Relative Morse Categorification Theory

TL;DR

This paper develops a relative Morse complex for manifolds with boundary, proves its homology matches relative singular homology, and constructs an $A_$-category structure, extending previous work to new settings.

Contribution

It introduces a relative Morse complex for manifolds with boundary and constructs an $A_$-category structure, generalizing prior results to manifolds with boundary.

Findings

Homology of the relative Morse complex is isomorphic to relative singular homology.

Constructed an $A_$-category structure on the relative Morse complex.

Generalized Fukaya's and Manabu's constructions to manifolds with boundary.

Abstract

In this paper, we define a relative Morse complex for manifold with boundary using the handlebody decomposition of the manifold. We prove that the homology of the relative Morse complex is isomorphic to the relative singular homology. Furthermore, we construct $A_\infty$-category structure on the relative Morse complex by counting the trajectory trees among the critical points of different Morse functions. Our result generalizes Fukaya's construction on closed manifold and Manabu's construction of absolute homology on manifold with boundary.