I want to tell you about Elmendorf’s theorem on equivariant homotopy theory. This theorem played a key role in a recent preprint I wrote with Hisham Sati and Urs Schreiber:

We figured out how to apply this theorem in mathematical physics. But Elmendorf’s theorem by itself is a gem of homotopy theory and deserves to be better known. Here’s what it says, roughly: given any G G -space X X , the equivariant homotopy type of X X is determined by the ordinary homotopy types of the fixed point subspaces X H X^H , where H H runs over all subgroups of G G . I don’t know how to intuitively motivate this fact; I would like to know, and if any of you have ideas, please comment. Below the fold, I will spell out the precise theorem, and show you how it gives us a way to define a G G -equivariant version of any homotopy theory.

We know that in ordinary homotopy theory, there are two kinds of spaces we can study. We can study CW-complexes up to homotopy equivalence, or we can study topological spaces up to weak homotopy equivalence. Weak homotopy equivalence is morally the right kind of equivalence, but Whitehead’s theorem tells us that for the nicer kind of space, the CW-complex, weak homotopy equivalence is the same as strong homotopy equivalence. Moreover, the CW-approximation theorem says that any space is weak homotopy equivalent to a CW-complex. So, they’re really two ways of studying the same thing. One is more flexible, the other more concrete.

NB. In this post, I’ll use the adjective “strong” to contrast homotopy equivalence with weak homotopy equivalence. People usually call strong homotopy equivalence just homotopy equivalence.

Now let G G be a compact Lie group. For G G -spaces, we can also define both strong and weak homotopy equivalence. The strong homotopy equivalence is the obvious thing: you have two equivariant maps f : X → Y f \colon X \to Y and g : Y → X g \colon Y \to X , that are inverse to each other up to equivariant homotopies η : f g ⇒ 1 Y \eta \colon f g \Rightarrow 1_Y and η ′ : g f ⇒ 1 X \eta' \colon g f \Rightarrow 1_X . This lets us consider G G -spaces up to homotopy equivalence. But as for spaces, the morally correct notion of equivalence is weak homotopy equivalence, and this is much stranger: a G G -equivariant map f : X → Y f \colon X \to Y is a equivariant weak homotopy equivalence if it restricts to an ordinary weak homotopy equivalence between the fixed points spaces, f : X H → Y H f \colon X^H \to Y^H , for all closed subgroups H ⊆ G H \subseteq G .

Why on earth should these two notions of equivalence be so different? The equivariant Whitehead theorem justifies this, though again I don’t have a good intuitive explanation for why it should be true. To state this theorem, first I have to tell you what a G G -CW-complex is. We can construct them much as we do ordinary CW-complexes, except they are built from cells of the form:

D n × G / H D^n \times G/H

where D n D^n is the n n -disk with the trivial G G action, and G / H G/H is a coset space of G G with the left G G action. These cells are then glued together by G G -equivariant attaching maps, just like an ordinary CW-complex. The result is a G G -CW-complex. The equivariant Whitehead theorem, due to Bredon, then says that for any pair of G G -CW-complexes, they are weak homotopy equivalent if and only if they are strong homotopy equivalent.

This suggests the key insight behind Elmendorf’s theorem: that we can study G G -spaces simply by looking at X H X^H for all closed subgroups H ⊆ G H \subseteq G . But this operation, of taking a subgroup H H to a space X H X^H , actually defines a functor:

X : Orb G op → Spaces . X \colon Orb_G^{op} \to Spaces .

Here, the domain of this contravariant functor is the orbit category Orb G Orb_G . This is the category with:

objects the coset spaces G / H G/H , for each closed subgroup H ⊆ G H \subseteq G .

, for each closed subgroup . morphisms the G G -equivariant maps.

This is called the orbit category thanks to the elementary fact that any orbit in any G G -space is of the form G / H G/H , for a closed subgroup H H the stabilizer of some point in the orbit.

Since the functor associated to X X is contravariant, it is a presheaf on the orbit category Orb G Orb_G , valued in the category of spaces, Spaces Spaces . The assignment taking a G G -space X X to the presheaf with value X H X^H on the orbit space G / H G/H defines an embedding:

y : G Spaces → PSh ( Orb G , Spaces ) y \colon G Spaces \to PSh(Orb_G, Spaces)

from the category G Spaces G Spaces of G G -spaces into the category of all presheaves on Orb G Orb_G . This is a souped up version of the Yoneda embedding: Orb G Orb_G is a subcategory of G Spaces G Spaces , and the embedding above is just Yoneda when restricted to this subcategory.

It turns out this embedding doesn’t change the homotopy theory at all, as long as we choose the correct weak equivalences on the right hand side: we choose them to be the levelwise weak equivalences. That is, two presheaves X X and Y Y are weak equivalent if there is a natural transformation f : X ⇒ Y f \colon X \Rightarrow Y whose components f H : X H → Y H f^H \colon X^H \to Y^H are ordinary weak equivalences of spaces. With this choice of weak equivalences, the homotopy theory of presheaves on Orb G Orb_G is the same as that of G Spaces G Spaces . That’s Elmendorf’s theorem:

Theorem (Elmendorf). There is an equivalence of homotopy theories G Spaces ≃ PSh ( Orb G , Spaces ) . G Spaces \simeq PSh(Orb_G, Spaces) . In the direction G Spaces → PSh ( Orb G , Spaces ) G Spaces \to PSh(Orb_G, Spaces) , this equivalence is simply the embedding y y .

You can read more about Elmendorf’s theorem in the original paper:

A much more modern treatment is in Andrew Blumberg’s lectures on equivariant homotopy theory. The theorem is so foundational to the topic that it first appears in Section 1.2 of these notes, and Section 1.3 is devoted to it:

Andrew Blumberg - Lectures on equivariant homotopy theory, UT Austin, Spring 2017. Notes by Arun Debray.

Let us step back and appreciate what this theorem has bought us. Besides being a really nice reformulation from a categorical point of view, it gives us a paradigm for constructing equivariant homotopy theories more generally. That is, if we have a homotopy theory in the guise of a category 𝒞 \mathcal{C} with weak equivalences, then you might go ahead and define the equivariant homotopy theory of 𝒞 \mathcal{C} to be: G 𝒞 = PSh ( Orb G , 𝒞 ) G \mathcal{C} = PSh(Orb_G, \mathcal{C}) where the weak equivalences are the levelwise weak equivalences, as in Elmendorf.

For instance, if 𝒞 \mathcal{C} is a model of rational homotopy theory Spaces ℚ Spaces_{\mathbb{Q}} , then G G -equivariant rational homotopy ought to be: PSh ( Orb G , Spaces ℚ ) . PSh(Orb_G, Spaces_{\mathbb{Q}}) . This is precisely what one finds in the literature, at least in the case when G G is a finite group:

This paper actually came before Elmendorf’s - perhaps it served as inspiration!

Or, if you want to get more adventurous, you can define “rational super homotopy theory”, a supersymmetric version of rational homotopy theory, modeled by some category with weak equivalences called SuperSpace ℚ SuperSpace_{\mathbb{Q}} . Then the G G -equivariant rational super homotopy theory ought to be: G SuperSpace ℚ = PSh ( Orb G , SuperSpace ℚ ) . G SuperSpace_{\mathbb{Q}} = PSh(Orb_G, SuperSpace_{\mathbb{Q}}) . This is the homotopy theory where the work in our preprint takes place! We use Elmendorf’s theorem to get our hands on what physicists call “black branes”. These turn out to be the fixed point subspaces X H X^H , for X X a particular rational superspace equipped with an action.

To close, let me ask if you or anyone you know has a nice conceptual explanation for Elmendorf’s theorem, or at the very least for the equivariant Whitehead theorem: