Let $G$ a finite group. I've seen three options discussed for making $G$-cell complexes: in increasing generality, one might allow $X_n$ to be constructed from $X_{n-1}$ by attaching cells of the form

$G/H\times D^n$, where $D^n$ has trivial $G$-action, or

$G/H\times D(V)$, where $D(V)$ is the unit disk of a $G$-representation $V$, or

$G\times_H D(V)$, where $D(V)$ is the unit disk of an $H$-representation $V$

(The subgroup $H$ and representation $V$ is allowed to vary for different cells. Cells of type 1 are what are used in the standard definition of $G$-CW complex.)

Megan Shulman, on p.47 of her thesis (link goes directly there), says regarding standard $G$-CW complexes that

If we are interested in putting CW structures on spaces found in nature, this is a very restrictive definition...

However, I seem to remember that even for a $G$-cell complex $X$ constructed with cells of type 3, we can always rearrange (triangulate?) somehow to give $X$ a standard $G$-CW complex structure. Is that true or false? Which papers look into this?

Ferland and Lewis, on p.23 of their paper (link goes directly there), say that cells of type 3

... are of interest because they arise naturally from equivariant Morse theory (see, for example, [

21]). Further, if $G$ is a finite abelian group, then the usual Schubert cell structure on Grassmannian manifolds generalizes in an obvious way to a generalized $G$-cell structure on the Grassmannian manifold $G(V, k)$ of $k$-planes in some $G$-representation $V$ (see Chapter 7).[

21] A. G. Wasserman, Equivariant differential topology, Topology 8 (1969), 127–150.

However, skimming over at that section of Wasserman's paper (link goes directly there), I must admit I don't see where anything like cells of type 3 show up, much less demonstrate their usefulness. Could anyone explain their relationship to equivariant Morse theory?

Also, in their Chapter 7 (link goes directly there; look at the bottom of p.83 & top of p.84 in particular), I'm not sure I see where cells of type 3 are necessary, because it seems like $W$ is inherently a $G$-representation, and hence they are only obtaining cells of type 2. Could anyone clarify this?

So to summarize, I'd greatly appreciate any of the following:

simple / natural examples that illustrate the additional flexibility of each generalized cell:

spaces which can easily be seen to have a cell structure when cells of type 3 are allowed, but not when only cells of type 2 are allowed

spaces which can easily be seen to have a cell structure when cells of type 2 are allowed, but not when only cells of type 1 are allowed

any other reasons why cells of type 2 and 3 are reasonable

a citation / proof / counterexample for this half-remembered "fact" that cells of type 1 suffice

Equivariant Algebraic Topology, Princeton University, 1972). I can't find his thesis online to check, though. $\endgroup$ – Arun Debray Dec 24 '17 at 3:28