hurewicz theorem

In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory. The theorem states that for a CW-complex X that is connected and simply connected, the least value of k ≥ 2 such that the homotopy group

π_k(X) ≠ {0}

is also the least value of k > 0 with the homology group (with integer coefficients)

H_k(X) ≠ {0};

and further that for this value, those two abelian groups are isomorphic. The theorem is due to Witold Hurewicz. The proof is based on the construction of the Hurewicz homomorphism

π_k(X) → H_k(X).

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