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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
Abstract

Let X be a closed m–dimensional spin manifold which admits a metric of positive scalar curvature and let +(X) be the space of all such metrics. For any g +(X), Hitchin used the KO–valued α–invariant to define a homomorphism An1: πn1(+(X),g) KOm+n. He then showed that A00 if m = 8k or 8k + 1 and that A10 if m = 8k 1 or 8k.

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

A8j+1m0andπ8j+1m(+(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(Γ8j58j+2){0} for j 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