Perturbed gradient flow trees and $A_\infty$-algebra structures on Morse cochain complexes

TL;DR

This paper develops a detailed framework for constructing $A_ abla$-algebra structures on Morse cochain complexes, based on perturbed gradient flow trees, extending ideas of Abouzaid and Fukaya.

Contribution

It provides a comprehensive and rigorous treatment of Abouzaid's approach to $A_ abla$-algebras on Morse complexes, including analytic foundations and detailed constructions.

Findings

Established a coherent method for $A_ abla$-algebra structures on Morse cochains.

Extended Fukaya's Morse-$A_ abla$-categories to a more detailed analytic setting.

Provided new insights into perturbed gradient flow trees and their algebraic implications.

Abstract

We elaborate on an idea of M. Abouzaid of equipping the Morse cochain complex of a smooth Morse function on a closed oriented manifold with the structure of an $A_\infty$-algebra. This is a variation on K. Fukaya's definition of Morse-$A_\infty$-categories for closed oriented manifolds involving families of Morse functions. The purpose of this article is to provide a coherent and detailed treatment of Abouzaid's approach including a discussion of all relevant analytic notions and results.