Involutive Heegaard Floer homology

TL;DR

This paper introduces involutive Heegaard Floer homology, a new three-manifold invariant based on conjugation symmetry, and develops related invariants for homology cobordism and knot concordance, with explicit calculations and applications.

Contribution

It defines involutive Heegaard Floer homology and new invariants, connecting it to Seiberg-Witten Floer homology and providing computational tools and applications.

Findings

Introduces involutive Heegaard Floer homology as a new invariant.

Defines four new invariants: , , ar V_0, and ar V_0.

Shows ar V_0 detects non-sliceness of the figure-eight knot.

Abstract

Using the conjugation symmetry on Heegaard Floer complexes, we define a three-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 $\bar{d}$, 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 non-sliceness 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.