Georgia Institute of Technology College of Sciences

In the past few years there have been a host of remarkable topological results arising from considering "real" versions of various gauge and Floer-theoretic invariants of three- and four-dimensional manifolds equipped with involutions. Recently Guth and Manolescu defined a real version of Lagrangian Floer theory, and applied it to Ozsváth and Szabó's three-manifold invariant Heegaard Floer homology, producing an invariant called real Heegaard Floer homology associated to a 3-manifold together with an orientation-preserving involution whose fixed set is codimension two (for example a branched double cover). We review the construction of real Heegaard Floer theory and use tools from equivariant Lagrangian Floer theory, originally developed by Seidel-Smith and Large in a somewhat different context, to produce a spectral sequence from the ordinary to real Heegaard Floer homologies in their simplest "hat" version, in particular proving the existence of a rank inequality between the theories. Our results apply more generally to the real Lagrangian Floer homology of exact symplectic manifolds with antisymplectic involutions. Along the way we give a little history and context for this kind of result in Heegaard Floer theory. This is a series of two talks; the first "prep" talk will discuss some background and context that might be helpful to (for example) graduate students in attendance.