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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

For dimensions n 3, we show that the space Riem+(Sn) of metrics of positive scalar curvature on the sphere Sn 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 3, that the path component of Riem+(Sn) containing the round metric is weakly homotopy equivalent to an n–fold loop space. Furthermore, we show that when n = 3 or n 5, the space Riem+(Sn) 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.

positive scalar curvature, iterated loop space, $H$–space, connected sum, operad
Mathematical Subject Classification 2010
Primary: 53C99
Secondary: 55S99
Received: 6 February 2013
Revised: 3 January 2014
Accepted: 15 February 2014
Published: 2 October 2014
Proposed: Bill Dwyer
Seconded: John Lott, Peter Teichner
Mark Walsh
Department of Mathematics, Statistics and Physics
Wichita State University
1845 Fairmount
Wichita, KS 67260