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
a representative singular $k$-sphere $f$ in $X$ to the push-forward along $f$ of the fundamental class $[S_k] \in H_k(S^k) \simeq \mathbb{Z}$.
If a topological space (or infinity-groupoid) $X$ is (n-1)-connected for $n \geq 2$ then the Hurewicz homomorphism, def. 1
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: