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 path-connected space X and positive integer n there exists a group homomorphism called the Hurewicz homomorphism, from the n-th homotopy group to the n-th homology group (with integer coefficients). It is given in the following way: choose a canonical generator, then a homotopy class of maps is taken to. The Hurewicz theorem states cases in which the Hurewicz homomorphism is an isomorphism.

Relative version

For any pair of spaces (X,A) and integer k>1 there exists a homomorphism from relative homotopy groups to relative homology groups. The Relative Hurewicz Theorem states that if both X and A are connected and the pair is (n-1)-connected then H_k(X,A)=0 for k<n and H_n(X,A) is obtained from \pi_n(X,A) by factoring out the action of \pi_1(A). This is proved in, for example, by induction, proving in turn the absolute version and the Homotopy Addition Lemma. This relative Hurewicz theorem is reformulated by as a statement about the morphism where CA denotes the cone of A. 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.

Triadic version

For any triad of spaces (X;A,B) (i.e., a space X and subspaces A, B) and integer k>2 there exists a homomorphism from triad homotopy groups to triad homology groups. Note that The Triadic Hurewicz Theorem states that if X, A, B, and C=A\cap B are connected, the pairs (A,C) and (B,C) are (p-1)-connected and (q-1)-connected, respectively, and the triad (X;A,B) is (p+q-2)-connected, then for k<p+q-2 and is obtained from by factoring out the action of 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 -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 for i\leq r. Then the Hurewicz map induces an isomorphism for and a surjection for i = 2r+1.