Allowing $G$-CW complexes to have more general cells

Question

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

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

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

3. $G\times_H D(V)$, 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.)

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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?

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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?

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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

Answer 1

Let me start with the fact that, in one sense, it's true that Type 1 complexes are all that are "needed." That's true in the sense that complexes built from Type 2 and 3 cells have the $G$-homotopy type of $G$-CW complexes (meaning ones built from Type 1 cells). I'm not sure of a reference, but the proof is relatively straightforward: Cells of Types 2 and 3 can be triangulated, hence given $G$-CW structures, and the attaching maps are homotopic to cellular maps, assuming inductively that the lower skeleton has been replaced by a $G$-homotopy equivalent $G$-CW complex.

Aside: In your Type 2 cells, I think most people restrict to a single representation $V$ ($\pm$ trivial representations) rather than letting $V$ vary. This gives the notion of $G$-CW($V$) complex, originally developed by Stefan Waner in an unpublished manuscript, then used in various papers that he and I wrote together and also by Gaunce Lewis in his paper Equivariant Eilenberg-MacLane spaces and the equivariant Seifert-van Kampen and suspension theorems. Stefan and I give a detailed account of $G$-CW($V$) complexes and a certain kind of Type 3 complex ($G$-CW($\gamma$) complexes) in our recent LNM volume "Equivariant Ordinary Homology and Cohomology."

So why should we be interested in cell complexes built out of cells of Type 2 or 3? Mainly because their combinatorics (meaning chain complexes they give rise to) give useful invariants. Equivariant Bredon ordinary homology and cohomology are defined in integer grading using $G$-CW structures. It's known that, if the coefficient system used is a Mackey functor, then Bredon homology can be extended to an $RO(G)$-graded theory. But what do those additional groups mean geometrically? It turns out that the $V$th homology group of $X$ is what you get if you approximate $X$ by a $G$-CW($V$) complex instead of a $G$-CW complex.

For example, if $X$ is a $G$-CW complex, then the suspension $\Sigma^V X_+$ has an obvious (based) $G$-CW($V$) structure, and its combinatorics give the $V+$integer homology groups of $\Sigma^V X_+$, which are the same as the integer-graded homology groups of $X$, as they ought to be by the suspension isomorphism.

More generally, if $\xi$ is a $G$-vector bundle over a $G$-CW complex $X$ and each fiber $\xi_b$ of $\xi$ is $G_b$-isomorphic to (a fixed) $V$ (we say that $\xi$ is a $V$-bundle), then the Thom space $T\xi$ has a corresponding $G$-CW($V$) structure, and this observation is essentially the equivariant Thom isomorphism theorem for $V$-bundles.

As another example, suppose $M$ is a smooth $G$-manifold and that its tangent bundle is a $V$-bundle for some $V$. (We say that $M$ is a $V$-manifold.) Illman showed in general that we can triangulate $M$, hence give it a $G$-CW structure. If we form the dual cell structure, in which the top-dimensional cells are the stars of the original vertices in the first barycentric subdivision, this dual structure is a $G$-CW($V$) structure (assuming $G$ is finite). This observation is essentially the equivariant Poincaré duality theorem for $V$-manifolds.

But what if you wanted a Thom isomorphism for general $G$-vector bundles, or a Poincaré duality theorem for general $G$-manifolds? If you try to form the corresponding cell structure on the Thom space, or the dual cell structure on a manifold, what you get will be a cell structure with Type 3 cells in general (but with some control; the representations showing up are not completely unrelated). Where do the corresponding invariants live? Giving a detailed answer to that question is one of the main reasons Stefan and I wrote LNM 2178.

Answer 2

Just a quick note adding to Steve's answer, with a focus on the stable setting. In the Hill-Hopkins-Ravenel paper on the Kervaire problem, they introduce the "positive complete model structure" on G-spectra, and the "complete" part gives you cells of type 3. This is on page 161 of the last arxiv version (which I think is super close to the published annals version) https://arxiv.org/pdf/0908.3724v4.pdf

They needed these cells in many places in the paper, but most notably to prove that, for every cofibrant spectrum $X$, the natural map $(E\Sigma_n)_+ \wedge X^{\wedge n} \to X^{\wedge n}/\Sigma_n$ is a weak equivalence (where $\Sigma_n$ denotes the symmetric group on $n$ letters). They use this property to set up a good homotopy theory of commutative monoids. This property was claimed without proof in earlier sources setting up the homotopy theory of equivariant spectra, but the proof is really non-trivial. It follows for the positive and positive flat model structures, but I only know how to prove it using the positive complete model structure. I believe Doug Ravenel has continued to investigate this model structure. I guess my point is that you should not restrict yourself to looking at classical sources, because the decision about which cells to use was still in flux till very recently. Ravenel gave a great talk a few times in 2015 where he argued that this positive complete model structure is the definitive right answer (at least, for equivariant stable homotopy theory).