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Hurewicz theorem
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===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>.
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