Geometry and Topology Seminar (2/2)

Real Heegaard Floer homology is an invariant recently defined by Gary Guth and Ciprian Manolescu, which takes as input a 3-manifold equipped with a branched cover involution, and outputs a graded Z/2-vector space. It is defined in a way that parallels Jiakai Li's construction of Real Monopole Floer homology, by restricting attention only to Heegaard diagrams, intersection points, and pseudo-holomorphic curves that respect the involution. In this talk, we explore the question of orientability for the Floer moduli spaces, which would be required to lift the Guth-Manolescu theory to integer coefficients, and its consequences to being able to define a Heegard-Floer theoretic analogue of Miyazawa's invariant for surfaces in 4-manifolds (defined by counting real solutions to the Seiberg-Witten equations). In the case of roll-spun knots, this numerical invariant should be computable from the functorial maps of the 3-manifold invariant. Time permitting, we may also discuss the question of orientability in generalized cohomology theories, with the idea of defining Heegaard-Floer homotopy types analogous to the Seriberg-Witten story.