Stable Invariant Morse-Bott Functions

• Stable invariant Morse-Bott functions are smooth, G-invariant functions whose critical sets decompose into a finite number of G-orbits with nondegenerate Hessians.
• They are constructed by perturbing any G-invariant function via careful stabilization techniques that guarantee Morse-Bott conditions and stability in the slice directions.
• These functions play a central role in equivariant Morse theory by ensuring transversality in gradient flows and underpinning the construction of spectrum-valued homotopy invariants.

A stable invariant Morse-Bott function is a $G$-invariant smooth function on a closed manifold $M$ with an action of a compact Lie group $G$, whose critical set decomposes into finitely many $G$-orbits, and for which the Hessian is nondegenerate and stable in precise equivariant senses. These functions arise as fundamental objects in equivariant Morse theory, equivariant cohomology, and the construction of spectrum-valued invariants extending Morse-Bott theory, and they play a central role in the modern understanding of equivariant transversality and structural stability.

1. Definitions and Foundational Structure

Let $M$ be a closed smooth manifold with a smooth action by a compact Lie group $G$. The space of $G$-invariant smooth functions is $$C^\infty(M)^G = \{f \mid f(g \cdot x) = f(x), \forall g \in G\}$$.

A function $f \in C^\infty(M)^G$ is a $G$-Morse-Bott function if:

• Its critical locus $\Crit(f)$ is a finite disjoint union of $G$-invariant connected submanifolds, each of which is a single $G$-orbit: $S = G \cdot p$ for some $p \in M$.
• At every $p \in S$, the tangent space $T_p S$ coincides with the kernel of the Hessian: $T_pS = \ker(\operatorname{Hess}_f(p))$.

A $G$-Morse-Bott function is stable if for each critical point $p$, the normal space to the orbit (slice) decomposes as $N_p = N_p' \oplus N_p''$, with $N_p'$ the fixed subspace for the stabilizer $H = \mathrm{Stab}(p)$ and $N_p''$ its complement, and the Hessian $\operatorname{Hess}_f(p)$ is positive definite on $N_p''$ (Bao et al., 22 Jan 2026).

Equivalently, stability means that all negative and zero eigenspaces of the Hessian lie within the $H$-fixed subspace; all nontrivial slice directions contribute only positive eigenvalues.

2. Construction and Existence via Stabilization

For any $G$-invariant function $f_0 \in C^\infty(M)^G$, the existence of a stable $G$-Morse-Bott function arbitrarily close to $f_0$ is established via stabilization. For every $\varepsilon > 0$, one constructs $h \in C^\infty(M)^G$ with $\|h\|_{C^1} < 1$ such that $f_\varepsilon = f_0 + \varepsilon h$ is a stable $G$-Morse-Bott function and $f_\varepsilon$ converges to $f_0$ in $C^1$ as $\varepsilon \to 0$ (Theorem 3.7) (Bao et al., 22 Jan 2026).

The construction is performed slice-by-slice:

• For each critical orbit $G \cdot p$, one selects a $G$-invariant tubular neighborhood and models $f_0$ locally using the equivariant Morse-Bott lemma.
• One then perturbs in the normal directions using carefully chosen bump functions and Morse-Bott models on small spheres, ensuring all new critical submanifolds remain $G$-invariant and the instability is resolved.
• The perturbation is then extended to the $G$-orbit, patched globally, and repeated for all critical loci.

This method ensures the resulting function is both $C^1$-close and stably $G$-Morse-Bott.

3. Hessian Structure and Transversality

At each critical orbit $S = G \cdot p$, the Hessian at $p$ decomposes in accordance with the slice representation. Stability ensures that the restriction of the Hessian to $N''_p$ is positive definite.

For generic $G$-invariant Riemannian metrics, the pair $(f_\varepsilon, g)$ satisfies the Morse-Bott-Smale transversality condition: all stable and unstable manifolds for the gradient flow of $f_\varepsilon$ intersect transversely (Bao et al., 22 Jan 2026). This transversality is essential for the construction of well-defined moduli spaces of gradient flow lines, equivariant cohomology operations, and Floer theoretic invariants.

Moreover, for stable invariant Morse-Bott functions, all negative normal bundles at critical orbits are trivial and orientable, a property crucial for orientability considerations and cohomological constructions (Lemma 5.3 in (Bao et al., 22 Jan 2026)).

4. Explicit Examples: Non-Linear Morse-Bott Functions

An important class of explicit stable invariant Morse-Bott functions comes from non-linear quadratic models on symmetric spaces, such as the quaternionic Stiefel manifold $X_{n,k} \cong Sp(n) / Sp(n-k)$.

In this context, the function $h(x) = \operatorname{Tr}(P^*P)$ is $Sp(n)$-invariant and Morse-Bott. Its critical sets decompose according to the rank $q$ of the projector $P^*P$, yielding a dichotomy in the structure of critical manifolds governed by the relation of $n$ and $2k$. The critical submanifolds are total spaces of fibrations over products of quaternionic Grassmannians, exhibiting the rich geometry inherently tied to the symmetry group action.

The gradient and Hessian can be computed explicitly, and any small $Sp(n)$-invariant perturbation preserves the Morse-Bott property due to the nondegeneracy of the Hessian in normal directions and the $G$-equivariance (Macías-Virgós et al., 2020).

5. Applications to Equivariant Morse Theory and Homotopy Invariants

The construction of stable invariant Morse-Bott functions underlies the rigorization of equivariant Morse-theoretic models for Borel equivariant cohomology as in the Austin-Braam framework: choosing a stable invariant Morse-Bott function ensures all the necessary transversality, compactness, and orientability assumptions (Bao et al., 22 Jan 2026).

In the context of Morse-Bott theory on closed manifolds, such functions enable the definition of flow categories whose stable normal framings are inputs to the Cohen-Jones-Segal (CJS) construction. For generic pseudo-gradient flows associated to a Morse-Bott function, the CJS methodology yields a stable homotopy type equivalent to $\Sigma_+^\infty M$ (Bonciocat, 2024). The flexibility and stability of the Morse-Bott data allow for the recovery of all Thom spectra $M^E$ for $KO$-theory classes $E$, broadening the impact well beyond the classical scope.

In addition, in the special context of surfaces with semi-free $S^1$ actions, a normal form theorem identifies all Morse-Bott-type circle-invariant functions up to diffeomorphism and reparametrization, and small $C^\infty$-perturbations preserve the normal form structure and invariants, conferring a strong form of structural stability within this class (Feshchenko, 2024).

6. Local Models and Explicit Construction Techniques

Locally near a critical orbit $G \cdot p$, a stable invariant Morse-Bott function admits a normal form: in $G$-invariant coordinates, the function takes the shape

$$f(p + x) = f(p) - \|x_-\|^2 + \|x_+\|^2 + O(\|x\|^3)$$

where $x_-$ are coordinates in the negative normal directions, and $x_+$ in the positive (Bao et al., 22 Jan 2026). By stabilization, one replaces the quadratic form in $x_-$ with a "dimpled" Morse-Bott profile on a small sphere to eliminate nontrivial negative and zero directions except along the $G$-orbit.

Gradient flow equations are explicitly computable, particularly in symmetric space examples, supporting explicit integration of flows and construction of moduli spaces. For instance, in the quaternionic Stiefel manifold example, integrating the quadratic gradient flow shows every trajectory converges to a critical submanifold, demonstrating robust convergence properties (Macías-Virgós et al., 2020).

7. Relation to Classification and Structural Stability

A key property of stable invariant Morse-Bott functions is their robustness under small equivariant perturbations. The classification of such functions, particularly in the presence of large symmetry groups or on specific surface models, can be achieved in terms of topological data (orbit types, fibration structure, and critical value combinatorics).

In circle-valued Morse-Bott-type models on surfaces with semi-free $S^1$-actions, the normal form $f = x \circ f_0 \circ h^{-1}$ is stable under small perturbations in the specified function class, providing invariants that control the combinatorial and jet-type structure of singularities (Feshchenko, 2024). For general $G$, the stabilization and perturbation techniques ensure the persistence of Morse-Bott and stability properties for dense open families of $G$-invariant functions (Bao et al., 22 Jan 2026).

• "Equivariant Morse-Bott cohomology through stabilization" (Bao et al., 22 Jan 2026)
• "Non-linear Morse-Bott functions on quaternionic Stiefel manifolds" (Macías-Virgós et al., 2020)
• "Normal forms of functions with degenerate singularities on surfaces equipped with semi-free circle actions" (Feshchenko, 2024)
• "Revisiting the Cohen-Jones-Segal construction in Morse-Bott theory" (Bonciocat, 2024)