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Hurewicz theorem
From Wikipedia, the free encyclopedia
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.
For any n-connected CW-complex or Kan complex X and integer k ≥ 1 such that n ≥ 0, there exists a homomorphism
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)]\,](../../../math/9/6/4/964b212804ab8047359ebab2fab8e43b.png)
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).](../../../math/5/d/f/5dffcbd3578fd35c9648ed41c2e59fe9.png)
The first reduced homology group therefore vanishes if π1 is perfect and X is connected.