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 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

$h_{*}:{\pi }_n(X)\to H_n(X),$

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 $\widetilde{H_i}(X)=0$ for all $i<n$, and the Hurewicz map $h_{*}:{\pi }_n(X)\to H_n(X)$ is an isomorphism.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 $h_{*}:{\pi }_{n+1}(X)\to H_{n+1}(X)$ is an epimorphism in this case.Template:Rp
• For $n=1$, the Hurewicz homomorphism induces an isomorphism ${\tilde{h}}_{*}:{\pi }_1(X)/[{\pi }_1(X),{\pi }_1(X)]\to H_1(X)$, 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

$h_{*}:{\pi }_k(X,A)\to H_k(X,A)$

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 $k<n$ and $H_n(X,A)$ is obtained from ${\pi }_n(X,A)$ by factoring out the action of ${\pi }_1(A)$. 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

${\pi }_n(X,A)\to {\pi }_n(X\cup CA),$

where $CA$ denotes the cone of $A$. This statement is a special case of a homotopical excision theorem, involving induced modules for $n>2$ (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

$h_{*}:{\pi }_k(X;A,B)\to H_k(X;A,B)$

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 ${\mathrm{cat}}^n$-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 ${\pi }_i(X)\otimes \mathbb{Q}=0$ for $i\le r$. Then the Hurewicz map

$h\otimes \mathbb{Q}:{\pi }_i(X)\otimes \mathbb{Q}⟶H_i(X;\mathbb{Q})$

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

References

• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation

• Template:Citation
• Template:Citation