The Gromoll filtration, KO-characteristic classes and metrics of positive scalar curvature

Abstract

Let X be a closed m-dimensional spin manifold which admits a metric of positive scalar curvature and let R+ (X) be the space of all such metrics. For any g ∈ R+ (X),Hitchin used the KO-valued α-invariant to define a homomorphism An-1: πn-1 (R+ (X), g) → KOm+1. He then showed that A0 ≠0 if m = 8k or 8k + 1 and that A1 ≠ 0 if m = 8k - 1 or 8k. In this paper we use Hitchin's methods and extend these results by proving that A8j+1-m ≠ 0 and π8j+1-m(R+(X)) ≠ 0 whenever m ≥ 7 and 8j - m ≥ 0. The new input are elements with nontrivial α-invariant deep down in the Gromoll filtration of the group Γn+1 = π0(Diff(Dn; ∂)). We show that α(Γ8j+28j-5) ≠ {0} for j ≥ 1. This information about elements existin..