nLab
Hurewicz theorem

Not to be confused with the Hurwitz theorem.

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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

For topological spaces

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.

For spectra

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

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

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

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

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

Hurewicz theorem

Theorem

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

$$\Phi \;\colon\; \pi_n(X,x) \longrightarrow H_n(X)$$

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

• Hopf degree theorem

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

References

Named after Witold Hurewicz.

The basic statement is for instance in

• Allen Hatcher, Algebraic Topology (web)

Lecture notes:

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

• Andrew Kobin, Section 7.3 of: Algebraic Topology, 2016 (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

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:

• Daniel Christensen, Luis Scoccola, The Hurewicz theorem in Homotopy Type Theory (arXiv:2007.05833)