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 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 π_{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 Γ^{n+1} = π_0(\Diff(D^n, \del)). We show that α(Γ^{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.

14 pages, ams-latex. v2: corrections. Based on a referee's report we added Lemma 2.5 v2 and we gave more details in the proof of Lemma 2.14 v2. We also removed Corollary 1.2 v1 since we found a gap in the proof. v3: typo (wrong index) in Lemma 2.14 corrected

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