nLab
Hurewicz theorem

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 $H^n(X,A)$.

Related concepts

• Boardman homomorphism

• The Adams spectral sequence is a vast generalization of the computation of homotopy groups from cohomology groups via the Hurewicz theorem.

References

The basic statement is for instance in

• Allen Hatcher, Algebraic Topology (web)

Lecture notes include

• Michael Hutchings, Introduction to higher homotopy groups and obstruction theory (2011) (pdf)

For discussion in stable homotopy theory modeled on symmetric spectra is in

• Stefan Schwede, part II, prop. 6.30 of Symmetric spectra, 2012 (pdf)

See also

• wikipedia, Hurewicz theorem

In the generality of the Boardman homomorphism:

• Frank Adams, Part II.6 of Stable homotopy and generalised homology, 1974