Instanton construction of the mapping cone Thom-Smale complex

TL;DR

This paper constructs an instanton cochain complex for the mapping cone Thom-Smale complex on a Riemannian manifold, proving it is cochain isomorphic to the topologically defined complex, thus linking analytical and topological methods.

Contribution

It introduces an instanton construction of the mapping cone Thom-Smale complex using eigenvalues of a deformed Laplacian, establishing a cochain isomorphism with the topological complex.

Findings

The instanton complex is cochain isomorphic to the topological mapping cone Thom-Smale complex.

The construction uses eigenspaces of a deformed Laplacian influenced by Morse function parameters.

The method bridges analytical and topological approaches in Morse theory.

Abstract

The wedge by a smooth closed $\ell$-form induces the mapping cone de Rham cochain complex. This complex is quasi-isomorphic to the mapping cone Thom-Smale cochain complex. We give an instanton construction of the mapping cone Thom-Smale complex in this paper. More precisely, for a Morse function with the transversality condition on a closed oriented Riemannian manifold, we construct an instanton cochain complex using the eigenspaces of the mapping cone Laplacian deformed by the Morse function and two parameters. As the main result, we prove that our instanton complex is cochain isomorphic to the topologically constructed mapping cone Thom-Smale complex.