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 $X{\times }_{G}EG$, 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 $\rho :G\times X\to X$ a continuous $G$-action, the Borel construction of $\rho$ is the topological space $X{\times }_{G}EG$, hence quotient of the product of $X$ with the total space of the $G$-universal principal bundle $EG$ 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

$$(X//G{)}_{•}=(\cdots X\times G\times G\overrightarrow{\overrightarrow{\to }}X\times G\stackrel{\stackrel{\rho }{\to }}{\underset{p_1}{\to }}X).$$(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 $EG=G//G$ itself as a groupoid has the nerve

$$(EG{)}_{•}=(\cdots G\times G\times G\overrightarrow{\overrightarrow{\to }}G\times G\stackrel{\stackrel{\cdot }{\to }}{\underset{p_1}{\to }}G)$$(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

$$(X//G{)}_{•}{\simeq }_{\mathrm{iso}}X{\times }_G(EG{)}_{•}.$$(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 $\mid -\mid :{\mathrm{Top}}^{{\Delta }^{\mathrm{op}}}\to \mathrm{Top}$ 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 $G\to \mathrm{Top}$ 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.

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)