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Let $n$ be a positive integer, and let $\ell>1$ be square-free odd. Weclassify the set of equivariant homeomorphism classes of free $C_\ell$-actionson 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 homotopytheory, and we perform a careful study of $h$-cobordisms. The $\ell=2$ case wascompleted 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 ofequivariant self-equivalences with quadratic growth in $\ell$. The latter ishandled by the composition formula for structure groups of A Ranicki (2009).

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