Bordism, rho-invariants and the Baum-Connes conjecture

By P. Piazza, Thomas Schick. 2004

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Letbe a finitely generated discrete group. In this paper we establish vanishing results for rho-invariants associated to (i) the spin Dirac operator of a spin manifold with positive scalar curvature and fundamental group� ; (ii) the signature operator of the disjoint union of a pair of homotopy equivalent oriented manifolds with fundamental group� . The invariants we consider are more precisely � the Atiyah-Patodi-Singer (� APS) rho-invariant associated to a pair of finite dimensional unitary representations� 1;� 2W� ! U.d/, � theL 2 -rho-invariant of Cheeger-Gromov, � the delocalized eta-invariant of Lott for a non-trivial conjugacy class ofwhich is finite. We prove that all these rho-invariants vanish if the groupistorsion-free and the Baum-Connes map for the maximal group C*-algebra is bijective. This condition is satisfied, for example, by torsion-free amenable groups or by torsion-free discrete subgroups of SO.n;1/ and SU.n;1/. For the delocalized invariant we only assume the validity of the Baum-Connes conjecture for the reduced C*-algebra. In addition to the examples above, this condition is satisfied e.g. by Gromov hyperbolic groups or by cocompact discrete subgroups of SL.3; C/. In particular, the three rho-invariants associated to the signature operator are, for such groups, homotopy invariant. For the APS and the Cheeger-Gromov rho-invariants the latter result had been established by Navin Keswani. Our proof reestablishes this result and also extends it to the delocalized eta-invariant of Lott. The proof exploits in a fundamental way results from bordism theory as well as various generalizations of the APS-index theorem; it also embeds these results in general vanishing phenomena for degree zero higher rho-invariants (taking values in A=ŒA;Afor suitable C*-algebras A). We also obtain precise information about the eta-invariants in question themselves, which are usually much more subtle objects than the rho-invariants.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/81f46c09c6065d07c4ef10f1670333fc97f49022

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Citations in the corpus

K-Homology, Assembly and Rigidity Theorems for Relative Eta Invariants (2010, result): “Our interest is in the following results, originally due in various formulations to Mathai [29], Weinberger [37] and Keswani [27] (for an interesting recent treatment, see the article [32] of Piazza and Schick).”
Rho‐classes, index theory and Stolz’ positive scalar curvature sequence (2012, methodology): “One can use the homotopy equivalences built in the definition of the L-groups and the structure set in order to obtain a smoothing perturbation which makes the signature operator invertible, as in [28].”
Delocalized Eta Invariants, Algebraicity, and K-Theory of Group C*-Algebras (2018, result): “Some of the main results in this paper are inspired by previous work of Lott, Leichtnam, Piazza and Schick [22, 23][25][21].”