Hurewicz and Whitehead Theorems

The Hurewicz theorem bridges homotopy and homology, while the Whitehead theorem shows that weak homotopy equivalences between CW complexes are genuine homotopy equivalences.

The Hurewicz Homomorphism

Definition

The Hurewicz homomorphism $h : \pi_n(X, x_0) \to H_n(X; \mathbb{Z})$ sends the homotopy class of a map $f : S^n \to X$ to the image of the fundamental class: $h([f]) = f_*([S^n]) \in H_n(X; \mathbb{Z})$ where $[S^n] \in H_n(S^n) \cong \mathbb{Z}$ is the canonical generator. For $n = 1$ , the Hurewicz map is the abelianization $\pi_1(X) \to \pi_1(X)^{\text{ab}} \cong H_1(X)$ .

The Hurewicz homomorphism is natural: if $g : X \to Y$ is a continuous map, then $h \circ g_* = g_* \circ h$ .

The Hurewicz Theorem

Theorem9.5Hurewicz Theorem

Let $X$ be an $(n-1)$ -connected space with $n \geq 2$ (i.e., $\pi_k(X) = 0$ for $k < n$ ). Then $H_k(X; \mathbb{Z}) = 0$ for $1 \leq k < n$ and the Hurewicz map $h : \pi_n(X) \xrightarrow{\;\cong\;} H_n(X; \mathbb{Z})$ is an isomorphism.

ExampleHurewicz for spheres

Since $S^n$ is $(n-1)$ -connected, the Hurewicz theorem gives $\pi_n(S^n) \cong H_n(S^n) \cong \mathbb{Z}$ . The generator is the identity map $\text{id} : S^n \to S^n$ , confirming that the degree of a map $f : S^n \to S^n$ determines its homotopy class (when $n \geq 2$ ).

The Whitehead Theorem

Theorem9.6Whitehead Theorem

Let $f : X \to Y$ be a continuous map between connected CW complexes. If $f$ induces isomorphisms $f_* : \pi_n(X) \to \pi_n(Y)$ for all $n \geq 0$ (i.e., $f$ is a weak homotopy equivalence), then $f$ is a homotopy equivalence.

The proof uses the cellular approximation theorem and a careful inductive construction of a homotopy inverse on the skeleta of $X$ and $Y$ .

ExampleCW hypothesis is necessary

The Warsaw circle $W$ (a compact subspace of $\mathbb{R}^2$ that is path-connected but not locally path-connected) has $\pi_n(W) = 0$ for all $n$ , yet $W$ is not contractible. The map $W \to \{*\}$ is a weak homotopy equivalence but not a homotopy equivalence. This shows the CW complex hypothesis in Whitehead's theorem cannot be removed.

RemarkHomological Whitehead theorem

By combining the Hurewicz and Whitehead theorems: if $f : X \to Y$ is a map between simply-connected CW complexes that induces isomorphisms $f_* : H_n(X; \mathbb{Z}) \to H_n(Y; \mathbb{Z})$ for all $n$ , then $f$ is a homotopy equivalence. This is the homological Whitehead theorem, which allows one to detect homotopy equivalences using homology alone (in the simply-connected case).