Editing 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 group]]s and [[homology group]]s.

===Absolute version===
For any [[Path connected|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 (mathematics)|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 [[group isomorphism|isomorphism]].

* For <math>n\ge 2</math>, if ''X'' is [[N-connected|<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">{{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}}</ref>{{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 <math>h_* \colon \pi_{n+1}(X) \to H_{n+1}(X)</math> is an [[epimorphism]] in this case.<ref name=":0" />{{Rp|page=390|location=?}}
* For <math>n=1</math>, the Hurewicz homomorphism induces an [[group isomorphism|isomorphism]] <math>\tilde{h}_* \colon \pi_1(X)/[ \pi_1(X), \pi_1(X)] \to H_1(X)</math>, between the [[Commutator subgroup|abelianization]] of the first homotopy group (the [[fundamental group]]) and the first homology group.

===Relative version===
For any [[topological pair|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, {{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 

:<math>\pi_n(X,A) \to \pi_n(X \cup CA),</math>

where <math>CA</math> denotes the [[Cone (topology)|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 module]]s 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 version===
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 

:<math>H_k(X;A,B) \cong H_k(X\cup (C(A\cup B))).</math>

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 version===
The Hurewicz theorem for topological spaces can also be stated for ''n''-connected [[simplicial set]]s satisfying the Kan condition.<ref>{{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</ref>

===Rational Hurewicz theorem===

'''Rational Hurewicz theorem:<ref>{{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 }}</ref><ref>{{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}}</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>.

==Notes==
{{Reflist}}

==References==
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[[Category:Theorems in homotopy theory]]
[[Category:Homology theory]]
[[Category:Theorems in algebraic topology]]