Floer homology and covering spaces

Lidman, Tye; Manolescu, Ciprian

doi:10.2140/gt.2018.22.2817

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Floer homology and covering spaces

Tye Lidman and Ciprian Manolescu

Geometry & Topology 22 (2018) 2817–2838

DOI: 10.2140/gt.2018.22.2817

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 $ℤ ∕ p ℤ$ – $L$ –space (for $p$ prime), then $Y$ is a $ℤ ∕ p ℤ$ – $L$ –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.

Keywords

Smith inequality, Seiberg–Witten, Heegaard Floer homology, virtually cosmetic, L–spaces

Mathematical Subject Classification 2010

Primary: 57R58

Secondary: 57M10, 57M60

References

Publication

Received: 12 February 2017

Accepted: 5 November 2017

Published: 1 June 2018

Proposed: András I Stipsicz

Seconded: Peter Ozsváth, Ian Agol

Authors

Tye Lidman
Department of Mathematics North Carolina State University Raleigh, NC United States

Ciprian Manolescu
Department of Mathematics UCLA Los Angeles, CA United States

Volume 22, issue 5 (2018)