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

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.

100 pages, 18 figures. The contents of this article are parts of the author's Ph.D. thesis, submitted to the University of Leipzig in September 2016