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 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
The Hurewicz theorem states cases in which the Hurewicz homomorphism is an isomorphism.
For any pair of spaces
there exists a homomorphism from relative homotopy groups to relative homology groups.
The Relative Hurewicz Theorem states that if both
are connected and the pair is
by factoring out the action of
This is proved in, for example, Whitehead (1978) by induction, proving in turn the absolute version and the Homotopy Addition Lemma.
This relative Hurewicz theorem is reformulated by Brown & Higgins (1981) as a statement about the morphism where
denotes the cone of
This statement is a special case of a homotopical excision theorem, involving induced modules for
(crossed modules if
), 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
(i.e., a space X and subspaces A, B) and integer
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
are connected, the pairs
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.
The Hurewicz theorem for topological spaces can also be stated for n-connected simplicial sets satisfying the Kan condition.
[2] Rational Hurewicz theorem:[3][4] Let X be a simply connected topological space with
Then the Hurewicz map induces an isomorphism for