Fixed points of unitary $\textbf {Z}/p^ s$-manifolds

Abstract:

Let $G = {\mathbf {Z}}/{p^s}$ ($p$ an odd prime). We show that restricting the local representations in a unitary $G$-manifold $M$ with isolated fixed points results in severe restrictions on the number of fixed points (counted with the sign of their orientation), paralleling results obtained by Conner and Floyd in the case $G = {\mathbf {Z}}/p$. Specifically, the number of noncancelling fixed points is either zero or divisible by ${p^n}$, where $n \to \infty$ as the dimension of $M \to \infty$. This result also parallels phenomena in framed $G$-manifolds, as discussed by the first author in a previous paper.