Rank inequalities for the Heegaard Floer homology of branched covers

Kristen Hendricks; Tye Lidman; Robert Lipshitz

doi:10.4171/dm/878

JournalsdmVol. 27pp. 581–612

Rank inequalities for the Heegaard Floer homology of branched covers

Kristen Hendricks
Mathematics Department, Rutgers University, New Brunswick, NJ 08854, USA
Tye Lidman
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
Robert Lipshitz
Department of Mathematics, University of Oregon, Eugene, OR 97403, USA

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Abstract

We show that if $L$ is a nullhomologous link in a 3-manifold $Y$ and $Σ (Y, L)$ is a double cover of $Y$ branched along $L$ then for each spin $^{c}$ -structure $s$ on $Y$ there is an inequality

dim H F (Σ (Y, L), π^{*} s; F_{2}) \geq dim H F (Y, s; F_{2}) .

We discuss the relationship with the $L$ -space conjecture and give some other topological applications, as well as an analogous result for sutured Floer homology.

Cite this article

Kristen Hendricks, Tye Lidman, Robert Lipshitz, Rank inequalities for the Heegaard Floer homology of branched covers. Doc. Math. 27 (2022), pp. 581–612

DOI 10.4171/DM/878