Floer homology and covering spaces

TL;DR

This paper establishes a Smith-type inequality in monopole Floer homology for regular covering spaces and derives implications for the structure of 3-manifolds and knot surgeries, linking covering space properties to Floer homology invariants.

Contribution

It introduces a new inequality in monopole Floer homology for covering spaces and connects it to properties of 3-manifolds and knot surgeries via the Floer correspondence.

Findings

Regular covers impose constraints on Floer homology invariants.

If a manifold admits a certain regular cover, it inherits L-space properties.

Constraints on knot surgeries relate to covering space structures.

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 Z/pZ-L-space (for p prime), then Y is a Z/pZ-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.