Duke Mathematical Journal

Involutive Heegaard Floer homology

Kristen Hendricks and Ciprian Manolescu

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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 Z4-equivariant Seiberg–Witten Floer homology. Further, we obtain two new invariants of homology cobordism, d̲ and d¯, and two invariants of smooth knot concordance, V̲0 and 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 V̲0 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.

Article information

Duke Math. J., Volume 166, Number 7 (2017), 1211-1299.

Received: 24 November 2015
Revised: 12 July 2016
First available in Project Euclid: 11 January 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R58: Floer homology
Secondary: 57M27: Invariants of knots and 3-manifolds

Heegaard Floer homology cobordism 3-manifold knot


Hendricks, Kristen; Manolescu, Ciprian. Involutive Heegaard Floer homology. Duke Math. J. 166 (2017), no. 7, 1211--1299. doi:10.1215/00127094-3793141. https://projecteuclid.org/euclid.dmj/1484103841

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