Using the conjugation symmetry on Heegaard Floer complexes, we define a -manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to -equivariant Seiberg–Witten Floer homology. Further, we obtain two new invariants of homology cobordism, and , and two invariants of smooth knot concordance, and . 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 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.
"Involutive Heegaard Floer homology." Duke Math. J. 166 (7) 1211 - 1299, 15 May 2017. https://doi.org/10.1215/00127094-3793141