Floer homology and covering spaces

Abstract

We prove a Smith-type inequality for regular covering spaces in monopole Floer homology. Using the monopole Floer/Heegaard Floer correspondence, we deduce that if a $3$–manifold $Y$ admits a $p^n$–sheeted regular cover that is a $\mathbb{Z}∕p\mathbb{Z}$–$L$–space (for $p$ prime), then $Y$ is a $\mathbb{Z}∕p\mathbb{Z}$–$L$–space. Further, we obtain constraints on surgeries on a knot being regular covers over other surgeries on the same knot, and over surgeries on other knots.