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 space 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.

In case the first homotopy group is noncommutative, then this theorem says that its abelianization is isomorphic to the first homology group.

:π k ( X ) → H k ( X ).