Morse complexes and multiplicative structures

TL;DR

This paper details the construction of Fukaya's $A_$-structure on Morse complexes of manifolds with boundary, proving its independence from choices and emphasizing transversality for smooth fiber products.

Contribution

It provides a detailed construction of Fukaya's $A_$-structure on Morse complexes, establishing its homotopic independence from auxiliary choices.

Findings

Fukaya's $A_$-structure on Morse complexes is well-defined and homotopically independent.

Transversality arguments ensure smoothness of fiber products in the construction.

The paper clarifies the role of boundary conditions in Morse $A_$-structures.

Abstract

In this article we lay out the details of Fukaya's $A_\infty$-structure of the Morse complexe of a manifold possibly with boundary. We show that this $A_\infty$-structure is homotopically independent of the made choices. We emphasize the transversality arguments that make some fiber products smooth.