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| 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]].
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| ==Statement of the theorems==
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| The Hurewicz theorems are a key link between [[homotopy group]]s and
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| [[homology group]]s.
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| ===Absolute version===
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| For any space ''X'' and positive integer ''k'' there exists a [[group homomorphism]]
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| :<math>h_{\ast}\colon\, \pi_k(X) \to H_k(X) \,\!</math>
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| called the Hurewicz homomorphism from the ''k''-th [[homotopy group]] to the ''k''-th [[Homology (mathematics)|homology group]] (with integer coefficients), which for ''k'' = 1 is equivalent to the canonical [[Commutator subgroup|abelianization map]]
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| :<math>h_{\ast}\colon\, \pi_1(X) \to \pi_1(X)/[ \pi_1(X), \pi_1(X)] . \,\!</math>
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| The Hurewicz theorem states that if ''X'' is [[N-connected|(''n'' − 1)-connected]], the Hurewicz map is an [[isomorphism]] for all ''k'' ≤ ''n'' when ''n'' ≥ ''2'' and abelianization for ''n'' = ''1''. In particular, this theorem says that the abelianization of the first homotopy group (the [[fundamental group]]) is isomorphic to the first homology group:
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| :<math> H_1(X) \cong \pi_1(X)/[ \pi_1(X), \pi_1(X)] . \,\!</math>
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| The first homology group therefore vanishes if ''X'' is [[Connected space|path-connected]] and π<sub>1</sub>(''X'') is a [[perfect group]].
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| In addition, the Hurewicz homomorphism is an [[epimorphism]] from <math>\pi_{n+1}(X) \to H_{n+1}(X)</math> whenever X is (''n'' − 1)-connected, for <math>n \ge 2</math>.
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| The group homomorphism is given in the following way. Choose canonical generators <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>.
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| ===Relative version===
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| For any pair of spaces (''X'',''A'') and integer ''k'' > 1 there exists a homomorphism
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| :<math>h_{\ast}\colon \pi_k(X,A) \to H_k(X,A) \,\!</math>
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| from relative homotopy groups to relative homology groups. The Relative Hurewicz Theorem states that if each of ''X'', ''A'' are connected and the pair (''X'',''A'') is (''n''−1)-connected then ''H''<sub>''k''</sub>(''X'',''A'') = 0 for ''k'' < ''n'' and ''H''<sub>''n''</sub>(''X'',''A'') is obtained from π<sub>''n''</sub>(''X'',''A'') by factoring out the action of π<sub>1</sub>(''A''). This is proved in, for example, {{Harvtxt|Whitehead|1978}} by induction, proving in turn the absolute version and the Homotopy Addition Lemma.
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| This relative Hurewicz theorem is reformulated by {{Harvtxt|Brown|Higgins|1981}} as a statement about the morphism
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| :<math>\pi_n(X,A) \to \pi_n(X \cup CA) \,\!. </math>
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| 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.
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| ===Triadic version===
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| For any triad of spaces (''X'';''A'',''B'') (i.e. space ''X'' and subspaces ''A'',''B'') and integer ''k'' > 2 there exists a homomorphism
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| :<math>h_{\ast}\colon \pi_k(X;A,B) \to H_k(X;A,B) \,\!</math>
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| from triad homotopy groups to triad homology groups. Note that ''H''<sub>''k''</sub>(''X'';''A'',''B'') ≅ ''H''<sub>''k''</sub>(''X''∪(''C''(''A''∪''B'')). The Triadic Hurewicz Theorem states that if ''X'', ''A'', ''B'', and ''C'' = ''A''∩''B'' are connected, the pairs (''A'',''C''), (''B'',''C'') are respectively (''p''−1)-, (''q''−1)-connected, and the triad (''X'';''A'',''B'') is ''p''+''q''−2 connected, then ''H''<sub>''k''</sub>(''X'';''A'',''B'') = 0 for ''k'' < ''p''+''q''−2 and ''H''<sub>''p''+''q''−1</sub>(''X'';''A'') is obtained from π<sub>''p''+''q''−1</sub>(''X'';''A'',''B'') by factoring out the action of π<sub>1</sub>(''A''∩''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 cat<sup>''n''</sup>-group of an ''n''-cube of spaces.
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| ===Simplicial set version===
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| 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=P. G. | last2=Jardine | first2=J. F. | 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>
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| ===Rational Hurewicz theorem===
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| '''Rational Hurewicz theorem:<ref>{{Citation | last1=Klaus | first1=S. | last2=Kreck | first2=M. | 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 | pages=617–623}}</ref><ref>{{Citation | last1=Cartan | first1=H. | last2=Serre | first2=J. P. | title= Espaces fibres et groupes d'homotopie, II, Applications | journal= C. R. Acad. Sci. Paris | year=1952 | volume=2 | number=34 |pages=393–395}}</ref>''' Let ''X'' be a simply connected topological space with <math>\pi_i(X)\otimes \mathbb{Q} = 0</math> for <math>i\leq r</math>. Then the Hurewicz map
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| :<math>h\otimes \mathbb{Q} : \pi_i(X)\otimes \mathbb{Q} \longrightarrow H_i(X;\mathbb{Q})</math>
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| induces an isomorphism for <math>1\leq i \leq 2r</math> and a surjection for <math>i = 2r+1</math>.
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| ==References==
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| <references />
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| * {{citation
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| |last= Brown
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| |first= R.
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| |title= Triadic Van Kampen theorems and Hurewicz theorems
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| |journal= Contemporary Mathematics
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| |year= 1989
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| |volume= 96
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| |pages=39–57
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| |issn= 0040-9383
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| }}
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| <!--* R. Brown, ''Triadic Van Kampen theorems and Hurewicz theorems'', Algebraic Topology, Proc. Int. Conf. March 1988, Edited M.Mahowald and S.Priddy, Cont. Math. 96 (1989) 39-57.-->
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| * {{citation
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| |last1= Brown
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| |first1= Ronald
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| |last2= Higgins
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| |first2= P. J.
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| |title= Colimit theorems for relative homotopy groups
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| |journal= Journal of Pure and Applied Algebra
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| |year= 1981
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| |volume= 22
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| |pages= 11–41
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| |issn= 0022-4049
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| |doi= 10.1016/0022-4049(81)90080-3
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| }}
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| * {{citation
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| |last1= Brown
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| |first1= R.
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| |last2= Loday
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| |first2= J.-L.
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| |title= Homotopical excision, and Hurewicz theorems, for n-cubes of spaces
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| |journal= Proceedings of the London Mathematical Society. Third Series
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| |year= 1987
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| |volume= 54
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| |pages=176–192
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| |issn= 0024-6115
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| |doi= 10.1112/plms/s3-54.1.176
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| }}
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| * {{citation
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| |last1= Brown
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| |first1= R.
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| |last2= Loday
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| |first2= J.-L.
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| |title= Van Kampen theorems for diagrams of spaces
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| |journal= [[Topology (journal)|Topology]]
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| |year= 1987
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| |volume= 26
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| |pages=311–334
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| |issn= 0040-9383
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| |doi= 10.1016/0040-9383(87)90004-8
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| |issue= 3
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| }}
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| * {{citation
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| |last= Rotman
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| |first= Joseph J.<!--
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| |author-link= Joseph J. Rotman--><!-- missing link -->
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| |title= An Introduction to Algebraic Topology
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| |publisher= [[Springer-Verlag]]
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| |year= 1988
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| |publication-date= 1998-07-22
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| |series= [[Graduate Texts in Mathematics]]
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| |volume= 119
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| |isbn= 978-0-387-96678-6
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| }}
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| * {{citation
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| |last= Whitehead
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| |first= George W.
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| |author-link= George W. Whitehead
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| |title= Elements of Homotopy Theory
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| |publisher= [[Springer-Verlag]]
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| |year= 1978
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| |series= [[Graduate Texts in Mathematics]]
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| |volume= 61
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| |isbn= 978-0-387-90336-1
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| }}
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| [[Category:Homotopy theory]]
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| [[Category:Homology theory]]
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| [[Category:Theorems in algebraic topology]]
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