Involutive Heegaard Floer homology

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.

arXiv comment66 pages, 22 figures; final version, to appear in Duke Math. J

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Cited by in corpus (26)

• Link cobordisms and functoriality in link Floer homology
• Connected sums and involutive knot Floer homology
• A flexible construction of equivariant Floer homology and applications
• A splitting theorem for the Seiberg-Witten invariant of a homology $S^1\times S^3$
• Contact handles, duality, and sutured Floer homology
• Applications of involutive Heegaard Floer homology
• Chern-Simons Functional and the Homology Cobordism Group
• Duality and mapping tori in Heegaard Floer homology
• On homology cobordism and local equivalence between plumbed manifolds
• A survey of the homology cobordism group
• Linear independence of rationally slice knots
• Equivariant Seiberg-Witten-Floer cohomology
• Almost simple linear graphs, homology cobordism and connected Heegaard Floer homology
• Projective Naturality in Heegaard Floer Homology
• Irreducible 3-manifolds that cannot be obtained by 0-surgery on a knot
• Involutive Heegaard Floer homology and rational cuspidal curves
• On sums of torus knots concordant to alternating knots
• Upsilon type concordance invariants
• Correction to the article: Floer homology and splicing knot complements
• Connected Heegaard Floer homology of sums of Seifert fibrations
• On the nonorientable four-ball genus of torus knots
• Involutions, knots, and Floer K-theory
• Involutive Khovanov homology and equivariant knots
• On naturality of the Ozsvath-Szabo contact invariant
• Cobordism distance on the projective space of the knot concordance group
• On the involutive Heegaard Floer homology of negative semi-definite plumbed 3-manifolds with $b_1=1$