INVOLUTIVE HEEGAARD FLOER HOMOLOGY

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 Z(4)-equivariant Seiberg-Witten Floer homology. Further, we obtain two new invariants of homology cobordism, (d) under bar and (d) over bar, and two invariants of smooth knot concordance, (V) under bar (0) and (V) over bar (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 (V) under bar (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.