Not to be confused with the Hurwitz theorem.
homotopy theory, (∞,1)-category theory, homotopy type theory
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see also algebraic topology
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Definitions
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Basic facts
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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 include
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
Last revised on September 22, 2018 at 06:19:01. See the history of this page for a list of all contributions to it.