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

$\Phi : \pi_k(X,x) \to H_k(X)$

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

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

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

Remark

The Hurewicz homomorphism is a natural transformation between

$\Phi : \pi_k(-) \to H_k(-)$

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

Hurewicz theorem

Theorem

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

$\Phi : \pi_n(X,x) \to H_n(X)$

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