Geometry & Topology

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

Diarmuid Crowley and Thomas Schick

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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=8k1 or 8k.

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

A 8 j + 1 m 0 a n d π 8 j + 1 m ( + ( X ) ) 0

whenever m7 and 8jm0. 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 j1. This information about elements existing deep in the Gromoll filtration is the second main new result of this note.

Article information

Geom. Topol., Volume 17, Number 3 (2013), 1773-1789.

Received: 18 September 2012
Revised: 16 January 2013
Accepted: 8 April 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R60: Homotopy spheres, Poincaré conjecture
Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] 53C27: Spin and Spin$^c$ geometry 58B20: Riemannian, Finsler and other geometric structures [See also 53C20, 53C60]

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


Crowley, Diarmuid; Schick, Thomas. The Gromoll filtration, $\mathit{KO}$–characteristic classes and metrics of positive scalar curvature. Geom. Topol. 17 (2013), no. 3, 1773--1789. doi:10.2140/gt.2013.17.1773.

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