Heegaard Floer homology, degree-one maps and splicing knot complements

Abstract

Let $K$ denote a knot inside the homology sphere $Y$ and $K'$ denote a knot inside a homology sphere $L$-space. Let $X=Y(K,K')$ denote the 3-manifold obtained by splicing the complements of $K$ and $K'$. We show that $\text{rank}(\widehat{HF}(X)) \ge \text{rank}(\widehat{HF}(Y))$.