---
tag: ZA11N5XPZM
kind: statement
role: theorem
name: Hurewicz theorem
msc: [55Q05, 55N10]
uses: [P976WYEFMB, M0Q5CFJ7ZH, NJQMWVK3QK, 1A943BWBQT, S1XK3HXHFX, 0AQZV0905N, 0WB6DH499T]
about: [P976WYEFMB, M0Q5CFJ7ZH]
sources:
  - url: https://en.wikipedia.org/wiki/Hurewicz_theorem
same_as:
  wikidata: Q2394548
  wikipedia: Hurewicz theorem
---

## Statement

Let $X$ be a [[path-connected|0WB6DH499T]] space with base point $x_0$, and let $n \ge 1$.
Fix a generator $u_n$ of $H_n(S^n; \Z) \cong \Z$.
The *Hurewicz map* $h \colon \pi_n(X, x_0) \to H_n(X; \Z)$ sends the class of $f \colon S^n \to X$ to $f_*(u_n)$.
Here $\pi_n$ is the [[homotopy group|P976WYEFMB]] and $H_n$ is [[singular homology|M0Q5CFJ7ZH]].

1. For $n = 1$, $h$ is surjective and its kernel is the commutator subgroup of the [[fundamental group|S1XK3HXHFX]]. Hence $H_1(X; \Z)$ is the [[abelianisation|0AQZV0905N]] of $\pi_1(X, x_0)$.
2. For $n \ge 2$, assume that $X$ is [[$(n-1)$-connected|1A943BWBQT]]. Then the [[reduced homology|NJQMWVK3QK]] $\tilde H_k(X; \Z)$ vanishes for $k < n$, and $h \colon \pi_n(X, x_0) \to H_n(X; \Z)$ is an isomorphism.

## Remarks

The map $h$ is a homomorphism, because a sum of maps $S^n \to X$ factors through the pinch map $S^n \to S^n \vee S^n$.
For $n \ge 2$, one replaces $X$ by a weakly equivalent CW complex with a single $0$-cell and no cells of dimensions $1$ to $n - 1$.
Then both groups are computed from the $n$-cells and the attaching maps of the $(n+1)$-cells.
Example: $S^n$ is $(n-1)$-connected and $H_n(S^n) \cong \Z$, so $\pi_n(S^n) \cong \Z$, generated by the identity.
The isomorphism stops at the first nonzero degree: $S^2$ is $1$-connected, yet $\pi_3(S^2) \cong \Z$ while $H_3(S^2) = 0$.

## Explanations

### L1 human

Homotopy groups are hard to compute, and homology groups are much easier.
The theorem says that the two agree in the first dimension where something happens.
In dimension one, homology is the fundamental group made commutative.

### L2 human

The theorem gives the first nonzero homotopy group of a simply connected space from its homology.
With the [[Whitehead theorem|6P1E2FJPSD]], it shows that a homology isomorphism between simply connected CW complexes is a homotopy equivalence.
A relative version for pairs of spaces extends it further.
It is also the starting point for Serre's mod $\mathcal{C}$ theory, which compares homotopy and homology modulo classes of abelian groups.