Involutive Heegaard Floer homology

Abstract

Using the conjugation symmetry on Heegaard Floer complexes, we define a $3$-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to ${\mathbb{Z}}_4$-equivariant Seiberg–Witten Floer homology. Further, we obtain two new invariants of homology cobordism, $\underline{d}$ and $\overline{d}$, and two invariants of smooth knot concordance, ${\underline{V}}_0$ and ${\overline{V}}_0$. We also develop a formula for the involutive Heegaard Floer homology of large integral surgeries on knots. We give explicit calculations in the case of L-space knots and thin knots. In particular, we show that ${\underline{V}}_0$ detects the nonsliceness of the figure-eight knot. Other applications include constraints on which large surgeries on alternating knots can be homology-cobordant to other large surgeries on alternating knots.