nLab
Borel construction

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, 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.

Properties

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

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

Observing that $E G = G//G$ 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.

Related concepts

• Borel-equivariant cohomology?

References

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

• Alejandro Adem, Michele Klaus, Lectures on orbifolds and group cohomology (pdf)

• Rick Jardine, Stacks and the homotopy theory of simplicial sheaves (pdf)