Geometry and Topology Seminar

There has been a burst of interest in gauge theoretic invariants of 3- and 4-manifolds equipped with an involution, developed in various contexts by Tian-Wang, Nakamura, Konno-Miyazawa-Taniguchi, and Li. Notably, Miyazawa proved the existence of an infinite family of exotic RP^2-knots using real Seiberg-Witten theory. In joint work with Ciprian Manolescu, we construct an invariant of based 3-manifolds with an involution, called real Heegaard Floer homology. This is the  analogue of Li's real monopole Floer homology.  Our construction is a particular case of a real version of Lagrangian Floer homology, which may be of independent interest to symplectic geometers. In the case of Heegaard Floer homology, the construction starts from a Heegaard diagram where the involution swaps the alpha and beta curves. We prove that real Heegaard Floer homology is indeed a topological invariant of the underlying pointed real 3-manifold. Further, we study the Euler characteristic of our theory, which is the Heegaard Floer analogue of Miyazawa's invariant for twist-spun 2-knots. This quantity is algorithmically computable and, indeed, appears to agree with Miyazawa's invariant.