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Hurewicz theorem

Contents

Idea

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

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

a representative singular kk-sphere ff in XX to the push-forward along ff of the fundamental class [S k]H k(S k)[S_k] \in H_k(S^k) \simeq \mathbb{Z}.

Remark

The Hurewicz homomorphism is a natural transformation between

Φ:π k()H k() \Phi : \pi_k(-) \to H_k(-)

between functors Top */Top^{*/} \to Ab.

Hurewicz theorem

Theorem

If a topological space (or infinity-groupoid) XX is (n-1)-connected for n2n \geq 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)H^n(X,A).

References

This appears for instance as theorem 4.32 in

Lecture notes include

See also

Revised on December 28, 2013 13:19:23 by archipelago? (46.5.2.218)