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Hurewicz theorem
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===Triadic version=== For any triad of spaces <math>(X;A,B)</math> (i.e., a space ''X'' and subspaces ''A'', ''B'') and integer <math>k>2</math> there exists a homomorphism :<math>h_*\colon \pi_k(X;A,B) \to H_k(X;A,B)</math> from triad homotopy groups to triad homology groups. Note that :<math>H_k(X;A,B) \cong H_k(X\cup (C(A\cup B))).</math> The Triadic Hurewicz Theorem states that if ''X'', ''A'', ''B'', and <math>C=A\cap B</math> are connected, the pairs <math>(A,C)</math> and <math>(B,C)</math> are <math>(p-1)</math>-connected and <math>(q-1)</math>-connected, respectively, and the triad <math>(X;A,B)</math> is <math>(p+q-2)</math>-connected, then <math>H_k(X;A,B)=0</math> for <math>k<p+q-2</math> and <math>H_{p+q-1}(X;A)</math> is obtained from <math>\pi_{p+q-1}(X;A,B)</math> by factoring out the action of <math>\pi_1(A\cap B)</math> 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 <math>\operatorname{cat}^n</math>-group of an ''n''-cube of spaces.
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