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-complexX that is connected and simply connected, the least value of k ≥ 2 such that the homotopy group
πk(X) ≠ {0}
is also the least value of k > 0 with the homology group (with integer coefficients)
Hk(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
Although Hurewicz knew intimately the topology that was being studied in Poland he chose to go to Vienna to continue his studies.
Hurewicz was awarded a Rockefeller scholarship which allowed him to spend the year 1927-28 in Amsterdam.
Hurewicz is best remembered for two remarkable contributions to mathematics, his discovery of the higher homotopy groups in 1935-36, and his discovery of exact sequences in 1941.
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