# 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.

## Statement of the theorems

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

### Absolute version

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

$h_{\ast }\colon \,\pi _{k}(X)\to H_{k}(X)\,\!$ 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

$h_{\ast }\colon \,\pi _{1}(X)\to \pi _{1}(X)/[\pi _{1}(X),\pi _{1}(X)].\,\!$ 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:

$H_{1}(X)\cong \pi _{1}(X)/[\pi _{1}(X),\pi _{1}(X)].\,\!$ The first homology group therefore vanishes if X is path-connected and π1(X) is a perfect group.

### Relative version

For any pair of spaces (X,A) and integer k > 1 there exists a homomorphism

$h_{\ast }\colon \pi _{k}(X,A)\to H_{k}(X,A)\,\!$ from relative homotopy groups to relative homology groups. The Relative Hurewicz Theorem states that if each of X, A are connected and the pair (X,A) is (n−1)-connected then Hk(X,A) = 0 for k < n and Hn(X,A) is obtained from πn(X,A) by factoring out the action of π1(A). This is proved in, for example, Template:Harvtxt by induction, proving in turn the absolute version and the Homotopy Addition Lemma.

This relative Hurewicz theorem is reformulated by Template:Harvtxt as a statement about the morphism

$\pi _{n}(X,A)\to \pi _{n}(X\cup CA)\,\!.$ This statement is a special case of a homotopical excision theorem, involving induced modules for n>2 (crossed modules if n=2), which itself is deduced from a higher homotopy van Kampen theorem for relative homotopy groups, whose proof requires development of techniques of a cubical higher homotopy groupoid of a filtered space.

For any triad of spaces (X;A,B) (i.e. space X and subspaces A,B) and integer k > 2 there exists a homomorphism

$h_{\ast }\colon \pi _{k}(X;A,B)\to H_{k}(X;A,B)\,\!$ from triad homotopy groups to triad homology groups. Note that Hk(X;A,B) ≅ Hk(X∪(C(AB)). The Triadic Hurewicz Theorem states that if X, A, B, and C = AB are connected, the pairs (A,C), (B,C) are respectively (p−1)-, (q−1)-connected, and the triad (X;A,B) is p+q−2 connected, then Hk(X;A,B) = 0 for k < p+q−2 and Hp+q−1(X;A) is obtained from πp+q−1(X;A,B) by factoring out the action of π1(AB) and the generalised Whitehead products. The proof of this theorem uses a higher homotopy van Kampen type theorem for triadic homotopy groups, which requires a notion of the fundamental catn-group of an n-cube of spaces.

### Simplicial set version

The Hurewicz theorem for topological spaces can also be stated for n-connected simplicial sets satisfying the Kan condition.

### Rational Hurewicz theorem

Rational Hurewicz theorem: Let X be a simply connected topological space with $\pi _{i}(X)\otimes \mathbb {Q} =0$ for $i\leq r$ . Then the Hurewicz map

$h\otimes \mathbb {Q} :\pi _{i}(X)\otimes \mathbb {Q} \longrightarrow H_{i}(X;\mathbb {Q} )$ 