Injective Satellites, Immersed Curves, and Involutive Floer Homology

Work of Hanselman-Rasmussen-Watson has shown that the bordered Floer invariants for a 3-manifold with torus boundary can be represented as a collection of (decorated) immersed curves in the punctured torus; moreover, the Heegaard Floer homology of the closed manifold obtained by gluing two such bordered manifolds together along their common boundary can be computed as the Lagrangian intersection Floer homology of their immersed curve invariants. In other words, the bordered modules correspond to objects in the Fukaya category and the bordered pairing theorem is given by the hom-pairing. We show that (as expected) four-manifold invariants can be extracted from composition. We leverage this relationship to address questions about exotic phenomena in the four-ball. Time permitting, we will also discuss a structural result about the involutive extension of bordered Floer homology and constructions of 1-stably exotic contractible 4-manifolds. This draws on work with J. Cohen and S. Kang.