Editing Hurewicz theorem (section)

===Absolute version===
For any [[Path connected|path-connected]] space ''X'' and positive integer ''n'' there exists a [[group homomorphism]]

:<math>h_* \colon \pi_n(X) \to H_n(X),</math>

called the '''Hurewicz homomorphism''', from the ''n''-th [[homotopy group]] to the ''n''-th [[Homology (mathematics)|homology group]] (with integer coefficients).  It is given in the following way: choose a canonical generator <math>u_n \in H_n(S^n)</math>, then a homotopy class of maps <math>f \in \pi_n(X)</math> is taken to <math>f_*(u_n) \in H_n(X)</math>.

The Hurewicz theorem states cases in which the Hurewicz homomorphism is an [[group isomorphism|isomorphism]].

* For <math>n\ge 2</math>, if ''X'' is [[N-connected|<math>(n-1)</math>-connected]] (that is: <math>\pi_i(X)= 0</math> for all <math>i < n</math>), then <math>\tilde{H_i}(X)= 0</math> for all <math>i < n</math>, and the Hurewicz map <math>h_* \colon \pi_n(X) \to H_n(X)</math> is an isomorphism.<ref name=":0">{{citation |last=Hatcher |first=Allen |title=Algebraic Topology |page= |year=2001 |publisher=[[Cambridge University Press]] |isbn=978-0-521-79160-1 |author-link=Allen Hatcher}}</ref>{{Rp|page=366|location=Thm.4.32}} This implies, in particular, that the [[homological connectivity]] equals the [[homotopical connectivity]] when the latter is at least 1. In addition, the Hurewicz map <math>h_* \colon \pi_{n+1}(X) \to H_{n+1}(X)</math> is an [[epimorphism]] in this case.<ref name=":0" />{{Rp|page=390|location=?}}
* For <math>n=1</math>, the Hurewicz homomorphism induces an [[group isomorphism|isomorphism]] <math>\tilde{h}_* \colon \pi_1(X)/[ \pi_1(X), \pi_1(X)] \to H_1(X)</math>, between the [[Commutator subgroup|abelianization]] of the first homotopy group (the [[fundamental group]]) and the first homology group.