A coincidence theorem for commuting involutions (Q2845068)

Under which conditions do two smooth involutions \(S, T\) which commute have a coincidence? Further, in some sense, what is the size of the coincidence set of the two involutions? The paper under review gives a contribution to this question. It generalizes the following example: Let \(S^n\) denote the \(n\)-sphere and consider \(S\) the antipodal involution and the involution given by \(T(x_0, x_1, ..., x_n)=(-x_0,-x_1, ...,-x_m)\). The set \(Coin(S, T)\) is clearly an \((n-1)\)-sphere. The main result of the paper is:NEWLINENEWLINE{Theorem:} Let \(M^n\) be an \(m\)-dimensional, closed and smooth manifold, and \(S\) and \(T\) two smooth commuting involutions on \(M^n\) with \(S \neq T\) on each component of \(M^n\). Suppose that \(F_T\) is empty and the number of points of \(F_S\) is of the form \(2p\) with \(p\) odd. Then \(Coin(S,T)\) has at least one component of dimension \(m-1\); that is, \(\dim(F_{ST})=m-1\).NEWLINENEWLINETo prove the result the author uses bordism and singular bordism theory in a nontrivial way. Several examples are provided which illustrate the necessity of the hypotheses of the Theorem, but even so leaving open the possibility of generalization of his result.