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 strictly positive integer n there exists a group homomorphism

:<math>h_* \colon \pi_n(X) \to H_n(X),</math>

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 <math>u_n \in H_n(S^n)</math>, then a homotopy class of maps <math>f \in \pi_n(X)</math> is taken to <math>f_*(u_n) \in H_n(X)</math>.

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

For <math>n\ge 2</math>, if X is <math>(n-1)</math>-connected (that is: <math>\pi_i(X)= 0</math> for all <math>i < n</math>), then <math>\tilde{H_i}(X)= 0</math> for all <math>i < n</math>, and the Hurewicz map <math>h_* \colon \pi_n(X) \to H_n(X)</math> 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 <math>h_* \colon \pi_{n+1}(X) \to H_{n+1}(X)</math> is an epimorphism in this case.

Rational Hurewicz theorem

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

:<math>h\otimes \Q \colon \pi_i(X)\otimes \Q \longrightarrow H_i(X;\Q )</math>

induces an isomorphism for <math>1\leq i \leq 2r</math> and a surjection for <math>i = 2r+1</math>.

Notes

References