Hurewicz theorem
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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
- 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
- πk(X) → Hk(X).