CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-14

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The classifying space of a discrete group is a K(G,1)

Statement

Assume AC. For a discrete group $G$ , Milnor's $B G$ is connected and

$π_{1} (B G) ≅ G, π_{n} (B G) = 0 (n > 1) .$

Consequently any connected CW model of $B G$ is an Eilenberg--Mac Lane space $K (G, 1)$ .

Facts & Assumptions

[F1]

The loop comparison gives $π_{n} (B G) ≅ π_{n - 1} (G)$ for $n \geq 2$ and identifies $π_{1} (B G)$ with $π_{0} (G)$ (The based loop space of BG recovers G weakly).

[F2]

A discrete group has components indexed by its elements and has zero positive homotopy groups.

[F3]

$E G$ is contractible and its orbit map is surjective (Milnor's join model is a contractible free G-space).

[A1]

AC is inherited exactly from the loop comparison and its numerable-bundle lifting construction (The Axiom of Choice).

Proof

Given: A discrete group $G$ and [A1].

1.1

Assume AC, exactly as required by the loop comparison [F1]. By [F3], $E G$ is path connected. Its continuous surjective image $B G$ is therefore path connected. By [F1, F2], for $n > 1$ we have $π_{n} (B G) ≅ π_{n - 1} (G) = 0$ , and the component part of the same fiber sequence gives $π_{1} (B G) ≅ π_{0} (G) = G$ . With right-action conventions this identification may differ from the chosen concatenation convention by inversion, which is the canonical isomorphism $G^{o p} ≅ G$ .

F1F2F3

2.1

A connected CW model preserves all these homotopy groups. It therefore has fundamental group $G$ and no higher positive homotopy groups, exactly the definition of $K (G, 1)$ . The trivial group gives a contractible connected model and is included. $□$

A1F1step 1.1

Depends on

Used by

Real projective infinity as BZ/2 Example

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

James Davis and Paul Kirk, Lecture Notes in Algebraic Topology (standard reference, not scraped)
Dale Husemoller, Fibre Bundles, Third Edition (standard reference, not scraped)