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

Philipp Reiser

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.

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

Abstract

We classify which of the 672 oriented diffeomorphism types of closed, simply-connected spin 7-manifolds with the cohomology ring of 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.

Paper Structure

This paper contains 14 sections, 31 theorems, 60 equations, 3 tables.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Free circle actions
  4. Simply-connected 6-manifolds
  5. Classification of spin cohomology $S^2\times S^5$s
  6. The diffeomorphism and homeomorphism classifications
  7. The homotopy classification
  8. Principal circle bundles
  9. Topology of the quotient space
  10. $s$-invariants of the total space
  11. Proof of the main results
  12. Proof of Proposition \ref{['P:pi4']}
  13. Further results
  14. Tables

Key Result

Theorem 1.2

All $28$ oriented diffeomorphism types of closed $7$-manifolds that are homeomorphic to $S^2\times S^5$ admit a free circle action.

Theorems & Definitions (61)

  • Theorem 1.2: Xu25
  • Definition 1.3
  • Theorem 1
  • Remark 1.4
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Definition 2.1
  • Proposition 2.2
  • proof
  • ...and 51 more