Free transformations of $S^1 \times S^n$ of square-free odd period

TL;DR

This paper classifies free circle actions on products of spheres with odd square-free periods using surgery and homotopy theory, revealing new complexities due to number theory and group actions.

Contribution

It provides a classification of equivariant homeomorphism classes of free $C_ ext{ell}$-actions on $S^1 imes S^n$ for odd square-free $ ext{ell}$, extending previous work for $ ext{ell}=2$ with new number-theoretic insights.

Findings

Classification expressed via number theory.

Identification of new issues for odd $ ext{ell}$ involving ideal class groups.

Application of structure group composition formula to handle quadratic growth.

Abstract

Let $n$ be a positive integer, and let $\ell>1$ be square-free odd. We classify the set of equivariant homeomorphism classes of free $C_\ell$-actions on the product $S^1 \times S^n$ of spheres, up to indeterminacy bounded in $\ell$. The description is expressed in terms of number theory. The techniques are various applications of surgery theory and homotopy theory, and we perform a careful study of $h$-cobordisms. The $\ell=2$ case was completed by B Jahren and S Kwasik (2011). The new issues for the case of $\ell$ odd are the presence of nontrivial ideal class groups and a group of equivariant self-equivalences with quadratic growth in $\ell$. The latter is handled by the composition formula for structure groups of A Ranicki (2009).