Groups with torsion, bordism and rho-invariants

Abstract

Let G be a discrete group, and let M be a closed spin manifold of dimension m>3 with pi_1(M)=G. We assume that M admits a Riemannian metric of positive scalar curvature. We discuss how to use the L2-rho invariant and the delocalized eta invariant associated to the Dirac operator on M in order to get information about the space of metrics with positive scalar curvatur1e. In particular we prove that, if G contains torsion and M is congruent 3 mod 4 then M admits infinitely many different bordism classes of metrics with positive scalar curvature. We show that this is true even up to diffeomorphism. If G has certain special properties then we obtain more refined information about the ``size'' of the space of metric of positive scalar curvature, and these results also apply if the dimension is congruent to 1 mod 4. For example, if G contains a central element of odd order, then the moduli space of metrics of positive scalar curvature has infinitely many components, if it is not empty. Some of our invariants are the delocalized eta-invariants introduced by John Lott. These invariants are defined by certain integrals whose convergence is not clear in general, and we show, in effect, that examples exist where this integral definitely does not converge, thus answering a question of Lott. We also discuss the possible values of the rho invariants of the Dirac operator and show that there are certain global restrictions (provided the scalar curvature is positive).

21 pages; comma in metadata (author field) added. final version to appear in Pacific Journal of Mathematics

References in corpus (3)

• Bordism, rho-invariants and the Baum-Connes conjecture
• On Invariants of Hirzebruch and Cheeger-Gromov
• Real versus complex K-theory using Kasparov's bivariant KK-theory

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• Infinite loop spaces and positive scalar curvature in the presence of a fundamental group
• Positive scalar curvature and low-degree group homology
• Positive Scalar Curvature due to the Cokernel of the Classifying Map
• Positive scalar curvature via end-periodic manifolds
• Positive Scalar Curvature on Spin Pseudomanifolds: the Fundamental Group and Secondary Invariants
• L^2 rho form for normal coverings of fibre bundles
• Relative eta invariant and uniformly positive scalar curvature on non-compact manifolds