Let $G$ be a Lie group acting properly on a smooth manifold $M$. The (non-equivariant) definition of a Morse function does not carry over to equivariant functions $M rightarrow mathbb{R}$ (where $mathbb{R}$ has the trivial action) since critical points are no longer isolated: Every orbit of a critical point will consist entirely of critical points.

However, there is an adaption via Morse-Bott functions $f: M rightarrow mathbb{R}$ whose set of critical points is a disjoint union of submanifolds, so called critical submanifolds. Instead of demanding the Hessian at a critical point to be non-degenerate, one requires the Hessian to be non-degenerate on the normal bundle of these submanifolds. We call such submanifolds non-degenerate.

Let us say that a $G$-equivariant map $f : M rightarrow mathbb{R}$ is a $G$-equivariant Morse function if it is a Morse-Bott function whose critical submanifolds are non-degenerate orbits.*

If I understood it correctly, in the late 60’s Wasserman proved the following (see also lemma 4.8 here).

Let $G$ be a compact Lie group. The set of $G$-equivariant Morse functions is dense in the space of all smooth $G$-equivariant maps $C^{infty, G}(M, mathbb{R})$ (equipped with the subspace topology where $C^infty(M,mathbb{R})$ has the strong topology).

This result horribly fails when $G$ is non-compact. Indeed, in this paper Illman and Kankaanrinta prove that if $G$ is a non-compact Lie group acting properly on $M$; and $N$ is any $G$-manifold, then the topology on

$C^{infty, G}(M, N)$ induced from the strong topology on $C^{infty}(M, N)$ is discrete! In their paper, they introduce the so called “strong-weak” topology which sits inbetween the weak and the strong topology on $C^infty(M,N)$. In particular, if $G$ is compact or acts trivial the strong-weak topology is the strong topology, whereas it is the weak topology if $G$ acts cocompactly.

The strong-weak topology seems to be very adequate for the study of equivariant maps, for instance the authors prove that equivariant proper embeddings between proper $G$-manifolds are open in $C^{infty, G}(M,N)$ for the strong-weak topology. Similar results can be found in the linked paper.

Of course now I wonder:

Does Wasserman’s result remain true in the non-compact case if we consider proper Lie group actions and equip $C^{infty, G}(M, mathbb{R})$ with the strong-weak topology?

*(in Wasserman’s paper these are precisely the functions in a space called $mathfrak{M}(M,M)$. His definition of Morse function makes a further technical assumption which I do not need).