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Homotopy groups of the moduli space of metrics of positive scalar curvature

Boris Botvinnik, Bernhard Hanke, Thomas Schick and Mark Walsh

Geometry & Topology 14 (2010) 2047–2076

DOI: 10.2140/gt.2010.14.2047

Abstract

We show by explicit examples that in many degrees in a stable range the homotopy groups of the moduli spaces of Riemannian metrics of positive scalar curvature on closed smooth manifolds can be non-trivial. This is achieved by further developing and then applying a family version of the surgery construction of Gromov–Lawson to certain nonlinear smooth sphere bundles constructed by Hatcher.

Keywords

metrics of positive scalar curvature, moduli space of positive scalar curvature metrics, classifying space of a diffeomorphism group, Gromov–Lawson surgery parametrized by a Morse function, rational homotopy type, Hatcher map

Mathematical Subject Classification

Primary: 53-02

Secondary: 55-02

References
Publication

Received: 30 July 2009
Revised: 19 October 2009
Accepted: 7 July 2010
Published: 29 August 2010
Proposed: Steve Ferry
Seconded: Ralph Cohen, Benson Farb

Authors
Boris Botvinnik
Department of Mathematics
University of Oregon
Eugene OR 97403
USA
Bernhard Hanke
Institut für Mathematik
Universität Augsburg
86135 Augsburg
Germany
Thomas Schick
Mathematisches Institut
Georg-August-Universität Göttingen
Bunsenstr. 3
37073 Göttingen
Germany
Mark Walsh
Mathematisches Institut
WWU Münster
Einsteinstr. 62
48149 Münster
Germany