Real Heegaard Floer Homology

Abstract

We define an invariant of three-manifolds with an involution with non-empty fixed point set of codimension $2$; in particular, this applies to double branched covers over knots. Our construction gives the Heegaard Floer analogue of Li's real monopole Floer homology. It is a special case of a real version of Lagrangian Floer homology, which may be of independent interest to symplectic geometers. The Euler characteristic of the real Heegaard Floer homology is the analogue of Miyazawa's invariant, and can be computed combinatorially for all knots.

54 pages. Revised based on referee reports. Minor errors fixed. Redefined the H_1 action and added a bounding chain to the differential for multi-pointed diagrams. Final version to appear in GAFA