Volume 22, issue 5 (2018)

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Floer homology and covering spaces

Tye Lidman and Ciprian Manolescu

Geometry & Topology 22 (2018) 2817–2838
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.

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Keywords
Smith inequality, Seiberg–Witten, Heegaard Floer homology, virtually cosmetic, L–spaces
Mathematical Subject Classification 2010
Primary: 57R58
Secondary: 57M10, 57M60