Hurewicz theorem

Template:Short description 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 theoremsEdit

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

Absolute versionEdit

For any path-connected space X and 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.<ref name=":0">Template:Citation</ref>Template:Rp 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.<ref name=":0" />Template:Rp
For <math>n=1</math>, the Hurewicz homomorphism induces an isomorphism <math>\tilde{h}_* \colon \pi_1(X)/[ \pi_1(X), \pi_1(X)] \to H_1(X)</math>, between the abelianization of the first homotopy group (the fundamental group) and the first homology group.

Relative versionEdit

For any pair of spaces <math>(X,A)</math> and integer <math>k>1</math> there exists a homomorphism

<math>h_* \colon \pi_k(X,A) \to H_k(X,A)</math>

from relative homotopy groups to relative homology groups. The Relative Hurewicz Theorem states that if both <math>X</math> and <math>A</math> are connected and the pair is <math>(n-1)</math>-connected then <math>H_k(X,A)=0</math> for <math>k<n</math> and <math>H_n(X,A)</math> is obtained from <math>\pi_n(X,A)</math> by factoring out the action of <math>\pi_1(A)</math>. This is proved in, for example, Template:Harvtxt by induction, proving in turn the absolute version and the Homotopy Addition Lemma.

This relative Hurewicz theorem is reformulated by Template:Harvtxt as a statement about the morphism

where <math>CA</math> denotes the cone of <math>A</math>. This statement is a special case of a homotopical excision theorem, involving induced modules for <math>n>2</math> (crossed modules if <math>n=2</math>), 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 versionEdit

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

<math>h_*\colon \pi_k(X;A,B) \to H_k(X;A,B)</math>

from triad homotopy groups to triad homology groups. Note that

The Triadic Hurewicz Theorem states that if X, A, B, and <math>C=A\cap B</math> are connected, the pairs <math>(A,C)</math> and <math>(B,C)</math> are <math>(p-1)</math>-connected and <math>(q-1)</math>-connected, respectively, and the triad <math>(X;A,B)</math> is <math>(p+q-2)</math>-connected, then <math>H_k(X;A,B)=0</math> for <math>k<p+q-2</math> and <math>H_{p+q-1}(X;A)</math> is obtained from <math>\pi_{p+q-1}(X;A,B)</math> by factoring out the action of <math>\pi_1(A\cap B)</math> 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 <math>\operatorname{cat}^n</math>-group of an n-cube of spaces.

Simplicial set versionEdit

The Hurewicz theorem for topological spaces can also be stated for n-connected simplicial sets satisfying the Kan condition.<ref>Template:Citation, III.3.6, 3.7</ref>

Rational Hurewicz theoremEdit

Rational Hurewicz theorem:<ref>Template:Citation</ref><ref>Template:Citation</ref> 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>.

NotesEdit

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