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.

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_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_1(X).]

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

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