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 (with integer coefficients). It is given in the following way: choose a canonical generator formula_2, then a homotopy class of maps formula_3 is taken to formula_4.
For formula_5 this homomorphism induces an isomorphism 
between the abelianization of the first homotopy group (the fundamental group) and the first homology group.
If formula_7 and "X" is formula_8-connected, the Hurewicz map formula_9 is an isomorphism. In addition, the Hurewicz map formula_10 is an epimorphism in this case.
Relative version.
For any pair of spaces formula_11 and integer formula_12 there exists a homomorphism
from relative homotopy groups to relative homology groups. The Relative Hurewicz Theorem states that if both formula_14 and formula_15 are connected and the pair is formula_8-connected then formula_17 for formula_18 and formula_19 is obtained from formula_20 by factoring out the action of formula_21. 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 formula_23 denotes the cone of formula_15. This statement is a special case of a homotopical excision theorem, involving induced modules for formula_25 (crossed modules if formula_26), 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 formula_27 (i.e., a space "X" and subspaces "A", "B") and integer formula_28 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 formula_31 are connected, the pairs formula_32 and formula_33 are formula_34-connected and formula_35-connected, respectively, and the triad formula_27 is formula_37-connected, then formula_38 for formula_39 is obtained from formula_40 by factoring out the action of formula_41 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 formula_42-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 formula_43 for formula_44. Then the Hurewicz map 
induces an isomorphism for formula_46 and a surjection for formula_47.