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 named after Witold Hurewicz, and generalizes earlier results of Henri Poincaré.

The Hurewicz theorems are a key link between homotopy groups and homology groups.

For any space X and positive integer k there exists a group homomorphism

called the Hurewicz homomorphism from the k-th homotopy group to the k-th homology group (with integer coefficients), which for k = 1 and X path-connected is equivalent to the canonical abelianization map

The Hurewicz theorem states that if X is (n − 1)-connected, the Hurewicz map is an isomorphism for all k ≤ n when n ≥ 2 and abelianization for n = 1. In particular, this theorem says that the abelianization of the first homotopy group (the fundamental group) is isomorphic to the first homology group:

The first homology group therefore vanishes if X is path-connected and π₁(X) is a perfect group.

In addition, the Hurewicz homomorphism is an epimorphism from ${\displaystyle \pi _{n+1}(X)\to H_{n+1}(X)}$ whenever X is (n − 1)-connected, for ${\displaystyle n\geq 2}$.

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