H-Spaces, Loop Spaces and the Space of Positive Scalar Curvature Metrics on the Sphere

Abstract

For dimensions n greater than or equal to 3, we show that the space of metrics of positive scalar curvature on the n-sphere is homotopy equivalent to a subspace which takes the form of a 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 of the little n-disks operad on this subspace which, using results of Boardman, Vogt and May implies that when n=3 or n is at least 5, the space of metrics of positive scalar curvature on the n-sphere is weakly homotopy equivalent to an n-fold loop space.

arXiv comment43 pages, 32 figures. In version 2 we added a line to the introduction acknowledging a relevant new result in the field. In version 3, we correct an error in the proof of the second main result