Immersed Curves and 4-Manifold Invariants

TL;DR

This paper connects immersed curve invariants of 3-manifolds with torus boundary to cobordism maps in 4-manifold topology, providing a geometric framework for understanding 4-manifold invariants.

Contribution

It establishes a relationship between morphisms of immersed curve invariants and cobordism maps, linking Fukaya category composition to Floer invariants.

Findings

Relates Fukaya category composition to Floer cobordism maps

Provides a method to obstruct smooth equivalences of 4-manifolds

Interprets bordered Floer invariants as immersed curves with local systems

Abstract

For 3-manifolds with torus boundary, the bordered Heegaard Floer invariants of Lipshitz--Ozsv\'ath--Thurston have a geometric interpretation as immersed multi-curves with local systems in the punctured torus according to the work of Hanselman--Rasmussen--Watson. We consider morphisms between these immersed curve invariants and show that they compute certain cobordism maps. More precisely, we relate composition in the Fukaya category of immersed curves in the punctured torus to composition of morphisms between the bordered Floer invariants, which have interpretations in terms of certain cobordism maps. We make use of this formalism to obstruct smooth equivalences between 4-manifolds with boundary, and between surfaces with boundary in the 4-ball.