Contents

Idea

For $G$ a topological group acting on a topological space $X$, its Borel construction or Borel space is another topological space $X \times_G E G$, also known as the homotopy quotient. In many cases, its ordinary cohomology is the $G$-equivariant cohomology of $X$.

Definition

For $X$ a topological space, $G$ a topological group and $\rho\colon G \times X \to X$ a continuous $G$-action (a topological G-space), the Borel construction of $\rho$ is the topological space $X \times_G E G$, hence quotient of the product of $X$ with the total space of the $G$-universal principal bundle $E G$ by the diagonal action of $G$ on both.

Analogously, for $\mathcal{G}$ a simplicial group and $\mathcal{G} \times X \xrightarrow{\;\;} X$ a simplicial group action, its Borel construction is the quotient

(1)$\frac { W \mathcal{G} \times X} {\mathcal{G}} \;\;\; \in \;\; SimplicialSets$

of the Cartesian product of $X$ with the universal principal simplicial complex $W \mathcal{G}$ by the diagonal action of $\mathcal{G}$ on these.

Properties

Homotopy-theoretic properties

In simplicial sets

Proposition

The simplicial Borel construction (1) is a right Quillen functor in a Quillen equivalence

$\big( W \mathcal{G} \times (-) \big) \;\; \colon \;\; \mathcal{G}Actions(sSet)_{proj} \xrightarrow{\;\;\;} \big(sSet_{Qu}\big)_{\overline W \mathcal{G}}$

from the projective model structure on simplicial group actions to the slice model structure of the classical model structure on simplicial sets over the simplicial classifying space of $\mathcal{G}$.

(DDK 80, Prop. 2.3, Prop. 2.4, see at Borel model structure for more)

As an example:

Example

If the underlying simplicial set $X \in sSet$ is a Kan complex, then the quotient projection from the Borel construction (1) to the simplicial classifying space is a Kan fibration

$\big( W\mathcal{G} \times X \big) \big/ \mathcal{G} \xrightarrow{\in Fib} \overline{W}\mathcal{G} \,,$

This follows because the morphism is the image of the terminal morphism $X \xrightarrow{\;} \ast$ under the right Quillen functor from Prop. , hence is a fibration if $X \to \ast$ is.

For generalization of this statement to simplicial presheaves see NSS 12b, Thm. 3.91.

In topological spaces

In the following, we write $TopSp$ for the convenient category of compactly generated weak Hausdorff spaces.

Let

Proposition

The topological Borel construction $(X \times E G)/G$ fits into a homotopy fiber sequence of the form

(2)$X \xrightarrow{\;\;} \frac{ X \times EG }{ G } \xrightarrow{\;\;} B G \,.$

Proof

The sequence (2) is the image under geometric realization of simplicial topological spaces of the sequence of simplicial topological spaces

(3)$const(X)_\bullet \xrightarrow{\;\;\;} X \times G^{\times^\bullet} \xrightarrow{\;\;\;} G^{\times^\bullet}$

which, in turn, is the image under passage to nerves of the sequence of topological groupoids

$(X \rightrightarrows X) \xrightarrow{\;\;} (X \times G \rightrightarrows X) \xrightarrow{\;\;} (G \rightrightarrows \ast) \,,$

with the action groupoid in the middle and the delooping groupoid on the right.

Therefore, by the general respect of geometric realization for homotopy fiber products (this Prop.) it is now sufficient to observe that

1. the simplicial topological spaces appearing in (3) are all good (so that their geometric realization models their homotopy colimit);

2. the sequence (3) of simplicial topological spaces is degreewise a homotopy fiber sequence of topological spaces;

3. the morphism $X \times G^{\times^\bullet} \to G^{\times^\bullet}$ is a homotopical Kan fibration (see here).

For the first condition observe that the degeneracy maps of the nerve of an action groupoid are all of the form $X \times G^{n} \times ( \{e\} \xhookrightarrow{\;} G)$, hence are closed cofibrations (by this Example).

For the second condition, observe that projections $X \times G^n \to G^n$ are evidently Serre fibrations whose ordinary fiber is $X$, so that $X \to X \times G^n \to G^n$ is a homotopy fiber sequence (by this Prop.).

For the third point, notice that in the present case the horn-filling map here are all isomorphisms, hence certainly surjective on connected components.

Proposition

(Borel construction on principal bundle is equivalent to plain quotient)
Assume that:

• $G$ and $X$ are both connected, $\pi_0 = \ast$;

• the coprojection $X \xrightarrow{\;q\;} X/G$ is a locally trivial $G$-fiber bundle (hence a $G$-principal bundle),

then the topological Borel construction is weakly homotopy equivalent to the plain quotient space:

$(X \times E G)/G \xrightarrow{ \in \,\mathrm{W}_{whe} } X/G \,.$

Proof

We need to show that the comparison morphism induces an isomorphisms on all homotopy groups $\pi_n$.

For $n = 0$ this is immediate, as passage to connected components is a coequalizer (here) and using that colimits commute with colimits. This implies that both $\pi_0(X/G)$ as well as $\pi_0\big( (X \times E G) / G\big)$ are the singleton set (by the assumption that $X$ is connected and since $E G$ is connected) and hence are isomorphic.

In order to compare the remaining higher homotopy groups, observe that, under the given assumption, with $X \to X/G$ also the coprojection $X \times E G \xrightarrow{ \;\; } (X \times E G)/G$ to the Borel construction itself is a locally trivial fiber bundle: Namely from this commuting diagram

one sees that for $\big\{ U_i \subset X/G\big\}_{i \in I}$ an open cover of $X/G$ over which $X \to X/G$ trivializes, we have that $\big\{ U_i \times E G \,\subset\, (X \times E G)/G \}_{i \in I}$ is an open cover over which the quotient coprojection into the Borel construction trivializes.

But since locally trivial bundles are Serre fibrations (this Prop.), it follows that the above diagram is a morphism of homotopy fiber sequences (by this Prop.), and hence induces a morphism of long exact sequences of homotopy groups, which contains the following segments, for all $n \in \mathbb{N}$:

Here the outer vertical morphisms are all isomorphisms by the fact that $E G \xrightarrow{\;} \ast$ is a weak homotopy equivalence. Therefore the five lemma (in its generality of possibly non-abelian groups, here) implies that also the middle vertical morphism is an isomorphism, for all $n$.

As the realization of the action groupoid

This Borel construction is naturally understood as being the geometric realization of the topological action groupoid $X // G$ of the action of $G$ on $X$:

the nerve of this topological groupoid is the simplicial topological space

$(X \sslash G)_\bullet = \left( \cdots X \times G \times G \stackrel{\to}{\stackrel{\to}{\to}} X \times G \stackrel{\overset{\rho}{\to}}{\underset{p_1}{\to}} X \right) \,.$

Observing that $E G = G \sslash G$ itself as a groupoid has the nerve

$(E G)_\bullet = \left( \cdots G \times G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \times G \stackrel{\overset{\cdot}{\to}}{\underset{p_1}{\to}} G \right)$

(where “$\cdot$” denotes the multiplication action of $G$ on itself) and regarding $X$ and $G$ as topological 0-groupoids ($G$ as a group object in topological 0-groupoids), hence with simplicially constant nerves, we have an isomorphism of simplicial topological spaces

$(X //G)_\bullet \simeq_{iso} X \times_G (E G)_\bullet \,.$

If this is set up in a sufficiently nice category of topological spaces, then, by the discussion at geometric realization of simplicial topological spaces, the geometric realization ${\vert{-}\vert}\colon Top^{\Delta^{op}} \to Top$ manifestly takes this to the Borel construction (since, by the discussion there, it preserves the product and the quotient).

As a homotopy colimit over the category associated to $G$

If $G$ is the topological category associated to the group $G$, then a $G$-space is precisely a Top-enriched functor $G\to Top$ in a similar fashion to the fact that an R-module is an Ab-enriched functor. If $X$ is a $G$-space, the ordinary quotient $X/G$ is the colimit of the diagram associated to $X$ and the Borel construction is (a model of) the homotopy colimit of that diagram. This is a reason for calling the Borel construction homotopy quotient in some contexts.

In rational homotopy theory

The image of the Borel construction in rational homotopy theory is the Weil model for equivariant de Rham cohomology. See there for more.

Literature

Lecture notes in classical topology:

General statement in simplicial homotopy theory of simplicial presheaves:

The nature of the Borel construction as the geometric realization of the action groupoid is mentioned for instance in