Real bordered Floer homology

Abstract

Fix a 3-manifold $Y$ with boundary $F⨿F$ and an orientation-preserving involution $\ddot{I}„:Y\to Y$ exchanging the boundary components, with nonempty fixed set. To an appropriate kind of Heegaard diagram for $Y$, we describe how to associate a module over the bordered Heegaard Floer algebra of $F$. These modules satisfy a gluing, or pairing, theorem, and extend the "hat" variant of Guth-Manolescu's real Heegaard Floer homology, $\widehat{HFR}(Y,\ddot{I}„)$. Using these modules, we give a practical algorithm to compute $\widehat{HFR}(Y,\ddot{I}„)$ for real 3-manifolds $(Y,\ddot{I}„)$ with connected fixed set.

75 pages, 33 figures