Not to be confused with the Hurwitz theorem.
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
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 $(X,x)$ a pointed topological space, the Hurewicz homomorphism is the function
from the $k$th homotopy group of $(X,x)$ to the $k$th singular homology group defined by sending
any representative singular $k$-sphere $f$ in $X$ to the push-forward along $f$ of the fundamental class $[S_k] \in H_k(S^k) \simeq \mathbb{Z}$.
The Hurewicz homomorphism is a natural transformation
between functors $Top^{\ast/}$ $\to$ Ab.
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 $R$:
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.
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.
(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:
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:
(in degree 1)
For $X$ a connected topological space the Hurewicz homomorphism (Def. ) in degree 1
is surjective. Its kernel is the commutator subgroup of $\pi_1(X,x)$. Therefore it induces an isomorphism from the abelianization $\pi_1(X,x)^{ab} \coloneqq \pi_1(X,x)/[\pi_1,\pi_1]$:
(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.
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 $H^n(X,A)$.
The Adams spectral sequence is a vast generalization of the computation of homotopy groups from cohomology groups via the Hurewicz theorem.
The original reference is:
This article is reproduced on pages 341–348 of
The simplicial version is due to
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
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:
Last revised on November 25, 2023 at 18:57:43. See the history of this page for a list of all contributions to it.