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

Crowley, Diarmuid; Schick, Thomas

doi:10.2140/gt.2013.17.1773

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The Gromoll filtration, $\mathit{KO}$–characteristic classes and metrics of positive scalar curvature

Diarmuid Crowley and Thomas Schick

Geometry & Topology 17 (2013) 1773–1789

DOI: 10.2140/gt.2013.17.1773

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 $g \in ℛ^{+} (X)$ , Hitchin used the $KO$ –valued $α$ –invariant to define a homomorphism $A_{n - 1} : π_{n - 1} (ℛ^{+} (X), g) \to {KO}_{m + n}$ . He then showed that $A_{0} \neq 0$ if $m = 8 k$ or $8 k + 1$ and that $A_{1} \neq 0$ if $m = 8 k - 1$ or $8 k$ .

In this paper we use Hitchin’s methods and extend these results by proving that

A_{8 j + 1 - m} \neq 0 a n d π_{8 j + 1 - m} (ℛ^{+} (X)) \neq 0

whenever $m \geq 7$ and $8 j - m \geq 0$ . The new input are elements with nontrivial $α$ –invariant deep down in the Gromoll filtration of the group $Γ^{n + 1} = π_{0} (Diff (D^{n}, \partial))$ . We show that $α (Γ_{8 j - 5}^{8 j + 2}) \neq {0}$ for $j \geq 1$ . This information about elements existing deep in the Gromoll filtration is the second main new result of this note.

Keywords

positive scalar curvature, $\alpha$–invariant, Gromoll filtration, exotic sphere

Mathematical Subject Classification 2010

Primary: 57R60

Secondary: 53C21, 53C27, 58B20

References

Publication

Received: 18 September 2012

Revised: 16 January 2013

Accepted: 8 April 2013

Published: 3 July 2013

Proposed: Bill Dwyer

Seconded: Steve Ferry, Haynes Miller

Authors

Diarmuid Crowley
Max Planck Institute for Mathematics Vivatsgasse 7 D-53111 Bonn Germany
http://www.dcrowley.net/
Thomas Schick
Mathematisches Institut Georg-August-Universität Göttingen Bunsenstrasse 3 D-37073 Göttingen Germany
http://www.uni-math.gwdg.de/schick

Volume 17, issue 3 (2013)