Free circle actions and positive Ricci curvature on manifolds with the cohomology ring of $S^2\times S^5$

TL;DR

This paper classifies 7-manifolds with the cohomology of S^2×S^5 that admit free circle actions, demonstrating their abundance and constructing positive Ricci curvature metrics in most cases.

Contribution

It provides a complete classification of free circle actions on these manifolds and constructs invariant metrics of positive Ricci curvature where possible.

Findings

Out of 672 diffeomorphism types, many admit free circle actions.

Existence of infinitely many non-equivalent free circle actions.

Positive Ricci curvature metrics are constructed in most cases.

Abstract

We classify which of the 672 oriented diffeomorphism types of closed, simply-connected spin 7-manifolds with the cohomology ring of $S^2\times S^5$ admit a free circle action. In addition, we show that whenever such an action exists, there exist infinitely many pairwise non-equivalent free circle actions. Finally, in almost all cases where such an action exists, we construct invariant Riemannian metrics of positive Ricci curvature.