Surface Gluing with Signs and Gradings in Decategorified Heegaard Floer Theory

Abstract

Abstract A previous result about the decategorified bordered (sutured) Heegaard Floer invariants of surfaces glued together along intervals, generalizing the decategorified content of Rouquier and the author’s higher-tensor-product-based gluing theorem in cornered Heegaard Floer homology, was proved only over ${\mathbb{F}}_2$ and without gradings. In this paper we add signs and prove a graded version of the interval gluing theorem over ${\mathbb{Z}}$, enabling a more detailed comparison of these aspects of decategorified Heegaard Floer theory with modern work on non-semisimple 3d TQFTs in mathematics and physics.