Groups with torsion, bordism and rho invariants

Paolo Piazza and Thomas Schick

Vol. 232 (2007), No. 2, 355–378

DOI: 10.2140/pjm.2007.232.355

Abstract

Let Γ be a discrete group, and let M be a closed spin manifold of dimension m > 3 with π₁(M) = Γ. We assume that M admits a Riemannian metric of positive scalar curvature. We discuss how to use the L²-rho invariant ρ₍₂₎ and the delocalized eta invariant η_<g> associated to the Dirac operator on M to get information about the space of metrics with positive scalar curvature.

In particular, we prove that if Γ contains torsion and m ≡ 3 (mod 4) then M admits infinitely many different bordism classes of metrics with positive scalar curvature. This implies that there exist infinitely many concordance classes; we show that this is true even up to diffeomorphism.

If Γ has certain special properties, for example, if it contains polynomially growing conjugacy classes of finite order elements, then we obtain more refined information about the “size” of the space of metrics of positive scalar curvature, and these results also apply if the dimension is congruent to 1 mod 4. For example, if dimM ≡ 1 (mod 4) and Γ contains a central element of odd order, then the moduli space of metrics of positive scalar curvature (modulo the action of the diffeomorphism group) 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 that the scalar curvature is positive).

Keywords

spin manifolds, positive scalar curvature, Dirac operators, rho invariants, bordism

Mathematical Subject Classification 2000

Primary: 58J28

Milestones

Received: 14 June 2006

Revised: 21 September 2006

Accepted: 25 September 2006

Published: 1 October 2007