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…
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see also algebraic topology
Introductions
Definitions
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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
a representative singular $k$-sphere $f$ in $X$ to the push-forward along $f$ of the fundamental class $[S_k] \in H_k(S^k) \simeq \mathbb{Z}$.
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.
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.
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
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.