nLab Hurewicz theorem

Contents

Not to be confused with the Hurwitz theorem.


Context

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

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

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)(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}.

Remark

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

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

[π 0(X)]H 0(X). \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 XX is connected:

Proposition

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

Φ:π 1(X,x)H 1(X) \Phi \colon \pi_1(X,x) \longrightarrow H_1(X)

is surjective. Its kernel is the commutator subgroup of π 1(X,x)\pi_1(X,x). Therefore it induces an isomorphism from the abelianization π 1(X,x) abπ 1(X,x)/[π 1,π 1]\pi_1(X,x)^{ab} \coloneqq \pi_1(X,x)/[\pi_1,\pi_1]:

π 1(X,x) abH 1(X). \pi_1(X,x)^{ab} \overset{\simeq}{\longrightarrow} H_1(X) \,.

Theorem

(in degree 2\geq 2)
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 \;\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)H^n(X,A).

References

The original reference is:

  • Witold Hurewicz, Koninklijke Akademie van Wetenschappen: Proceedings of the Section of Sciences 38 (1935), 112–119. Mathematics. — Beiträge zur Topologie der Deformationen (I. Höherdimensionale Homotopiegruppen). PDF

This article is reproduced on pages 341–348 of

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

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

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:

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]]

Last revised on June 2, 2022 at 07:14:01. See the history of this page for a list of all contributions to it.