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

$$\Phi \;\colon\; \pi_k(X,x) \to H_k(X)$$

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

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

any 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

$$\Phi \;\colon\; \pi_k(-) \to H_k(-)$$

between functors $Top^{\ast/}$ $\to$ 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

In general, homology is a coarser invariant than homotopy, and ordinary homology is the coarsest of all generalized homology-invariants. Therefore the Hurewicz homomorphism (Def. ) is bound to lose information, in general.

Indeed, the Hurewicz homomorphism exhibits a kind of abelianization of the homotopy type (in the sense of stable homotopy theory, see at Boardman homomorphism for more on this), a statement that in low degrees is true in the plain sense of abelianization: this is the content of Prop. and Prop. below.

While in higher degrees the Hurewicz homomorphism is in general far from being an isomorphism, the thrust of the Hurewicz theorem is to show that high connectivity is a sufficient condition to ensure that it is. This is Theorem below.

Proposition

(in degree 0)
For $X$ a topological space, the Hurewicz homomorphism (Def. ) in degree 0 exhibits an isomorphism between the free abelian group $\mathbb{Z}[\pi_0(X)]$ on the set of connected components of $X$ and the degree-0 singular homlogy:

$$\mathbb{Z}[\pi_0(X)] \simeq H_0(X) \,.$$

Since a homotopy group in positive degree depends on the homotopy type of the connected component of the base point, while the ordinary homology does not depend on a basepoint, it is interesting to compare these groups only for the case that $X$ is connected:

Proposition

(in degree 1)
For $X$ a connected topological space the Hurewicz homomorphism (Def. ) in degree 1

$$\Phi \colon \pi_1(X,x) \longrightarrow H_1(X)$$

is surjective. Its kernel is the commutator subgroup of $\pi_1(X,x)$. Therefore it induces an isomorphism from the abelianization $\pi_1(X,x)^{ab} \coloneqq \pi_1(X,x)/[\pi_1,\pi_1]$:

$$\pi_1(X,x)^{ab} \overset{\simeq}{\longrightarrow} H_1(X) \,.$$

Theorem

(in degree $\geq 2$)
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

The original reference:

• Witold Hurewicz: Beiträge zur Topologie der Deformationen (II. Homotopie- und Homologigruppen), Proc. Akad. Wet. Amsterdam 38 (1935) 521–528 [pdf, pdf]

reproduced on pages 341–348 of:

• Krystyna Kuperberg (ed.): Collected Works of Witold Hurewicz, American Mathematical Society (1995) [ISBN 0-8218-0011-6, ams:cworks-4]

The simplicial version is due to

• Daniel M. Kan, The Hurewicz theorem, 1958 Symposium internacional de topología algebraica (International symposium on algebraic topology), pp. 225–231 Universidad Nacional Autónoma de México and UNESCO, Mexico City.

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)

Discussion of persistent homotopy with focus on the van Kampen theorem, excision and the Hurewicz theorem:

• Mehmet Ali Batan, Mehmetcik Pamuk, Hanife Varli, Persistent Homotopy $[$arXiv:1909.08865$]$