(L^2\) rho-invariant. This is defined as the difference between the eta invariant of the operator \(D\) on \(M\), and the integral on a fundamental domain \(\mathcal F\) for \(\Gamma\) in \(M'\) of the pointwise trace which usually defines the eta invariant, corresponding to the lift \(D'\) of the operator \(D\) to \(M'\): \(\eta_{(2)}(D)= \int_0^\infty (\pi s)^{-1/2} \int_{\mathcal F} \mathrm{tr}(\kappa_{D'\exp(-s{D'}^2)}(m,m))\,dm\,ds.\)
Analogous results are obtained for the so-called delocalized eta invariant corresponding to finite conjugacy classes in \(\Gamma\). Here, the hypothesis is that the assembly map with values in the \(K\) theory of the reduced \(C^*\) algebra of \(\Gamma\) is an isomorphism.

MSC:

58J28 Eta-invariants, Chern-Simons invariants
19K56 Index theory
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