Hurewicz theorem

In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is due to Witold Hurewicz.

Statement of the theorem

For any n-connected CW-complex or Kan complex X and integer k ≥ 1 such that n ≥ 0, there exists a homomorphism

$h_{*}:\pi _{k}(X)\rightarrow {\tilde {H}}_{k}(X)$

called the Hurewicz homomorphism from homotopy to reduced homology (with integer coefficients), which turns out to be isomorphic to the canonical abelianization map

$\pi _{1}(X)\rightarrow \pi _{1}(X)/[\pi _{1}(X),\pi _{1}(X)]\,$

if k = 1. The Hurewicz theorem states that under the above conditions, the Hurewicz map is an isomorphism if k = n + 1 and an epimorphism if k = n + 2.

In particular, if the first homotopy group (the fundamental group) is nonabelian, this theorem says that its abelianization is isomorphic to the first reduced homology group:

$\pi _{1}(X)/[\pi _{1}(X),\pi _{1}(X)]\cong {\tilde {H}}_{1}(X).$

The first reduced homology group therefore vanishes if π₁ is perfect and X is connected.