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class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#HurewiczHomomorphismSection'>Hurewicz homomorphism</a></li> <ul> <li><a href='#for_topological_spaces'>For topological spaces</a></li> <li><a href='#ForSpectra'>For spectra</a></li> </ul> <li><a href='#HurewiczTheorem'>Hurewicz theorem</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The first nonzero <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a> and <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary</a>/<a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a> <a class="existingWikiWord" href="/nlab/show/homology+group">group</a> of a <a class="existingWikiWord" href="/nlab/show/simply-connected+topological+space">simply-connected topological space</a> occur in the same dimension and are <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a>.</p> <h2 id="HurewiczHomomorphismSection">Hurewicz homomorphism</h2> <h3 id="for_topological_spaces">For topological spaces</h3> <div class="num_defn" id="HurewiczHomomorphism"> <h6 id="definition">Definition</h6> <p><strong>(Hurewicz homomorphism)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,x)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed</a> <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, the <strong>Hurewicz homomorphism</strong> is the <a class="existingWikiWord" href="/nlab/show/function">function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Φ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>H</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Phi \;\colon\; \pi_k(X,x) \to H_k(X) </annotation></semantics></math></div> <p>from the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>th <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,x)</annotation></semantics></math> to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>th <a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a> group defined by sending</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Φ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>f</mi><mo lspace="verythinmathspace">:</mo><msup><mi>S</mi> <mi>k</mi></msup><mo>→</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>∼</mo></msub><mo>↦</mo><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">[</mo><msub><mi>S</mi> <mi>k</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \Phi \;\colon\; (f \colon S^k \to X)_{\sim} \mapsto f_*[S_k] </annotation></semantics></math></div> <p>any representative singular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-sphere <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the push-forward along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/fundamental+class">fundamental class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>S</mi> <mi>k</mi></msub><mo stretchy="false">]</mo><mo>∈</mo><msub><mi>H</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">[S_k] \in H_k(S^k) \simeq \mathbb{Z}</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The Hurewicz homomorphism is a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Φ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>→</mo><msub><mi>H</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Phi \;\colon\; \pi_k(-) \to H_k(-) </annotation></semantics></math></div> <p>between <a class="existingWikiWord" href="/nlab/show/functors">functors</a> <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup> </mrow> <annotation encoding="application/x-tex">Top^{\ast/}</annotation> </semantics> </math></a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>.</p> </div> <h3 id="ForSpectra">For spectra</h3> <p>The above construction has an immediate analog in <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a>:</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/ring">ring</a>, its <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">Eilenberg-MacLane spectrum</a> is an <a class="existingWikiWord" href="/nlab/show/E-infinity+ring">E-infinity ring</a> and hence receives a canonical <a class="existingWikiWord" href="/nlab/show/unit">unit</a> homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕊</mi><mo>⟶</mo><mi>H</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">\mathbb{S} \longrightarrow H R</annotation></semantics></math> from the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a>.</p> <p>Under <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> and passing to <a class="existingWikiWord" href="/nlab/show/stable+homotopy+group">stable homotopy group</a>, this induces a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> from <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups">stable homotopy groups</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (its <a class="existingWikiWord" href="/nlab/show/stable+homotopy+homology+theory">stable homotopy homology theory</a>) to <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>π</mi> <mo>•</mo> <mi>st</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝕊</mi><mo>∧</mo><msub><mi>X</mi> <mo>+</mo></msub><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>H</mi><mi>R</mi><mo>∧</mo><msub><mi>X</mi> <mo>+</mo></msub><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi^{st}_\bullet(X) \;\simeq\; \pi_\bullet( \mathbb{S} \wedge X_+ ) \longrightarrow \pi_\bullet( H R \wedge X_+ ) \simeq H_\bullet(X,R) \,. </annotation></semantics></math></div> <p>If here the <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">Eilenberg-MacLane spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">H R</annotation></semantics></math> is replaced by any other <a class="existingWikiWord" href="/nlab/show/E-infinity+ring+spectrum">E-infinity ring spectrum</a> the analogous construction is called the <em><a class="existingWikiWord" href="/nlab/show/Boardman+homomorphism">Boardman homomorphism</a></em>.</p> <h2 id="HurewiczTheorem">Hurewicz theorem</h2> <p>In general, <a class="existingWikiWord" href="/nlab/show/homology+theory">homology</a> is a coarser invariant than <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy</a>, and <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a> is the coarsest of all <a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a>-invariants. Therefore the Hurewicz homomorphism (Def. <a class="maruku-ref" href="#HurewiczHomomorphism"></a>) is bound to lose information, in general.</p> <p>Indeed, the Hurewicz homomorphism exhibits a kind of abelianization of the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> (in the sense of <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a>, see at <em><a class="existingWikiWord" href="/nlab/show/Boardman+homomorphism">Boardman homomorphism</a></em> for more on this), a statement that in low degrees is true in the plain sense of <em><a class="existingWikiWord" href="/nlab/show/abelianization">abelianization</a></em>: this is the content of Prop. <a class="maruku-ref" href="#HurewiczTheoremInDegreeZero"></a> and Prop. <a class="maruku-ref" href="#HurewiczTheoremInDegreeOne"></a> below.</p> <p>While in higher degrees the Hurewicz homomorphism is in general far from being an isomorphism, the thrust of the Hurewicz theorem is to show that high connectivity is a sufficient condition to ensure that it is. This is Theorem <a class="maruku-ref" href="#HurewiczTheoremInDegreeTwoAndHigher"></a> below.</p> <p> <div class='num_prop' id='HurewiczTheoremInDegreeZero'> <h6>Proposition</h6> <p><strong>(in degree 0)</strong> <br /> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a topological space, the Hurewicz homomorphism (Def. <a class="maruku-ref" href="#HurewiczHomomorphism"></a>) in degree 0 exhibits an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> between the <a class="existingWikiWord" href="/nlab/show/free+abelian+group">free abelian group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[\pi_0(X)]</annotation></semantics></math> on the set of <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and the degree-0 singular homlogy:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>≃</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{Z}[\pi_0(X)] \simeq H_0(X) \,. </annotation></semantics></math></div> <p></p> </div> </p> <p>Since a <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a> in <a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> degree depends on the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of the <a class="existingWikiWord" href="/nlab/show/connected+component">connected component</a> of the base point, while the <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a> does not depend on a basepoint, it is interesting to compare these groups only for the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is connected:</p> <p> <div class='num_prop' id='HurewiczTheoremInDegreeOne'> <h6>Proposition</h6> <p><strong>(in degree 1)</strong> <br /> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected topological space</a> the <a class="existingWikiWord" href="/nlab/show/Hurewicz+homomorphism">Hurewicz homomorphism</a> (Def. <a class="maruku-ref" href="#HurewiczHomomorphism"></a>) in degree 1</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Φ</mi><mo lspace="verythinmathspace">:</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Phi \colon \pi_1(X,x) \longrightarrow H_1(X) </annotation></semantics></math></div> <p>is <a class="existingWikiWord" href="/nlab/show/surjection">surjective</a>. Its <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> is the <a class="existingWikiWord" href="/nlab/show/commutator+subgroup">commutator subgroup</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(X,x)</annotation></semantics></math>. Therefore it induces an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> from the <a class="existingWikiWord" href="/nlab/show/abelianization">abelianization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mi>ab</mi></msup><mo>≔</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">[</mo><msub><mi>π</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\pi_1(X,x)^{ab} \coloneqq \pi_1(X,x)/[\pi_1,\pi_1]</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mi>ab</mi></msup><mover><mo>⟶</mo><mo>≃</mo></mover><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_1(X,x)^{ab} \overset{\simeq}{\longrightarrow} H_1(X) \,. </annotation></semantics></math></div> <p></p> </div> </p> <p> <div class='num_theorem' id='HurewiczTheoremInDegreeTwoAndHigher'> <h6>Theorem</h6> <p><strong>(in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\geq 2</annotation></semantics></math>)</strong> <br /> If a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> (or <a class="existingWikiWord" href="/nlab/show/infinity-groupoid">infinity-groupoid</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/n-connected+object+in+an+%28infinity%2C1%29-topos">(n-1)-connected</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n \geq 2</annotation></semantics></math> then the <a class="existingWikiWord" href="/nlab/show/Hurewicz+homomorphism">Hurewicz homomorphism</a>, Def. <a class="maruku-ref" href="#HurewiczHomomorphism"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Φ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Phi \;\colon\; \pi_n(X,x) \longrightarrow H_n(X) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> <p></p> </div> </p> <p>A proof is spelled out for instance with theorem 2.1 in (<a href="#Hutchings">Hutchings</a>).</p> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>With the <a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a> a corresponding statement follows for the <a class="existingWikiWord" href="/nlab/show/cohomology+group">cohomology group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^n(X,A)</annotation></semantics></math>.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Boardman+homomorphism">Boardman homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf+degree+theorem">Hopf degree theorem</a></p> </li> <li> <p>The <em><a class="existingWikiWord" href="/nlab/show/Adams+spectral+sequence">Adams spectral sequence</a></em> is a vast generalization of the computation of <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> from <a class="existingWikiWord" href="/nlab/show/cohomology+groups">cohomology groups</a> via the Hurewicz theorem.</p> </li> </ul> <h2 id="references">References</h2> <p>The original reference:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Witold+Hurewicz">Witold Hurewicz</a>: <em>Beiträge zur Topologie der Deformationen (II. Homotopie- und Homologigruppen)</em>, Proc. Akad. Wet. Amsterdam <strong>38</strong> (1935) 521–528 [<a href="https://dwc.knaw.nl/DL/publications/PU00016726.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Hurewicz-BeitraegeII.pdf" title="pdf">pdf</a>]</li> </ul> <p>reproduced on pages 341–348 of:</p> <ul> <li>Krystyna Kuperberg (ed.): <em>Collected Works of Witold Hurewicz</em>, American Mathematical Society (1995) [ISBN 0-8218-0011-6, <a href="https://bookstore.ams.org/cworks-4">ams:cworks-4</a>]</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a> version is due to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Daniel+M.+Kan">Daniel M. Kan</a>, <em>The Hurewicz theorem</em>, 1958 Symposium internacional de topología algebraica (International symposium on algebraic topology), pp. 225–231 Universidad Nacional Autónoma de México and UNESCO, Mexico City.</li> </ul> <p>The basic statement is for instance in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Allen+Hatcher">Allen Hatcher</a>, <em>Algebraic Topology</em> (<a href="http://www.math.cornell.edu/~hatcher/AT/ATpage.html">web</a>)</li> </ul> <p>Lecture notes:</p> <ul> <li id="Hutchings"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Hutchings">Michael Hutchings</a>, <em>Introduction to higher homotopy groups and obstruction theory</em> (2011) (<a href="http://math.berkeley.edu/~hutching/teach/215b-2011/homotopy.pdf">pdf</a>)</p> </li> <li id="Kobin16"> <p>Andrew Kobin, Section 7.3 of: <em>Algebraic Topology</em>, 2016 (<a class="existingWikiWord" href="/nlab/files/KobinAT2016.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>For discussion in <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> modeled on <a class="existingWikiWord" href="/nlab/show/symmetric+spectra">symmetric spectra</a> is in</p> <ul> <li id="Schwede12"><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, part II, prop. 6.30 of <em><a class="existingWikiWord" href="/nlab/show/Symmetric+spectra">Symmetric spectra</a></em>, 2012 (<a href="http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf">pdf</a>)</li> </ul> <p>See also:</p> <ul> <li>Wikipedia: <em><a href="http://en.wikipedia.org/wiki/Hurewicz_theorem">Hurewicz theorem</a></em></li> </ul> <p>In the generality of the <a class="existingWikiWord" href="/nlab/show/Boardman+homomorphism">Boardman homomorphism</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Frank+Adams">Frank Adams</a>, Part II.6 of <em><a class="existingWikiWord" href="/nlab/show/Stable+homotopy+and+generalised+homology">Stable homotopy and generalised homology</a></em>, 1974</li> </ul> <p>Discussion of the stable Hurewicz homomorphism includes</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Akhil+Mathew">Akhil Mathew</a>, <em>Torsion exponents in stable homotopy and the Hurewicz homomorphism</em>, Algebr. Geom. Topol. 16 (2016) 1025-1041 (<a href="https://arxiv.org/abs/1501.07561">arXiv:1501.07561</a>)</li> </ul> <p>Proof of the <a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a> in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>, hence in general <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-toposes">(∞,1)-toposes</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Daniel+Christensen">Daniel Christensen</a>, <a class="existingWikiWord" href="/nlab/show/Luis+Scoccola">Luis Scoccola</a>, <em>The Hurewicz theorem in Homotopy Type Theory</em> (<a href="https://arxiv.org/abs/2007.05833">arXiv:2007.05833</a>)</li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/persistent+homotopy">persistent homotopy</a> with focus on the <a class="existingWikiWord" href="/nlab/show/van+Kampen+theorem">van Kampen theorem</a>, <a class="existingWikiWord" href="/nlab/show/excision">excision</a> and the <a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a>:</p> <ul> <li>Mehmet Ali Batan, Mehmetcik Pamuk, Hanife Varli, <em>Persistent Homotopy</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/1909.08865">arXiv:1909.08865</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 26, 2024 at 18:10:34. 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