Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Instanton construction of the mapping cone Thom-Smale complex

Hao Zhuang

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.

Instanton construction of the mapping cone Thom-Smale complex

Abstract

The wedge by a smooth closed -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.
Paper Structure (6 sections, 24 theorems, 156 equations)

This paper contains 6 sections, 24 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Review of the topological side
  3. Mapping cone Hodge theory
  4. Harmonic oscillators around critical points
  5. Isomorphism between cochain complexes
  6. Decomposition of cohomology

Key Result

Theorem 1.5

We perturb $(f, g)$ subject to perturbation 1, perturbation 2, and perturbation 3. Then, we have constants $C_0>1$ and $T_0>0$ such that when $S > e^{C_0T} > e^{C_0T_0}$, there is a cochain isomorphism between the instanton complex of $(f,g,\omega)$ and the mapping cone Thom-Smale complex of $(f,g)$

Theorems & Definitions (51)

  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7: Clausen-Tang-Tseng clausen_tang_tseng_2024_mappingconemorsetheory, 2026
  • Corollary 1.8
  • Definition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Proposition 2.4
  • ...and 41 more