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Free torus actions and twisted suspensions

Fernando Galaz-García, Philipp Reiser

TL;DR

This work develops a framework to realize free torus actions on closed simply-connected manifolds by representing total spaces of principal $S^1$-bundles over connected sums in terms of twisted suspensions. It introduces $\Sigma_e M$ and $\widetilde{\Sigma}_e M$ and proves diffeomorphism formulas expressing $P$ as a connected sum with these twisted suspensions under precise hypotheses, generalizing Duan's suspensions. The authors apply these tools to classify total spaces of free circle/torus actions, especially for connected sums of products of spheres, and obtain a topological classification for manifolds with free cohomogeneity-four torus actions, along with corollaries asserting the existence of invariant metrics of positive Ricci curvature. By linking principal-bundle topology, twisted suspensions, and high-symmetry actions, the paper advances understanding of how symmetry constraints shape manifold topology and geometry.

Abstract

We express the total space of a principal circle bundle over a connected sum of two manifolds in terms of the total spaces of circle bundles over each summand, provided certain conditions hold. We then apply this result to provide sufficient conditions for the existence of free circle and torus actions on connected sums of products of spheres and obtain a topological classification of closed, simply-connected manifolds with a free cohomogeneity-four torus action. As a corollary, we obtain infinitely-many manifolds with Riemannian metrics of positive Ricci curvature and isometric torus actions.

Free torus actions and twisted suspensions

TL;DR

This work develops a framework to realize free torus actions on closed simply-connected manifolds by representing total spaces of principal -bundles over connected sums in terms of twisted suspensions. It introduces and and proves diffeomorphism formulas expressing as a connected sum with these twisted suspensions under precise hypotheses, generalizing Duan's suspensions. The authors apply these tools to classify total spaces of free circle/torus actions, especially for connected sums of products of spheres, and obtain a topological classification for manifolds with free cohomogeneity-four torus actions, along with corollaries asserting the existence of invariant metrics of positive Ricci curvature. By linking principal-bundle topology, twisted suspensions, and high-symmetry actions, the paper advances understanding of how symmetry constraints shape manifold topology and geometry.

Abstract

We express the total space of a principal circle bundle over a connected sum of two manifolds in terms of the total spaces of circle bundles over each summand, provided certain conditions hold. We then apply this result to provide sufficient conditions for the existence of free circle and torus actions on connected sums of products of spheres and obtain a topological classification of closed, simply-connected manifolds with a free cohomogeneity-four torus action. As a corollary, we obtain infinitely-many manifolds with Riemannian metrics of positive Ricci curvature and isometric torus actions.
Paper Structure (8 sections, 43 theorems, 98 equations, 1 table)

This paper contains 8 sections, 43 theorems, 98 equations, 1 table.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Auxiliary facts on principal torus bundles
  4. Auxiliary results on smooth manifolds and vector bundles
  5. Normally framed circles
  6. Surgery along framed circles
  7. Twisted suspensions
  8. Proof of Theorems \ref{['T:S1_BUNDLES']}, \ref{['T:FREE_CIRCLE']} and \ref{['T:COHOM_4']}, and Corollaries \ref{['C:FREE_TORUS']}, \ref{['C:COHOM_2']} and \ref{['C:RIC>0']}

Key Result

Theorem 1

Let $B_1$, $B_2$ be closed, oriented $n$-manifolds with $n\geq 5$ and let $P\xrightarrow{\pi}B_1\# B_2$ be a principal $S^1$-bundle. For $i=1,2$, denote by $e_i\in H^2(B_i)$ the restriction of the Euler class of $P$ to $B_i$ and by $P_i\xrightarrow{\pi_i}B_i$ the principal $S^1$-bundle with Euler cl

Theorems & Definitions (89)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Theorem 5
  • Theorem 6
  • Corollary 7
  • Corollary 8
  • Lemma 2.1
  • proof
  • ...and 79 more