Hurewicz theorem

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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.

[edit] 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 and an epimorphism if k = n + 1.

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:

$\pi_1(X)/[\pi_1(X), \pi_1(X)] \cong \tilde{H}_1(X).$