Geometry & Topology

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

Diarmuid Crowley and Thomas Schick

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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∈ℛ+(X), Hitchin used the KO–valued α–invariant to define a homomorphism An−1:πn−1(ℛ+(X),g)→KOm+n. He then showed that A0≠0 if m=8k or 8k+1 and that A1≠0 if m=8k−1 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 ( ℛ + ( 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(Γ8j−58j+2)≠{0} for j≥1. This information about elements existing deep in the Gromoll filtration is the second main new result of this note.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732618

Digital Object Identifier
doi:10.2140/gt.2013.17.1773

Mathematical Reviews number (MathSciNet)
MR3073935

Zentralblatt MATH identifier
1285.57015

Subjects
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]

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

Citation

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. https://projecteuclid.org/euclid.gt/1513732618


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