Borel construction



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


For XX a topological space, GG a topological group and ρ:G×XX\rho\colon G \times X \to X a continuous GG-action, the Borel construction of ρ\rho is the topological space X× GEGX \times_G E G, hence quotient of the product of XX with the total space of the GG-universal principal bundle EGE G by the diagonal action of GG on both.


As the realization of the action groupoid

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

the nerve of this topological groupoid is the simplicial topological space

(X//G) =(X×G×GX×Gp 1ρX). (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//GE G = G//G itself as a groupoid has the nerve

(EG) =(G×G×GG×Gp 1G) (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 GG on itself) and regarding XX and GG as topological 0-groupoids (GG as a group object in topological 0-groupoids), hence with simplicially constant nerves, we have an isomorphism of simplicial topological spaces

(X//G) isoX× 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 ||:Top Δ opTop{\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 GG

If GG is the topological category associated to the group GG, then a GG-space is precisely a Top-enriched functor GTopG\to Top in a similar fashion to the fact that an R-module is an Ab-enriched functor. If XX is a GG-space, the ordinary quotient X/GX/G is the colimit of the diagram associated to XX 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.

  • Borel-equivariant cohomology?


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)

Revised on April 14, 2014 05:49:58 by Urs Schreiber (