\(Z_ 2\)-smooth actions on the \(n\)-sphere \(S^ n\) (Q1200687)

This paper has two main results. The first says that an involution \((M^ n,T)\) bounds equivariantly if and only if all of the manifolds \(S^ m\times M/A\times T\) bound for \(m\geq 0\). The consequence is that every involution on a sphere is an equivariant boundary. The paper is quite attractive, and the approach is innovative, but the results are not really new. The first result is just a restatement of the ideas of Tammo tom Dieck. The classes \(S^ m\times M/A\times T\) determine the cobordism element of \(BZ_ 2\) associated to \((M,T)\). That every involution on a sphere bounds follows at once from a cohomological analysis of the fixed set using Smith theory. In all essential respects, the involution behaves as if it were linear.