Hurewicz theorem

In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results of Henri Poincaré.

Statement of the theorems
The Hurewicz theorems are a key link between homotopy groups and homology groups.

Absolute version
For any path-connected space X and positive integer n there exists a group homomorphism


 * $$h_* \colon \pi_n(X) \to H_n(X),$$

called the Hurewicz homomorphism, from the n-th homotopy group to the n-th homology group (with integer coefficients). It is given in the following way: choose a canonical generator $$u_n \in H_n(S^n)$$, then a homotopy class of maps $$f \in \pi_n(X)$$ is taken to $$f_*(u_n) \in H_n(X)$$.

The Hurewicz theorem states cases in which the Hurewicz homomorphism is an isomorphism.


 * For $$n\ge 2$$, if X is $(n-1)$-connected (that is: $$\pi_i(X)= 0$$ for all $$i < n$$), then $$\tilde{H_i}(X)= 0$$ for all $$i < n$$, and the Hurewicz map $$h_* \colon \pi_n(X) \to H_n(X)$$ is an isomorphism. This implies, in particular, that the homological connectivity equals the homotopical connectivity when the latter is at least 1. In addition, the Hurewicz map $$h_* \colon \pi_{n+1}(X) \to H_{n+1}(X)$$ is an epimorphism in this case.
 * For $$n=1$$, the Hurewicz homomorphism induces an isomorphism $$\tilde{h}_* \colon \pi_1(X)/[ \pi_1(X), \pi_1(X)] \to H_1(X)$$, between the abelianization of the first homotopy group (the fundamental group) and the first homology group.

Relative version
For any pair of spaces $$(X,A)$$ and integer $$k>1$$ there exists a homomorphism


 * $$h_* \colon \pi_k(X,A) \to H_k(X,A)$$

from relative homotopy groups to relative homology groups. The Relative Hurewicz Theorem states that if both $$X$$ and $$A$$ are connected and the pair is $$(n-1)$$-connected then $$H_k(X,A)=0$$ for $$k2$$ (crossed modules if $$n=2$$), which itself is deduced from a higher homotopy van Kampen theorem for relative homotopy groups, whose proof requires development of techniques of a cubical higher homotopy groupoid of a filtered space.

Triadic version
For any triad of spaces $$(X;A,B)$$ (i.e., a space X and subspaces A, B) and integer $$k>2$$ there exists a homomorphism


 * $$h_*\colon \pi_k(X;A,B) \to H_k(X;A,B)$$

from triad homotopy groups to triad homology groups. Note that


 * $$H_k(X;A,B) \cong H_k(X\cup (C(A\cup B))).$$

The Triadic Hurewicz Theorem states that if X, A, B, and $$C=A\cap B$$ are connected, the pairs $$(A,C)$$ and $$(B,C)$$ are $$(p-1)$$-connected and $$(q-1)$$-connected, respectively, and the triad $$(X;A,B)$$ is $$(p+q-2)$$-connected, then $$H_k(X;A,B)=0$$ for $$k<p+q-2$$ and $$H_{p+q-1}(X;A)$$ is obtained from $$\pi_{p+q-1}(X;A,B)$$ by factoring out the action of $$\pi_1(A\cap B)$$ and the generalised Whitehead products. The proof of this theorem uses a higher homotopy van Kampen type theorem for triadic homotopy groups, which requires a notion of the fundamental $$\operatorname{cat}^n$$-group of an n-cube of spaces.

Simplicial set version
The Hurewicz theorem for topological spaces can also be stated for n-connected simplicial sets satisfying the Kan condition.

Rational Hurewicz theorem
Rational Hurewicz theorem:  Let X be a simply connected topological space with $$\pi_i(X)\otimes \Q = 0$$ for $$i\leq r$$. Then the Hurewicz map


 * $$h\otimes \Q \colon \pi_i(X)\otimes \Q \longrightarrow H_i(X;\Q )$$

induces an isomorphism for $$1\leq i \leq 2r$$ and a surjection for $$i = 2r+1$$.