A note on the theorems of Lusternik–Schnirelmann and Borsuk–Ulam
Volume 111 / 2008
Colloquium Mathematicum 111 (2008), 35-42
MSC: Primary 55M20; Secondary 55M30, 55M35.
DOI: 10.4064/cm111-1-3
Abstract
Let $p$ be a prime number and $X$ a simply connected Hausdorff space equipped with a free $\mathbb Z_p$-action generated by $f_p:X\rightarrow X$. Let $\alpha:S^{2n-1}\rightarrow S^{2n-1}$ be a homeomorphism generating a free $\mathbb Z_p$-action on the $(2n-1)$-sphere, whose orbit space is some lens space. We prove that, under some homotopy conditions on $X$, there exists an equivariant map $F:(S^{2n-1},\alpha)\rightarrow (X,f_p)$. As applications, we derive new versions of generalized Lusternik–Schnirelmann and Borsuk–Ulam theorems.
