H–spaces, loop spaces and the space of positive scalar curvature metrics on the sphere

Walsh, Mark

doi:10.2140/gt.2014.18.2189

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$H\!$–spaces, loop spaces and the space of positive scalar curvature metrics on the sphere

Mark Walsh

Geometry & Topology 18 (2014) 2189–2243

DOI: 10.2140/gt.2014.18.2189

Abstract

For dimensions $n \geq 3$ , we show that the space ${Riem}^{+} (S^{n})$ of metrics of positive scalar curvature on the sphere $S^{n}$ is homotopy equivalent to a subspace of itself which takes the form of an $H$ –space with a homotopy commutative, homotopy associative product operation. This product operation is based on the connected sum construction. We then exhibit an action on this subspace of the operad obtained by applying the bar construction to the little $n$ –disks operad. Using results of Boardman, Vogt and May we show that this implies, when $n \geq 3$ , that the path component of ${Riem}^{+} (S^{n})$ containing the round metric is weakly homotopy equivalent to an $n$ –fold loop space. Furthermore, we show that when $n = 3$ or $n \geq 5$ , the space ${Riem}^{+} (S^{n})$ is weakly homotopy equivalent to an $n$ –fold loop space provided a conjecture of Botvinnik concerning positive scalar curvature concordance is resolved in the affirmative.

Keywords

positive scalar curvature, iterated loop space, $H$–space, connected sum, operad

Mathematical Subject Classification 2010

Primary: 53C99

Secondary: 55S99

References

Publication

Received: 6 February 2013

Revised: 3 January 2014

Accepted: 15 February 2014

Published: 2 October 2014

Proposed: Bill Dwyer

Seconded: John Lott, Peter Teichner

Authors

Mark Walsh
Department of Mathematics, Statistics and Physics Wichita State University 1845 Fairmount Wichita, KS 67260 USA

Volume 18, issue 4 (2014)