On the homotopy fixed point sets of circle actions on product spaces

Abstract

For arbitrary $S^{1}$-actions on $S^{m}_{{\mathbb {Q}}}$, $S^{n}_{{\mathbb {Q}}}$, and $S^{m}_{{\mathbb {Q}}}\times S^{n}_{{\mathbb {Q}}}$, we show the conditions for the tenability of the homotopy equivalence $(S^{m}_{{\mathbb {Q}}})^{hS^{1}}\times (S^{n}_{{\mathbb {Q}}})^{hS^{1}}\simeq (S^{m}_{{\mathbb {Q}}}\times S^{n}_{{\mathbb {Q}}})^{hS^{1}}$. Here, $X^{hS^1}$ denotes the homotopy fixed point set of an $S^1$-action on an space X.