nLab
Hurewicz theorem

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The first nonzero homotopy group and ordinary/singular homology group of a simply-connected topological space occur in the same dimension and are isomorphic.

Hurewicz homomorphism

Definition

(Hurewicz homomorphism)

For (X,x) a pointed topological space, the Hurewicz homomorphism is the function

Φ:π k(X,x)→H k(X)\Phi : \pi_k(X,x) \to H_k(X)

from the kth homotopy group of (X,x) to the kth singular homology group defined by sending

Φ:(f:S k→X) ∼↦f *[S k]\Phi : (f : S^k \to X)_{\sim} \mapsto f_*[S_k]

a representative singular k-sphere f in X to the push-forward along f of the fundamental class [S k]∈H k(S k)≃ℤ.

Remark

The Hurewicz homomorphism is a natural transformation between

Φ:π n(−)→H k(−)\Phi : \pi_n(-) \to H_k(-)

between functors Top */→ Ab.

Hurewicz theorem

Theorem

If a topological space (or infinity-groupoid) X is (n-1)-connected for n≥2 then the Hurewicz homomorphism, def. 1

Φ:π n(X,x)→H n(X)\Phi : \pi_n(X,x) \to H_n(X)

is an isomorphism.

A proof is spelled out for instance with theorem 2.1 in (Hutchings).

Remark

With the universal coefficient theorem a corresponding statement follows for the cohomology group H n(X,A).

References

This appears for instance as theorem 4.32 in

Lecture notes include

See also

Revised on February 23, 2013 16:07:23 by Anonymous Coward (85.5.61.83)
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