Equivariant Morse-Bott cohomology through stabilization

TL;DR

The paper addresses obstructions in defining equivariant Morse cohomology for compact Lie group actions by Austin-Braam, notably transversality and orientability. It introduces stabilization: a $C^1$-small equivariant perturbation that converts a $G$-invariant Morse-Bott function into a stable $G$-Morse-Bott function, enabling generic $G$-invariant metrics to satisfy Morse-Bott-Smale transversality. This yields well-defined ordinary and Cartan-model equivariant cochain complexes whose cohomologies recover $H^*(M)$ and $H_G^*(M)$, with orientability issues resolved by stability. The framework is illustrated through explicit stabilization procedures and examples, clarifying the link to Borel equivariant cohomology and enabling concrete calculations in stabilized settings.

Abstract

For closed manifolds with compact Lie group actions, we study Austin-Braam's Morse-theoretic construction of Borel equivariant cohomology using the technique of stabilization. We show that a $C^1$-small equivariant perturbation produces stable invariant Morse-Bott functions. This allows us to realize the equivariant transversality and orientability assumptions in Austin-Braam's framework by choosing generic invariant Riemannian metrics.

Figures (4)

• Figure 1: Various notions discussed above.
• Figure 2: The auxiliary functions $\Phi_{\lambda}$ (left) and $\Psi_\delta$ (right) appearing in Construction \ref{['construction:function']} for $\lambda=1,t_0=1.5,\delta=0.1$.
• Figure 3: Effects of stabilization on a torus fiber
• Figure :

Theorems & Definitions (45)

• Definition 1.1: Morse-Bott functions
• Definition 1.2: $G$-Morse-Bott functions
• Theorem 1.3: wasserman1969equivariant
• Definition 1.4: Stable
• Remark 1.5
• Theorem 1.6
• Definition 1.7: Morse-Bott-Smale
• Theorem 1.8
• Lemma 2.1
• proof