## June 27, 2018

### Elmendorf’s Theorem

#### Posted by John Huerta 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 $G$-space $X$, the equivariant homotopy type of $X$ is determined by the ordinary homotopy types of the fixed point subspaces $X^H$, where $H$ runs over all subgroups of $G$. 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 $G$-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 $G$ be a compact Lie group. For $G$-spaces, we can also define both strong and weak homotopy equivalence. The strong homotopy equivalence is the obvious thing: you have two equivariant maps $f \colon X \to Y$ and $g \colon Y \to X$, that are inverse to each other up to equivariant homotopies $\eta \colon f g \Rightarrow 1_Y$ and $\eta' \colon g f \Rightarrow 1_X$. This lets us consider $G$-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 $G$-equivariant map $f \colon X \to Y$ is a equivariant weak homotopy equivalence if it restricts to an ordinary weak homotopy equivalence between the fixed points spaces, $f \colon X^H \to Y^H$, for all closed subgroups $H \subseteq G$.

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 $G$-CW-complex is. We can construct them much as we do ordinary CW-complexes, except they are built from cells of the form:

$D^n \times G/H$

where $D^n$ is the $n$-disk with the trivial $G$ action, and $G/H$ is a coset space of $G$ with the left $G$ action. These cells are then glued together by $G$-equivariant attaching maps, just like an ordinary CW-complex. The result is a $G$-CW-complex. The equivariant Whitehead theorem, due to Bredon, then says that for any pair of $G$-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 $G$-spaces simply by looking at $X^H$ for all closed subgroups $H \subseteq G$. But this operation, of taking a subgroup $H$ to a space $X^H$, actually defines a functor:

$X \colon Orb_G^{op} \to Spaces .$

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

• objects the coset spaces $G/H$, for each closed subgroup $H \subseteq G$.
• morphisms the $G$-equivariant maps.

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

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

$y \colon G Spaces \to PSh(Orb_G, Spaces)$

from the category $G Spaces$ of $G$-spaces into the category of all presheaves on $Orb_G$. This is a souped up version of the Yoneda embedding: $Orb_G$ is a subcategory of $G Spaces$, 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 $X$ and $Y$ are weak equivalent if there is a natural transformation $f \colon X \Rightarrow Y$ whose components $f^H \colon X^H \to Y^H$ are ordinary weak equivalences of spaces. With this choice of weak equivalences, the homotopy theory of presheaves on $Orb_G$ is the same as that of $G Spaces$. That’s Elmendorf’s theorem:

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

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:

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 $\mathcal{C}$ with weak equivalences, then you might go ahead and define the equivariant homotopy theory of $\mathcal{C}$ to be: $G \mathcal{C} = PSh(Orb_G, \mathcal{C})$ where the weak equivalences are the levelwise weak equivalences, as in Elmendorf.

For instance, if $\mathcal{C}$ is a model of rational homotopy theory $Spaces_{\mathbb{Q}}$, then $G$-equivariant rational homotopy ought to be: $PSh(Orb_G, Spaces_{\mathbb{Q}}) .$ This is precisely what one finds in the literature, at least in the case when $G$ 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 $SuperSpace_{\mathbb{Q}}$. Then the $G$-equivariant rational super homotopy theory ought to be: $G SuperSpace_{\mathbb{Q}} = PSh(Orb_G, SuperSpace_{\mathbb{Q}}) .$ 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 $X^H$, for $X$ 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:

Question. What is an intuitive reason that equivariant homotopy types are captured by the homotopy types of their fixed point subspaces?

Posted at June 27, 2018 7:03 PM UTC

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### Re: Elmendorf’s theorem

In the case where $G$ is finite and the quotient $X/G$ is an orbifold, Elmendorf’s theorem seems to amount to the fact that the $G$-homotopy type of $X$ is equivalent to the homotopy type of all of the orbifold strata of $X/G$. I’m not sure if this counts as “intuitively motivating” Elmendorf or just a rephrasing it in the case where I can see things geometrically, but at least I can contemplate concrete examples:

Consider the unit circle $S^1\subset\mathbb{R}^2$ with $\mathbb{Z}/2\mathbb{Z}$ acting on it by reflection across the $y$-axis. Then, Elmendorf says that its $\mathbb{Z}/2\mathbb{Z}$-homotopy type is determined by the ordinary homotopy types of the fixed point subspaces $(S^1)^{\mathbb{Z}/2\mathbb{Z}}$ (which consists of the two points $(1,0)$ and $(-1,0)$ with nontrivial isotropy) and $S^1$ itself (the fixed point set of the identity subgroup). Furthermore, a $\mathbb{Z}/2\mathbb{Z}$-equivariant map $f$ between $S^1$ and some other $\mathbb{Z}/2\mathbb{Z}$-space $Y$ can be viewed in this way as well; that is, in addition to being continuous as a map between $S^1$ and $Y$, $f$ must also be a continuous map between the $\mathbb{Z}/2\mathbb{Z}$-fixed point set of $Y$ and $\{(1,0),(0,1)\}$. Then it kind of makes sense that for weak $\mathbb{Z}/2\mathbb{Z}$-homotopy equivalence, one simply considers weak homotopy equivalence of these maps of strata.

Posted by: j.c. on June 27, 2018 10:18 PM | Permalink | Reply to this

### Re: Elmendorf’s theorem

That’s definitely one of the main geometric conceptions we use in our paper: we think about orbifolds of spacetime by finite group actions. There are lots of ideas in the string theory literature about this. The key insight was to go the opposite direction from what you’re suggesting here, and start understanding these orbifolds as statements about equivariant homotopy types.

Though I like this way of thinking, I don’t feel like it gives me a deeper understanding of Elmendorf “in my bones”, so to speak.

Posted by: John Huerta on June 28, 2018 4:27 PM | Permalink | Reply to this

### Re: Elmendorf’s theorem

To me, the theorem starts to feel more natural when I start thinking of $G$-CW-complexes as a rather “restrictive” class of $G$-spaces.

For instance, suppose that $M$ is a (smooth) $G$-manifold (with smooth $G$-action). Then by equivariant Morse theory, we can decompose $M$ into a bunch of equivariant disk bundles over orbits $G/H$. But it’s not immediately obvious how to turn this into a $G$-CW-structure. So maybe I’ll pose this as a

Question: Is every $G$-manifold a $G$-CW-complex?

Posted by: Tim Campion on June 27, 2018 10:58 PM | Permalink | Reply to this

### Re: Elmendorf’s theorem

Of course, by the equivariant Whitehead theorem, every $G$-space is weakly $G$-homotopy equivalent to a $G$-CW complex. So the word “is” in my question means “up to $G$-homeomorphism” – I suppose it’s already interesting to ask whether it’s true up to strong $G$-homotopy equivalence.

Posted by: Tim Campion on June 27, 2018 11:01 PM | Permalink | Reply to this

### Re: Elmendorf’s theorem

My first instinct if someone hands me an $n$-disk bundle is to pass to a local trivialization, say on the cover $\{U_{\alpha}\}$, so that I can work with the trivial bundle $U_\alpha \times D^n$. But the cover is breaking up the the orbit space $G/H$, so this is not yet of the desired form $G/H \times D^n$. Hmmm…

Posted by: John Huerta on June 28, 2018 11:03 PM | Permalink | Reply to this

### Re: Elmendorf’s theorem

Denis Nardin points out that an affirmative answer for smooth $G$-manifolds (in the $G$-homeomorphism version) is Corollary 7.2 here – actually one gets triangulations here, not just $G$-CW structures.

Posted by: Tim Campion on June 29, 2018 3:22 PM | Permalink | Reply to this

### Re: Elmendorf’s theorem

I would argue that the reason fixed-point homotopy groups are enough has to do with the way the definition of $G$-CW-complex is rigged up.

First, it’s really easy to prove that a $G$-CW-inclusion is a weak $G$-homotopy equivalence iff it is a strong $G$-deformation retract: just induct on cells: if you attach a cell $G/H \times D^n$, the trivialization of the relative $H$-fixed-point homotopy group essentially hands you the data of the $G$-deformation retract because classes of $\pi_n(X^H)$ are essentially classes of maps $G/H \times S^n \to X$.

Now if you’re going to construct a model structure where the relative $G$-CW complexes are the cofibrations, your trivial fibrations are going to have the right lifting property against maps $G/H \times S^n \to G/H \times D^n$ – this is even better than being a weak $G$-homotopy equivalence – you can lift representatives of classes of $\pi_n(X^H)$ on the nose.

The yoga of model categories tells us that if we understand weak equivalences which are cofibrations and weak equivalences which are fibrations, then we understand all weak equivalences. Since both make sense in terms of weak $G$-equivalences, weak $G$-equivalences should be all there is to it.

Precisely, Jeff Smith’s theorem tells us (after replacing $Top$ with a convenient locally presentable category of spaces) that there is a model structure with

• cofibrations given by retracts of relative $G$-CW complexes and

• weak equivalences given by weak $G$-equivalences

as soon as we check that

• weak $G$-equivalences satisfy 2/3 (check!)

• every morphism with the right lifting property with respect to maps $G/H \times S^n \to G/H \times D^{n+1}$ is a weak $G$-equivalence (we just observed that even more is true – check)

• the weak $G$-equivalences which are relative $G$-CW complexes are closed under pushout and colimits of chains. The pushout part holds because these maps are strong $G$-deformation retracts, which are clearly closed under pushout. The chain part holds by compactness. (check.)

and a technical accessiblity condition on the $G$-weak equivalences which is clear.

The existence of such a model structure then gives us everything we want by the yoga of model categories.

Posted by: Tim Campion on June 28, 2018 12:39 AM | Permalink | Reply to this

### Re: Elmendorf’s theorem

Yes, that’s how I look at it. The appearance of closed subgroups in the notion of weak homotopy equivalence only seems magical until you realize they’re built into the notion of “$G$-CW-complex” for which the notion of weak homotopy equivalence is tailored to make the Whitehead theorem true. In other words, the answer to your question

What is an intuitive reason that equivariant homotopy types are captured by the homotopy types of their fixed point subspaces?

is that that’s basically how people have defined “equivariant (weak) homotopy type”.

If I recall correctly, you can even twiddle the knob: if you chose any well-behaved subclass $\mathcal{H}$ of closed subgroups of $G$, you can define a notion of $\mathcal{H}$-CW-complex and a notion of $\mathcal{H}$-weak equivalence that are connected by an $\mathcal{H}$-Whitehead theorem, and prove an Elemendorf’s theorem for diagrams on the category of $\mathcal{H}$-orbits. When $\mathcal{H}$ contains only the trivial subgroup, then you get the “levelwise” model structure that arises by considering $G Top$ as the diagram category $Top^{B G}$.

Personally, the conclusion I usually draw from Elmendorf’s theorem (pace my advisor) is that just as classical homotopy theory is often “really” the study of $\infty$-groupoids, not topological spaces, equivariant homotopy theory is often “really” the study of diagrams of $\infty$-groupoids over orbit categories. Topological spaces are important to classical homotopy theory mainly because they have fundamental $\infty$-groupoids; it turns out somewhat accidentally that (at least in classical foundations) every $\infty$-groupoid can be presented by some topological space, and so we can if we wish work entirely with those presentations. Similarly, a space with a $G$-action has a “fundamental orbit diagram of $\infty$-groupoids”, and it turns out somewhat accidentally that every such diagram can be presented by a $G$-space, so that we can if we wish work entirely with the latter as presentations too.

Posted by: Mike Shulman on June 28, 2018 5:00 PM | Permalink | Reply to this

### Re: Elmendorf’s theorem

…equivariant homotopy theory is often “really” the study of diagrams of ∞-groupoids over orbit categories…

Generalising those orbit categories is where Barwick et al. seem to be coming from in Parametrized higher category theory and higher algebra: A general introduction in looking to “untether equivariant homotopy theory from dependence upon a group”, generalising to variation over different categories.

There are Elmendorf theorems for these, such as mentioned on p. 8:

In , Farjoun builds on work of  and defines a model structure on the category of diagrams of spaces indexed on a small category I, called the I-equivariant model structure, which depends on the “I-orbits”: the diagrams $I \to Top$ whose strict (= 1-categorical) colimit is equal to a point. In particular if $I = G$ is a group these are precisely the $G$-orbits, and the resulting homotopy theory is the fixed-points model structure on $G$-spaces. Moreover Farjoun’s construction admits an Elmendorf-McClure theorem…

Posted by: David Corfield on June 28, 2018 8:09 PM | Permalink | Reply to this

### Re: Elmendorf’s theorem

Also related is Guillou-May’s Enriched model categories and presheaf categories in which they study categories $M$ equipped with an inclusion $\delta : D\to M$ with $D$ small, and ask when $M$ can be given a model structure so that the restricted Yoneda embedding $M \to Pre(D)$ is a Quillen equivalence.

Posted by: Mike Shulman on June 28, 2018 10:11 PM | Permalink | Reply to this

### Re: Elmendorf’s theorem

Mike wrote:

If I recall correctly, you can even twiddle the knob: if you chose any well-behaved subclass $\mathcal{H}$ of closed subgroups of $G$, you can define a notion of $\mathcal{H}$-CW-complex…

Should one think of this homotopy theory as the theory of $G$-spaces with orbits of type $\mathcal{H}$?

Personally, the conclusion I usually draw from Elmendorf’s theorem (pace my advisor) is that just as classical homotopy theory is often “really” the study of $\infty$-groupoids, not topological spaces, equivariant homotopy theory is often “really” the study of diagrams of $\infty$-groupoids over orbit categories.

That’s very $n$POV of you, and I am sure it’s morally right. But it prompted me to reformulate my question as follows: How about if we let $n = 1$, and ask about fundamental groupoids before going up to $n = \infty$. Why should a $G$-equivariant groupoid be the same as a diagram of groupoids over the orbit spaces?

Posted by: John Huerta on June 29, 2018 10:21 AM | Permalink | Reply to this

### Re: Elmendorf’s theorem

A partial answer to your question, John, may be found in the paper ‘A van Kampen theorem for equivariant fundamental groupoids’ by Manuel Bullejos, and Laura Scull, published in Journal of Pure and Applied Algebra 212 (2008) 2059 - 2068

Posted by: Tim Porter on June 29, 2018 2:33 PM | Permalink | Reply to this

### Re: Elmendorf’s theorem

Posted by: John Huerta on June 29, 2018 3:59 PM | Permalink | Reply to this

### Re: Elmendorf’s theorem

Should one think of this homotopy theory as the theory of $G$-spaces with orbits of type $\mathcal{H}$?

Well, if one is thinking of the case $\mathcal{H} =$ all closed subgroups as “really” being the homotopy theory of diagrams on the orbit category, as I suggested, then one would similarly think of this as being the homotopy theory of diagrams on the restricted category of orbits $G/H$ for $H\in\mathcal{H}$.

Another perspective on equivariant homotopy theory can be derived from the observation in Univalence for inverse EI diagrams that (for a compact Lie group $G$) the opposite of the orbit category is an inverse EI $(\infty,1)$-category. This means roughly that a diagram over it can be regarded as built up in stages from spaces with homotopy actions by $\infty$-groups (i.e. with the levelwise homotopy theory) that depend on each other in a specified way.

At the base is the subgroup $e\le G$, so that the first space $X_e$ has a homotopy action of $G/e = G$ itself. At the top is the subgroup $G\le G$, so that the space $X_G$ at the top has a homotopy action of the group $G/G=e$, i.e. no action at all. Since there is exactly one map of orbits $G/e \to G/G$, the dependence is that $X_G$ depends on $Hom_G(1, X_e)$, the homotopy fixed points of the $G$-action on $X_e$.

Thus, if $\mathcal{H}=\{G,e\}$ (and in particular if these are the only subgroups of $G$, such as if $G = C_p$), then an equivariant $G$-space consists of a space $X_e$ with a homotopy action by $G$, together with a specified “space of fixed points” of this action, such that every “specified fixed point” has an underlying actual (homotopy) fixed point. In other words, when passing from the levelwise homotopy theory of $G$-spaces to the equivariant homotopy theory, we make “being a fixed point” from a property into a structure (or I suppose more precisely a stuff). And when $\mathcal{H}$ is bigger than this, there is some kind of interpolation in between going on.

There are, of course, plenty of other cases where categorifying makes property into structure/stuff, e.g. in a monoid, associativity is a property, whereas in a monoidal category or $A_\infty$-space, we have to specify coherent associativity data. So this is one way to argue that equivariant homotopy theory really is a categorification of “ordinary” group actions, although I think it’s a bit unusual in that a $G$-space which is 0-truncated is still more general than an ordinary $G$-set (a diagram of sets on the orbit category need not be the diagram of fixed point sets of an ordinary $G$-set).

Why should a $G$-equivariant groupoid be the same as a diagram of groupoids over the orbit spaces?

For the same reasons as in the $\infty$-case. The category of groupoids with (strict) $G$-action, and strict $G$-morphisms between them, should admit multiple model structures parametrized by classes $\mathcal{H}$ of subgroups of $G$, etc. etc.

The only difference is that in this case we also have the 2-category of $G$-groupoids and pseudo $G$-morphisms. This should be equivalent to the homotopy theory of the “levelwise” model category of strict morphisms (the one where $\mathcal{H}=\{e\}$).

Posted by: Mike Shulman on June 29, 2018 7:34 PM | Permalink | Reply to this

### Re: Elmendorf’s theorem

So for all suitable classes of subgroups $\mathcal{H}$ of $G$, the homotopy theory is equivalent to the homotopy theory of $G$-spaces?

I guess this does make a lot of sense for the class $\mathcal{H} = \{ 1, G\}$, consisting of the trivial subgroup and $G$ itself. Then one is studying the homotopy theory of

$PSh(\mathcal{C}, Spaces)$

with the objectwise weak equivalences (“levelwise” seems to sound wrong to the ears of commenters here). Here, $\mathcal{C}$ is the full subcategory of $Orb_G$ on the objects $G/G$ and $G/1$. A presheaf $X$ indeed consists of a pair of spaces: X^G = X(G/G) and $X^1 = X(G/1)$. The morphisms $G/1 \to G/1$ become a $G$-action on $X^1$, with fixed points $X^G \subseteq X^1$. So, this is the homotopy theory of $G$-spaces once again.

Posted by: John Huerta on July 3, 2018 9:37 AM | Permalink | Reply to this

### Re: Elmendorf’s theorem

Well, I failed in a few minutes to find a citation for this, so I could be wrong; most of my reference books are in deep storage this summer due to construction on our building. But I believe that the model category on $G Top$ generated by orbits $G/H$ for all $H\in\mathcal{H}$ should be equivalent to the objectwise homotopy theory (e.g. projective, injective, or EI-Reedy model structure) on the category of orbits $G/H$ for all $H\in\mathcal{H}$, for any $\mathcal{H}$.

Posted by: Mike Shulman on July 3, 2018 5:00 PM | Permalink | Reply to this

### Re: Elmendorf’s theorem

Posted by: David Corfield on July 3, 2018 11:14 PM | Permalink | Reply to this

### Re: Elmendorf’s theorem

Awesome, thanks! He proves the version for general $\mathcal{H}$ as a Quillen equivalence of model categories.

Posted by: Mike Shulman on July 4, 2018 12:37 AM | Permalink | Reply to this

### Re: Elmendorf’s theorem

Thanks for this nice, detailed answer, Tim! Let me try to distill it into a rough conceptual outline: we are interested in studying the homotopy theory of $G$-spaces glued together from the cells

$G/H \times D^n$

consisting of a $G$ orbit $G/H$ and the contractible space $D^n$. Since taking presheaves on the orbit category is the free cocompletion of that category, it is not surprising that this homotopy theory coincides with

$PSh(Orb_G, Spaces) .$

Cool!

Now let me dig a little more into your comment. I’ll flesh some things out, both for my own edification and for that of anyone who happens to be reading this conversation.

You wrote:

classes of $\pi_n(X^H)$ are essentially classes of maps $G/H \times S^n \to X$.

This is a really key little fact in this game, so let me spell it out. First, there is a bijective correspondence between

• G-equivariant maps $G/H \to X$ and
• $H$-fixed points $x \in X^H$.

This bijection just sends our favorite coset $H \in G/H$ to $x \in X^H$, and conversely. Using this bijection, one obtains a bijective correspondence between:

• $G$-equivariant maps $G/H \times S^n \to X$ and
• maps $S^n \to X^H$.

Your observation tells us to use this bijection to pass between attaching maps in $G$-CW-complexes and classes in $\pi_n(X^H)$. That is indeed a really quick way to show you should start thinking about the homotopy type of $X^H$.

Next, I want to dig into the yoga of model categories a little bit. My model category knowledge is quite elementary. Of the three distinguished classes of morphisms defining a model structure:

• $W$ the weak equivalences
• $C$ the cofibrations
• $F$ the fibrations

I know it is enough to give $W$ and $C$ or $W$ and $F$, because

• fibrations are the maps with the right lifting property against trivial cofibrations,

and dually:

• cofibrations are the maps with the left lifting property against trivial fibrations.

However, you give a characterization of the trivial fibrations:

your trivial fibrations are going to have the right lifting property against maps $G/H \times S^n \to G/H \times D^n$

I feel sure I have seen this before, but I can’t find it at the moment. Can you elaborate?

Posted by: John Huerta on June 29, 2018 10:08 AM | Permalink | Reply to this

### Re: Elmendorf’s theorem

This just uses one of the basic axioms of a model category: trivial fibrations have the right lifting property with respect to cofibrations. Explicitly,

Claim: Assume there’s a model structure on $GTop$ whose cofibrations are the relative $G$-CW complexes. Let $p: X \overset{\sim}{\twoheadrightarrow} Y$ be a trivial fibration; Then $p$ is a weak $G$-equivalence.

Proof:

Injectivity: A fixedpoint homotopy group killed by $p$ can be represented by a commuting square thusly:

$\begin{matrix} G/H \times S^n & \to & X \\ \downarrow & & \downarrow \\ G/H \times D^{n+1} & \to & Y \end{matrix}$

This is a lifting problem with a cofibration against a trivial fibration, so we can solve it: there is a diagonal filler which tells us that the homotopy group was already zero. So $p$ is injective on fixedpoint homotopy.

Surjectivity: A fixedpoint homotopy group in $Y$ can be represented by a commutative square thusly:

$\begin{matrix} \emptyset & \to & X \\ \downarrow & & \downarrow \\ G/H \times S^n & \to & Y \end{matrix}$

Again, this is a lifting problem for a cofibration against a trivial fibration, so there exists a diagonal filler, telling us that the homotopy class has a preimage in $X$.

Remark: In both cases, we really only needed to find a diagonal filler making the diagram commute up to homotopy, but the lifting property actually gave us an on the nose filler, which is even better.

In order to see that there is a model structure where the weak equivalences are exactly the weak equivalences takes a bit more work, as in the argument I gave above using Jeff Smith’s theorem.

Posted by: Tim Campion on June 29, 2018 1:46 PM | Permalink | Reply to this

### Re: Elmendorf’s theorem

This just uses one of the basic axioms of a model category: trivial fibrations have the right lifting property with respect to cofibrations.

Whoops! You’re right! I hadn’t unwound the definition sufficiently to see that. In fact, now that I have unwound the defintion in Emily’s book, I can see a bit more is true: not only do trivial fibrations have the right lifting property with respect to cofibrations, they are equal to the class of maps with this right lifting property.

Let $p: X \overset{\sim}{\twoheadrightarrow} Y$ be a trivial fibration; Then $p$ is a weak $G$-equivalence.

Here, for the sake of people following along, I might rephrase “let $p$ be a trivial fibration, in the sense that it has the right lifting property against cofibrations; Then $p$ is a weak equivalence.” Since you’ve only given the weak equivalences and the cofibrations, we have to derive the fibrations from this information.

Thanks for the super quick proof! It’s lovely.

Posted by: John Huerta on June 29, 2018 3:58 PM | Permalink | Reply to this

### Re: Elmendorf’s Theorem

Perhaps clearer understanding may come from “taking the $G$ out of Genuine”, as Barwick et al. put it in their general introduction to Parametrized higher category theory and higher algebra.

Posted by: David Corfield on June 28, 2018 8:48 AM | Permalink | Reply to this

### Re: Elmendorf’s Theorem

Thanks, David! I haven’t had a chance yet, but eventually I will watch the video of Barwick’s lecture.

Posted by: John Huerta on July 3, 2018 9:42 AM | Permalink | Reply to this

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