Editing Hurewicz theorem (section)

===Relative version===
For any [[topological pair|pair of spaces]] <math>(X,A)</math> and integer <math>k>1</math> there exists a homomorphism

:<math>h_* \colon \pi_k(X,A) \to H_k(X,A)</math>

from relative homotopy groups to relative homology groups. The Relative Hurewicz Theorem states that if both <math>X</math> and <math>A</math> are connected and the pair is <math>(n-1)</math>-connected then <math>H_k(X,A)=0</math> for <math>k<n</math> and <math>H_n(X,A)</math> is obtained from <math>\pi_n(X,A)</math> by factoring out the action of <math>\pi_1(A)</math>. This is proved in, for example, {{Harvtxt|Whitehead|1978}} by induction, proving in turn the absolute version and the Homotopy Addition Lemma.

This relative Hurewicz theorem is reformulated by {{Harvtxt|Brown|Higgins|1981}} as a statement about the morphism 

:<math>\pi_n(X,A) \to \pi_n(X \cup CA),</math>

where <math>CA</math> denotes the [[Cone (topology)|cone]] of <math>A</math>. This statement is a special case of a [[homotopical excision theorem]], involving induced modules for <math>n>2</math> ([[crossed module]]s if <math>n=2</math>), 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.