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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

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 \;\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

Named after Witold Hurewicz.

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:

Last revised on January 5, 2021 at 04:13:42. See the history of this page for a list of all contributions to it.