Hurewicz theorem

{{short description|Gives a homomorphism from homotopy groups to homology groups}}

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.{{citation |last=Hatcher |first=Allen |title=Algebraic Topology |page= |year=2001 |publisher=Cambridge University Press |isbn=978-0-521-79160-1 |author-link=Allen Hatcher}}{{Rp|page=366|location=Thm.4.32}} 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.{{Rp|page=390|location=?}}
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 $k and H_n(X,A) is obtained from \pi_n(X,A) by factoring out the action of \pi_1(A) . This is proved in, for example, {{Harvtxt|Whitehead|1978}} by induction, proving in turn the absolute version and the Homotopy Addition Lemma.$

This relative Hurewicz theorem is reformulated by {{Harvtxt|Brown|Higgins|1981}} 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

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

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 $k and H_{p+q-1}(X;A) is obtained from \pi_{p+q-1}(X;A,B) by factoring out the action of \pi_1(A\cap B) and the generalised Whitehead products. The proof of this theorem uses a higher homotopy van Kampen type theorem for triadic homotopy groups, which requires a notion of the fundamental \operatorname{cat}^n -group of an$ 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.{{Citation | last1=Goerss | first1=Paul G. | last2=Jardine | first2=John Frederick | author-link2=Rick Jardine| title=Simplicial Homotopy Theory | publisher=Birkhäuser | location=Basel, Boston, Berlin | series=Progress in Mathematics | isbn=978-3-7643-6064-1 | year=1999 | volume=174}}, III.3.6, 3.7

=Rational Hurewicz theorem=

Rational Hurewicz theorem:{{Citation | last1=Klaus | first1=Stephan | last2=Kreck | first2=Matthias |author-link2=Matthias Kreck | title=A quick proof of the rational Hurewicz theorem and a computation of the rational homotopy groups of spheres | journal= Mathematical Proceedings of the Cambridge Philosophical Society | year=2004 | volume=136 | issue=3 | pages=617–623 | doi=10.1017/s0305004103007114| bibcode=2004MPCPS.136..617K | s2cid=119824771 }}{{Citation | last1=Cartan | first1=Henri |author-link1=Henri Cartan| last2=Serre | first2=Jean-Pierre | author-link2=Jean-Pierre Serre| title= Espaces fibrés et groupes d'homotopie, II, Applications | journal=Comptes rendus de l'Académie des Sciences | year=1952 | volume=2 | number=34 |pages=393–395}} 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 $i = 2r+1$ .

Notes

References

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Category:Theorems in homotopy theory

Category:Homology theory

Category:Theorems in algebraic topology