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

Abstract

For dimensions $n\ge 3$, we show that the space ${Riem}^{+}\left(S^n\right)$ 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 $n\ge 3$, that the path component of ${Riem}^{+}\left(S^n\right)$ containing the round metric is weakly homotopy equivalent to an $n$–fold loop space. Furthermore, we show that when $n=3$ or $n\ge 5$, the space ${Riem}^{+}\left(S^n\right)$ 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.