nLab
Borel construction

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Representation theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

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.

Definition

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.

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//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.

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.

References

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

Last revised on June 28, 2019 at 11:09:28. See the history of this page for a list of all contributions to it.