A coincidence theorem for commuting involutions

Abstract:

Let $M^{m}$ be an $m$-dimensional, closed and smooth manifold, and let $S, T:M^{m} \to M^{m}$ be two smooth and commuting diffeomorphisms of period $2$. Suppose that $S \not = T$ on each component of $M^{m}$. Denote by $F_{S}$ and $F_{T}$ the respective sets of fixed points. In this paper we prove the following coincidence theorem: if $F_{T}$ is empty and the number of points of $F_{S}$ is of the form $2p$, with $p$ odd, then $Coinc(S,T)=\{x \in M^{m} \ \vert \ S(x)=T(x) \}$ has at least some component of dimension $m-1$. This generalizes the classic example given by $M^{m}=S^{m}$, the $m$-dimensional sphere, $S(x_{0},x_{1},...,x_{m}) = (-x_{0},-x_{1},...,-x_{m-1},x_{m})$ and $T$ the antipodal map.