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 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. It is given in the following way: choose a canonical generator, then a homotopy class of maps is taken to.
For this homomorphism induces an isomorphism
between the abelianization of the first homotopy group and the first homology group.
If and X is -connected, the Hurewicz map is an isomorphism. In addition, the Hurewicz map is an epimorphism in this case.

Relative version

For any pair of spaces and integer there exists a homomorphism
from relative homotopy groups to relative homology groups. The Relative Hurewicz Theorem states that if both and are connected and the pair is -connected then for and is obtained from by factoring out the action of. 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 denotes the cone of. This statement is a special case of a homotopical excision theorem, involving induced modules for , 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 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 and are -connected and -connected, respectively, and the triad is -connected, then for 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. Then the Hurewicz map
induces an isomorphism for and a surjection for.