Rank inequalities for the Heegaard Floer homology of branched covers

We show that if L is a nullhomologous link in a 3-manifold Y and \Sigma(Y, L) is a double cover of Y branched along L then for each spin ^c -structure \mathfrak{s} on Y there is an inequality \dim\widehat{HF}(\Sigma(Y, L), \pi^\ast\mathfrak{s}; \mathbb{F}_2) \geq \dim \widehat{HF} (Y, \mathfrak{s}; \mathbb{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.