Hurewicz theorem

Not to be confused with the Hurwitz theorem.


Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




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


(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}.


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


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).


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


Named after Witold Hurewicz.

The basic statement is for instance in

Lecture notes include

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

See also

In the generality of the Boardman homomorphism:

Revised on December 14, 2016 15:29:55 by Urs Schreiber (