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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="homological_algebra">Homological algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></strong></p> <p>(also <a class="existingWikiWord" href="/nlab/show/nonabelian+homological+algebra">nonabelian homological algebra</a>)</p> <p><em><a class="existingWikiWord" href="/schreiber/show/Introduction+to+Homological+Algebra">Introduction</a></em></p> <p><strong>Context</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-additive+category">pre-additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+category">Grothendieck category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-abelian+category">semi-abelian category</a></p> </li> </ul> <p><strong>Basic definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>, <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex">complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential">differential</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homology+and+cohomology">chain homology and cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">simplicial homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>, <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a>, <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+resolution">flat resolution</a></p> </li> </ul> </li> </ul> <p><strong>Stable homotopy theory notions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>, <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated dg-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E-category">A-∞-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+chain+complexes">(∞,1)-category of chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>, <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">derived functor in homological algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>, <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lim%5E1+and+Milnor+sequences">lim^1 and Milnor sequences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fr-code">fr-code</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+complex">BRST-BV complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Serre+spectral+sequence">Serre spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Serre+spectral+sequence">Hochschild-Serre spectral sequence</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Lemmas</strong></p> <p><a class="existingWikiWord" href="/nlab/show/diagram+chasing">diagram chasing</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/3x3+lemma">3x3 lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/four+lemma">four lemma</a>, <a class="existingWikiWord" href="/nlab/show/five+lemma">five lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/snake+lemma">snake lemma</a>, <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/horseshoe+lemma">horseshoe lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baer%27s+criterion">Baer's criterion</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Schanuel%27s+lemma">Schanuel's lemma</a></p> <p><strong>Homology theories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> / <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal</a>, <a class="existingWikiWord" href="/nlab/show/operadic+Dold-Kan+correspondence">operadic</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a>, <a class="existingWikiWord" href="/nlab/show/Alexander-Whitney+map">Alexander-Whitney map</a>, <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+theorem">Eilenberg-Zilber theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">cochain on a simplicial set</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>:</mo><mi>sAb</mi><mo>→</mo><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup></mrow><annotation encoding="application/x-tex">C : sAb \to Ch_\bullet^+</annotation></semantics></math> be the chains/<a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a> functor of the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>sAb</mi><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(sAb, \otimes)</annotation></semantics></math> be the standard <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> structure given degreewise by the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> on <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Ch_\bullet^+, \otimes)</annotation></semantics></math> be the standard monoidal structure on the <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a>.</p> <div class="un_defn"> <h6 id="definition_2">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><mi>sAb</mi></mrow><annotation encoding="application/x-tex">A,B \in sAb</annotation></semantics></math> two abelian <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a>s, the <strong>Alexander-Whitney map</strong> is the <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> on <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>es</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>C</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Delta_{A,B} : C(A \otimes B) \to C(A) \otimes C(B) </annotation></semantics></math></div> <p>defined on two <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplices <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><msub><mi>A</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">a \in A_n</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msub><mi>B</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">b \in B_n</annotation></semantics></math> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mo>:</mo><mi>a</mi><mo>⊗</mo><mi>b</mi><mo>↦</mo><msub><mo>⊕</mo> <mrow><mi>p</mi><mo>+</mo><mi>q</mi><mo>=</mo><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><msup><mover><mi>d</mi><mo stretchy="false">˜</mo></mover> <mi>p</mi></msup><mi>a</mi><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><msubsup><mi>d</mi> <mn>0</mn> <mi>q</mi></msubsup><mi>b</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \Delta_{A,B} : a \otimes b \mapsto \oplus_{p + q = n} (\tilde d^p a) \otimes (d^q_0 b) \,, </annotation></semantics></math></div> <p>where the <em>front face map</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mover><mi>d</mi><mo stretchy="false">˜</mo></mover> <mi>p</mi></msup></mrow><annotation encoding="application/x-tex">\tilde d^p</annotation></semantics></math> is that induced by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>p</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo stretchy="false">]</mo><mo>:</mo><mi>i</mi><mo>↦</mo><mi>i</mi></mrow><annotation encoding="application/x-tex"> [p] \to [p+q] : i \mapsto i </annotation></semantics></math></div> <p>and the <em>back face</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mn>0</mn> <mi>q</mi></msubsup></mrow><annotation encoding="application/x-tex">d^q_0</annotation></semantics></math> map is that induced by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>q</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo stretchy="false">]</mo><mo>:</mo><mi>i</mi><mo>↦</mo><mi>i</mi><mo>+</mo><mi>p</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [q] \to [p+q] : i \mapsto i+p \,. </annotation></semantics></math></div></div> <div class="un_defn"> <h6 id="definition_3">Definition</h6> <p>This AW map restricts to the normalized chains complex</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mo>:</mo><mi>N</mi><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>N</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>N</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta_{A,B} : N(A \otimes B) \to N(A) \otimes N(B) \,. </annotation></semantics></math></div></div> <h2 id="properties">Properties</h2> <p> <div class='num_prop'> <h6>Proposition</h6> <p>The Alexander-Whitney map is an <a class="existingWikiWord" href="/nlab/show/oplax+monoidal+transformation">oplax monoidal transformation</a> that makes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> into <a class="existingWikiWord" href="/nlab/show/oplax+monoidal+functors">oplax monoidal functors</a>.</p> </div> </p> <p>Beware that the AW map is <em>not</em> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric</a>. For details see <em><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></em>.</p> <p> <div class='num_prop' id='EZAWDeformationRetract'> <h6>Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber%2FAlexander-Whitney+deformation+retraction">Eilenberg-Zilber/Alexander-Whitney deformation retraction</a>)</strong> <br /></p> <p>Let</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>sAb</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">A, B \,\in\, sAb = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SimplicialAbelianGroups">SimplicialAbelianGroups</a></li> </ul> <p>and denote</p> <ul> <li> <p>by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mi>N</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup><mo>=</mo></mrow><annotation encoding="application/x-tex">N(A), N(B) \,\in\, Ch^+_\bullet = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/ConnectiveChainComplexes">ConnectiveChainComplexes</a> their <a class="existingWikiWord" href="/nlab/show/normalized+chain+complexes">normalized chain complexes</a>,</p> </li> <li> <p>by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mi>B</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>sAb</mi></mrow><annotation encoding="application/x-tex">A \otimes B \,\in\, sAb</annotation></semantics></math> the degreewise <a class="existingWikiWord" href="/nlab/show/tensor+product+of+abelian+groups">tensor product of abelian groups</a>,</p> </li> <li> <p>by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>N</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(A) \otimes N(B)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+chain+complexes">tensor product of chain complexes</a>.</p> </li> </ul> <p>Then there is a <a class="existingWikiWord" href="/nlab/show/deformation+retraction">deformation retraction</a></p> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="397.591" height="65.705" viewBox="0 0 397.591 65.705"> <defs> <g> <g 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0, 0, -1, 293.81337, 16.668)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#0cqAJ165BMQrr_KWRcGUYokx0BQ=-glyph-5-1" x="252.973" y="10.025"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#0cqAJ165BMQrr_KWRcGUYokx0BQ=-glyph-4-1" x="261.79425" y="11.69875"></use> <use xlink:href="#0cqAJ165BMQrr_KWRcGUYokx0BQ=-glyph-4-2" x="268.699125" y="11.69875"></use> <use xlink:href="#0cqAJ165BMQrr_KWRcGUYokx0BQ=-glyph-4-3" x="271.535496" y="11.69875"></use> </g> <g clip-path="url(#0cqAJ165BMQrr_KWRcGUYokx0BQ=-clip-0)"> <path fill="none" stroke-width="2.79451" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -0.00115625 -10.98825 L 0.00275 -25.730438 " transform="matrix(1, 0, 0, -1, 186.466, 16.668)"></path> </g> <g clip-path="url(#0cqAJ165BMQrr_KWRcGUYokx0BQ=-clip-1)"> <path fill="none" stroke-width="1.83812" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(100%, 100%, 100%)" stroke-opacity="1" stroke-miterlimit="10" d="M -0.00115625 -10.98825 L 0.00275 -25.730438 " transform="matrix(1, 0, 0, -1, 186.466, 16.668)"></path> </g> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -1.553318 3.068445 C -0.799477 1.447281 1.227837 0.0565272 2.384108 0.00178083 C 1.227832 -0.0567563 -0.799622 -1.447308 -1.553624 -3.068396 " transform="matrix(0.00005, 0.99998, 0.99998, -0.00005, 186.46685, 42.39719)"></path> </svg> <p>where</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\nabla_{A,B}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta_{A,B}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Alexander-Whitney+map">Alexander-Whitney map</a>.</p> </li> </ul> <p></p> </div> </p> <p>For unnormalized chain complexes, where we have a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, this is the original <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+theorem">Eilenberg-Zilber theorem</a> (<a href="#EilenbergZilber53">Eilenberg & Zilber 1953</a>, <a href="#EilenbergMacLane54">Eilenberg & MacLane 1954, Thm. 2.1</a>). The above <a class="existingWikiWord" href="/nlab/show/deformation+retraction">deformation retraction</a> for normalized chain complexes is <a href="#EilenbergMacLane54">Eilenberg & MacLane 1954, Thm. 2.1a</a>. Both are reviewed in <a href="#May67">May 1967, Cor. 29.10</a>. Explicit description of the <a class="existingWikiWord" href="/nlab/show/homotopy+operator">homotopy operator</a> is given in <a href="#GonzalezDiazReal99">Gonzalez-Diaz & Real 1999</a>).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><strong>Alexander-Whitney map</strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a></p> </li> </ul> <h2 id="references">References</h2> <p>The <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+theorem">Eilenberg-Zilber theorem</a> is due to</p> <ul> <li id="EilenbergZilber53"> <p><a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, <a class="existingWikiWord" href="/nlab/show/Joseph+Zilber">Joseph Zilber</a>, <em>On Products of Complexes</em>, Amer. Jour. Math. 75 (1): 200–204, (1953) (<a href="https://www.jstor.org/stable/2372629">jstor:2372629</a>, <a href="https://doi.org/10.2307/2372629">doi:10.2307/2372629</a>)</p> </li> <li id="EilenbergMacLane54"> <p><a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, <a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, Section 2 of: <em>On the Groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>Π</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H(\Pi,n)</annotation></semantics></math>, II: Methods of Computation</em>, Annals of Mathematics, Second Series, Vol. 60, No. 1 (Jul., 1954), pp. 49-139 (<a href="https://www.jstor.org/stable/1969702">jstor:1969702</a>)</p> </li> </ul> <p>using the definition of the <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a> in:</p> <ul> <li id="EilenbergMacLane53"><a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, <a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, (5.3) of: <em>On the groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>Π</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H(\Pi,n)</annotation></semantics></math></em>, I<em>, Ann. of Math. (2) 58, (1953), 55–106. (<a href="https://www.jstor.org/stable/1969820">jstor:1969820</a>)</em></li> </ul> <p>Review:</p> <ul> <li id="May67"> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, Section 29 of: <em>Simplicial objects in algebraic topology</em> , Chicago Lectures in Mathematics, University of Chicago Press 1967 (<a href="https://press.uchicago.edu/ucp/books/book/chicago/S/bo5956688.html">ISBN:9780226511818</a>, <a href="http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu">djvu</a>, <a class="existingWikiWord" href="/nlab/files/May_SimplicialObjectsInAlgebraicTopology.pdf" title="pdf">pdf</a>)</p> </li> <li id="GonzalezDiazReal99"> <p>Rocio Gonzalez-Diaz, Pedro Real, <em>A Combinatorial Method for Computing Steenrod Squares</em>, Journal of Pure and Applied Algebra 139 (1999) 89-108 (<a href="https://arxiv.org/abs/math/0110308">arXiv:math/0110308</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 13, 2021 at 07:36:46. 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