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

For topological spaces


(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.

For spectra

The above construction has an immediate analog in stable homotopy theory:

For RR a ring, its Eilenberg-MacLane spectrum is an E-infinity ring and hence receives a canonical unit homomorphism 𝕊HR\mathbb{S} \longrightarrow H R from the sphere spectrum.

Under smash product and passing to stable homotopy group, this induces a natural transformation from stable homotopy groups of XX (its stable homotopy homology theory) to ordinary homology of XX with coefficients in RR:

π st(X)π (𝕊X +)π (HRX +)H (X,R). \pi^{st}_\bullet(X) \;\simeq\; \pi_\bullet( \mathbb{S} \wedge X_+ ) \longrightarrow \pi_\bullet( H R \wedge X_+ ) \simeq H_\bullet(X,R) \,.

If here the Eilenberg-MacLane spectrum HRH R is replaced by any other E-infinity ring spectrum the analogous construction is called the Boardman homomorphism.

Hurewicz theorem


If a topological space (or infinity-groupoid) XX is (n-1)-connected for n2n \geq 2 then the Hurewicz homomorphism, def.

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:

Discussion of the stable Hurewicz homomorphism includes

  • Akhil Mathew, Torsion exponents in stable homotopy and the Hurewicz homomorphism, Algebr. Geom. Topol. 16 (2016) 1025-1041 (arXiv:1501.07561)

Proof of the Hurewicz theorem in homotopy type theory, hence in general (∞,1)-toposes:

Last revised on July 14, 2020 at 08:19:26. See the history of this page for a list of all contributions to it.