Free circle actions on certain simply connected $7-$manifolds

TL;DR

This work characterizes exactly when manifolds of the form $kS^{2}\times S^{5}\#lS^{3}\times S^{4}\#\Sigma$ (with $\Sigma$ a homotopy $7$-sphere) admit free circle actions. The authors develop a framework based on orbit space analysis, circle bundles, and Kreck–Stolz $s$-invariants, distinguishing spin and nonspin cases and employing suspension techniques to handle arbitrary $k$ and $l$. They show that for $k\ge2$ or $l$ even, free circle actions exist for all $\Sigma$, while for $l$ odd the action exists precisely when the ambient homotopy $7$-sphere $\Sigma$ admits one; these conclusions tie to the $\mu$-invariant and the diffeomorphism types via the orbit-space invariants. The results extend known cases and provide a comprehensive classification of free circle actions on these simply connected $7$-manifolds, with implications for the topology of circle actions and exotic spheres.

Abstract

In this paper, we determine for which nonnegative integers $k$, $l$ and for which homotopy $7-$sphere $Σ$ the manifold $kS^{2}\times S^{5}\#lS^{3}\times S^{4}\#Σ$ admits a free smooth circle action.

Theorems & Definitions (32)

• Theorem 1
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Lemma 1
• Proof 1
• Lemma 2
• Proof 2
• Example 1