Integral Klein bottle surgeries and Heegaard Floer homology

Abstract

In this paper, we study which closed, connected, orientable three-manifolds $X$ containing a Klein bottle arise as integral Dehn surgery along a knot in $S^3$. Such $X$ are presentable as a gluing of the twisted $I$-bundle over the Klein bottle to a knot manifold, and we use a variety of Heegaard Floer-type invariants to generate surgery obstructions. Suppose that $X$ is $8$-surgery along a genus two knot and arises by gluing the twisted $I$-bundle over the Klein bottle to an $S^3$ knot complement. We show that $X$ is an $L$-space, it must be the dihedral manifold $(-1;\frac{1}{2},\frac{1}{2},\frac{2}{5})$, and the surgery knot must be $K=T(2,5)$.