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

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).

arXiv comment19 pages. Version 3: clarifications and corrections. Accepted to: Indiana University Mathematics Journal