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

TL;DR

This paper extends the understanding of positive scalar curvature metrics on spin manifolds by linking KO-characteristic classes, Gromoll filtration elements, and alpha-invariants, revealing non-trivial elements deep in the filtration.

Contribution

It proves new non-vanishing results for alpha-invariants associated with elements deep in the Gromoll filtration, expanding previous work on scalar curvature and KO-theory.

Findings

A_{8j+1-m} is non-zero for m>6 and 8j - m >= 0.

Alpha-invariant is non-zero for elements deep in the Gromoll filtration.

Extends Hitchin's results to a broader class of manifolds and filtration levels.

Abstract

Let X be a closed m-dimensional spin manifold which admits a metric of positive scalar curvature and let Pos(X) be the space of all such metrics. For any g in Pos(X), Hitchin used the KO-valued alpha-invariant to define a homomorphism A_{n-1} from \pi_{n-1}(Pos(X) to KO_{m+n}. He then showed that A_0 is not 0 if m = 8k or 8k+1 and that A_1 is not 0 if m = 8k-1 or 8$. In this paper we use Hitchin's methods and extend these results by proving that A_{8j+1-m} is not 0 whenever m>6 and 8j - m >= 0. The new input are elements with non-trivial alpha-invariant deep down in the Gromoll filtration of the group \Gamma^{n+1} = \pi_0(\Diff(D^n, \del)). We show that \alpha(\Gamma^{8j+2}_{8j-5}) is not 0 for j>0. This information about elements existing deep in the Gromoll filtration is the second main new result of this note.