Analytic and topological realizations of the invariant Thom-Smale complex

TL;DR

This work develops an analytic realization of the $G$-invariant Thom-Smale complex for Morse–Bott functions on closed oriented $G$-manifolds by pairing it with a $G$-invariant Witten instanton complex. The main result is a chain isomorphism between the topological Thom-Smale complex $C^*(M,f)^G$ and the analytic $F_T^*(M,f,\alpha)^G$ for sufficiently large deformation parameter $T$ above a spectral threshold $\alpha>\alpha_0$, established via a detailed spectral-gap analysis of the $G$-invariant Witten Laplacian. The framework relies on a carefully adjusted $G$-invariant metric near critical orbits and a horizontal–vertical decomposition that isolates contributions from the normal directions, thereby linking critical-orbit data to global cohomology $H^*(M)^G$. The results extend classical Morse theory to the equivariant Morse–Bott setting and yield $G$-invariant Morse–Bott inequalities with concrete computational tools, illustrated by explicit examples.

Abstract

With the smooth action of a connected compact Lie group G, we realize the G-invariant Thom-Smale complex in an analytic way using the G-invariant Witten instanton complex. Both complexes are associated to a specific Morse-Bott function on a closed oriented G-manifold. This result includes the influence from the horizontal direction around the critical set, generalizing the strict Morse case.

Theorems & Definitions (77)

• Remark 1.2
• Definition 1.3
• Theorem 1.4
• Definition 1.5
• Theorem 1.6
• Remark 1.7
• Lemma 2.1
• proof
• Lemma 2.2
• proof