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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='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#cwcomplex'>CW-Complex</a></li> <li><a href='#CellularHomology'>Cellular homology</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#cellular_chains'>Cellular chains</a></li> <li><a href='#relation_to_singular_homology'>Relation to singular homology</a></li> <li><a href='#RelationToSpectralSequenceOfFilteredSingularComplex'>Relation to the spectral sequence of the filtered singular complex</a></li> </ul> <li><a href='#software'>Software</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#early_references_on_cohomology'>Early references on (co)homology</a></li> <li><a href='#general'>General</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>Cellular homology is a very efficient tool for computing the <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary</a> <a class="existingWikiWord" href="/nlab/show/homology+groups">homology groups</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> which are <a class="existingWikiWord" href="/nlab/show/CW+complexes">CW complexes</a>, based on the <a class="existingWikiWord" href="/nlab/show/relative+singular+homology">relative singular homology</a> of their <a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>-decomposition and using <a class="existingWikiWord" href="/nlab/show/degree+of+a+continuous+function">degree</a>-computations.</p> <p>Hence cellular homology uses the combinatorial structure of a CW-complex to define, first a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> of <em>celluar chains</em> and then the corresponding <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a>. The resulting <em>cellular homology</em> of a CW-complex is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to its <a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a>, hence to its <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a> as a topological space, and hence provides an efficient method for computing the latter.</p> <h2 id="definition">Definition</h2> <h3 id="cwcomplex">CW-Complex</h3> <p>For definiteness and to fix notation which we need in the following, we recall the definition of <em><a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a></em>. The actual definition of cellular homology is <a href="#CellularHomology">below</a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> write</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mo>∈</mo></mrow><annotation encoding="application/x-tex">S^n \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> for the standard <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>-<a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup><mo>∈</mo></mrow><annotation encoding="application/x-tex">D^n \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> for the standard <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>-<a class="existingWikiWord" href="/nlab/show/disk">disk</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mo>↪</mo><msup><mi>D</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">S^n \hookrightarrow D^{n+1}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> that includes the <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>-sphere as the <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-disk.</p> </li> </ul> <p>Write furthermore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>≔</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">S^{-1} \coloneqq \emptyset</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/empty+set">empty</a> topological space and think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>→</mo><msup><mi>D</mi> <mn>0</mn></msup><mo>≃</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">S^{-1} \to D^0 \simeq *</annotation></semantics></math> as the boundary inclusion of the (-1)-sphere into the 0-disk, which is the <a class="existingWikiWord" href="/nlab/show/point">point</a>.</p> <div class="num_defn" id="CWComplex"> <h6 id="definition_2">Definition</h6> <p>A <strong><a class="existingWikiWord" href="/nlab/show/CW+complex">CW complex</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1)</annotation></semantics></math></strong> is the <a class="existingWikiWord" href="/nlab/show/empty+set">empty</a> <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>.</p> <p>By <a class="existingWikiWord" href="/nlab/show/induction">induction</a>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> a <strong><a class="existingWikiWord" href="/nlab/show/CW+complex">CW complex</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <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></strong> is a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_{n}</annotation></semantics></math> obtained from</p> <ol> <li> <p>a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CW</mi></mrow><annotation encoding="application/x-tex">CW</annotation></semantics></math>-complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_{n-1}</annotation></semantics></math> of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n-1</annotation></semantics></math>;</p> </li> <li> <p>an index set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cell</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">Cell(X)_n \in Set</annotation></semantics></math>;</p> </li> <li> <p>a set of <a class="existingWikiWord" href="/nlab/show/continuous+maps">continuous maps</a> (the <strong>attaching maps</strong>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>f</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>→</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>Cell</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\{ f_i \colon S^{n-1} \to X_{n-1}\}_{i \in Cell(X)_n}</annotation></semantics></math></p> </li> </ol> <p>as the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub><mo>≃</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>j</mi><mo>∈</mo><mi>Cell</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mrow></msub><msup><mi>D</mi> <mi>n</mi></msup><msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>j</mi><mo>∈</mo><mi>Cell</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></msub><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_n \simeq \coprod_{j \in Cell(X)_n} D^n \coprod_{j \in Cell(X)_n S^{n-1}} X_n</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>j</mi><mo>∈</mo><mi>Cell</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mrow></munder><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>j</mi><mo>∈</mo><mi>Cell</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mrow></munder><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>X</mi> <mi>n</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \coprod_{j \in Cell(X)_{n}} S^{n-1} &\stackrel{(f_j)}{\to}& X_{n-1} \\ \downarrow && \downarrow \\ \coprod_{j \in Cell(X)_{n}} D^{n} &\to& X_{n} } \,. </annotation></semantics></math></div> <p>By this construction an <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>-dimensional CW-complex is canonical a <a class="existingWikiWord" href="/nlab/show/filtered+topological+space">filtered topological space</a> with filter inclusion maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>∅</mi><mo>↪</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>↪</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>↪</mo><mi>⋯</mi><mo>↪</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>↪</mo><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> \emptyset \hookrightarrow X_0 \hookrightarrow X_1 \hookrightarrow \cdots \hookrightarrow X_{n-1} \hookrightarrow X_n </annotation></semantics></math></div> <p>the right vertical morphisms in these pushout diagrams.</p> <p>A general <strong><a class="existingWikiWord" href="/nlab/show/CW+complex">CW complex</a></strong> <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 <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> given as the <a class="existingWikiWord" href="/nlab/show/sequential+colimit">sequential colimit</a> over a <a class="existingWikiWord" href="/nlab/show/tower">tower</a> <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> each of whose morphisms is such a filter inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>∅</mi><mo>↪</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>↪</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>↪</mo><mi>⋯</mi><mo>↪</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \emptyset \hookrightarrow X_0 \hookrightarrow X_1 \hookrightarrow \cdots \hookrightarrow X \,. </annotation></semantics></math></div></div> <p>For the following a CW-complex is all this data: the chosen <a class="existingWikiWord" href="/nlab/show/filtered+topological+space">filtering</a> with the chosen attaching maps.</p> <h3 id="CellularHomology">Cellular homology</h3> <p>We define “<a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary</a>” cellular homology with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in the <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/integers">integers</a>. The analogous definition for other <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> is immediate.</p> <div class="num_defn" id="CellularChainComplex"> <h6 id="definition_3">Definition</h6> <p>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/CW-complex">CW-complex</a>, def. <a class="maruku-ref" href="#CWComplex"></a>, its <strong>cellular chain complex</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mo>•</mo> <mi>CW</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>Ch</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">H_\bullet^{CW}(X) \in Ch_\bullet</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> such that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math></p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> of <a class="existingWikiWord" href="/nlab/show/chains">chains</a> is the <a class="existingWikiWord" href="/nlab/show/relative+singular+homology">relative singular homology</a> group of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X_n \hookrightarrow X</annotation></semantics></math> relative to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X_{n-1} \hookrightarrow X</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>H</mi> <mi>n</mi> <mi>CW</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> H_n^{CW}(X) \coloneqq H_n(X_n, X_{n-1}) \,, </annotation></semantics></math></div></li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/differential">differential</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mo>∂</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow> <mi>CW</mi></msubsup><mo lspace="verythinmathspace">:</mo><msubsup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow> <mi>CW</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msubsup><mi>H</mi> <mi>n</mi> <mi>CW</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\partial^{CW}_{n+1} \colon H_{n+1}^{CW}(X) \to H_n^{CW}(X)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/composition">composition</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mo>∂</mo> <mi>n</mi> <mi>CW</mi></msubsup><mo lspace="verythinmathspace">:</mo><msub><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mo>∂</mo> <mi>n</mi></msub></mrow></mover><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></mover><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \partial^{CW}_n \colon H_{n+1}(X_{n+1}, X_n) \stackrel{\partial_n}{\to} H_n(X_n) \stackrel{i_n}{\to} H_n(X_n, X_{n-1}) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\partial_n</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> map of the <a class="existingWikiWord" href="/nlab/show/singular+chain+complex">singular chain complex</a> and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">i_n</annotation></semantics></math> is the morphism on <a class="existingWikiWord" href="/nlab/show/relative+homology">relative homology</a> induced from the canonical inclusion of pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo>,</mo><mi>∅</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X_n, \emptyset) \to (X_n, X_{n-1})</annotation></semantics></math>.</p> </li> </ul> </div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>The composition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mo>∂</mo> <mi>n</mi> <mi>CW</mi></msubsup><mo>∘</mo><msubsup><mo>∂</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow> <mi>CW</mi></msubsup></mrow><annotation encoding="application/x-tex">\partial^{CW}_{n} \circ \partial^{CW}_{n+1}</annotation></semantics></math> of two differentials in def. <a class="maruku-ref" href="#CellularChainComplex"></a> is indeed zero, hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mo>•</mo> <mi>CW</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^{CW}_\bullet(X)</annotation></semantics></math> is indeed a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>On representative singular <a class="existingWikiWord" href="/nlab/show/chains">chains</a> the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">i_n</annotation></semantics></math> acts as the identity and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mo>∂</mo> <mi>n</mi> <mi>CW</mi></msubsup><mo>∘</mo><msubsup><mo>∂</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow> <mi>CW</mi></msubsup></mrow><annotation encoding="application/x-tex">\partial^{CW}_{n} \circ \partial^{CW}_{n+1}</annotation></semantics></math> acts as the double singular boundary, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>n</mi></msub><mo>∘</mo><msub><mo>∂</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\partial_{n} \circ \partial_{n+1} = 0</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>By the discussion at <em><a href="relative%20homology#RelationToQuotientTopologicalSpaces">Relative homology - Relation to reduced homology of quotient spaces</a></em> the relative homology group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_n(X_n, X_{n-1})</annotation></semantics></math> is isomorphic to the the <a class="existingWikiWord" href="/nlab/show/reduced+homology">reduced homology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>H</mi><mo stretchy="false">˜</mo></mover> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">/</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde H_n(X_n/X_{n-1})</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">/</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_n/X_{n-1}</annotation></semantics></math>.</p> <p>This implies in particular that</p> <ul> <li> <p>a <strong>cellular <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>-chain</strong> is a singular <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>-chain required to sit in filtering degree <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>, hence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X_n \hookrightarrow X</annotation></semantics></math>;</p> </li> <li> <p>a <strong>cellular <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>-cycle</strong> is a singular <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>-chain whose singular boundary is not necessarily 0, but is contained in filtering degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-2)</annotation></semantics></math>, hence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X_{n-2} \hookrightarrow X</annotation></semantics></math>.</p> </li> </ul> </div> <h2 id="properties">Properties</h2> <h3 id="cellular_chains">Cellular chains</h3> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>For every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> we have an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>n</mi> <mi>CW</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi><mo stretchy="false">(</mo><mi>Cell</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^{CW}_n(X) \coloneqq H_n(X_n, X_{n-1}) \simeq \mathbb{Z}(Cell(X)_n) </annotation></semantics></math></div> <p>that the group of cellular <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>-chains with the <a class="existingWikiWord" href="/nlab/show/free+abelian+group">free abelian group</a> whose set of <a class="existingWikiWord" href="/nlab/show/basis">basis</a> elements is the set of <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>-<a class="existingWikiWord" href="/nlab/show/disks">disks</a> attached to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_{n-1}</annotation></semantics></math> to yield <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_n</annotation></semantics></math>.</p> </div> <p>This is discussed at <em><a href="relative%20homology#RelativeHomologyOfCWComplexes">Relative homology - Homology of CW-complexes</a></em>.</p> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>Thus, each cellular differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mo>∂</mo> <mi>n</mi> <mi>CW</mi></msubsup></mrow><annotation encoding="application/x-tex">\partial^{CW}_n</annotation></semantics></math> can be described as a <a class="existingWikiWord" href="/nlab/show/matrix">matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/integer">integer</a> entries <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">M_{i j}</annotation></semantics></math>. Here an index <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math> refers to the attaching map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>j</mi></msub><mo lspace="verythinmathspace">:</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>→</mo><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">f_j \colon S^n \to X_n</annotation></semantics></math> for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>j</mi> <mi>th</mi></msup></mrow><annotation encoding="application/x-tex">j^{th}</annotation></semantics></math> disk <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">D^{n+1}</annotation></semantics></math>. The integer entry <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">M_{i j}</annotation></semantics></math> corresponds to a map</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>≅</mo><msub><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msup><mi>D</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>,</mo><msup><mi>S</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>→</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>→</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>,</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>≅</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>≅</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z} \cong H_{n+1}(D^{n+1}, S^n) \to H_n(S^n) \to H_n(D^n, S^{n-1}) \cong H_n(S^n) \cong \mathbb{Z}</annotation></semantics></math></div> <p>and is computed as the <a class="existingWikiWord" href="/nlab/show/degree+of+a+continuous+function">degree of a continuous function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mover><mo>→</mo><mrow><msub><mi>f</mi> <mi>j</mi></msub></mrow></mover><msub><mi>X</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo>−</mo><msup><mi>D</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>≅</mo><msup><mi>D</mi> <mi>n</mi></msup><mo stretchy="false">/</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>≅</mo><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^n \stackrel{f_j}{\to} X_n \to X_n/(X_n - D^n) \cong D^n/S^{n-1} \cong S^n</annotation></semantics></math></div> <p>where the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub><mo>−</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>↪</mo><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_n - D^n \hookrightarrow X_n</annotation></semantics></math> corresponds to the attaching map for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mi>th</mi></msup></mrow><annotation encoding="application/x-tex">i^{th}</annotation></semantics></math> disk <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D^n</annotation></semantics></math>.</p> </div> <h3 id="relation_to_singular_homology">Relation to singular homology</h3> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>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/CW-complex">CW-complex</a>, its cellular homology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mo>•</mo> <mi>CW</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^{CW}_\bullet(X)</annotation></semantics></math> agrees with its <a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_\bullet(X)</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>H</mi> <mo>•</mo> <mi>CW</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^{CW}_\bullet(X) \simeq H_\bullet(X) \,. </annotation></semantics></math></div></div> <p>This appears for instance as (<a href="#Hatcher">Hatcher, theorem 2.35</a>). A proof is below as the proof of cor. <a class="maruku-ref" href="#CelluarEquivalentToSingularFromSpectralSequence"></a>.</p> <h3 id="RelationToSpectralSequenceOfFilteredSingularComplex">Relation to the spectral sequence of the filtered singular complex</h3> <p>The structure of a <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> on a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</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>, def. <a class="maruku-ref" href="#CWComplex"></a> naturally induces on its <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_\bullet(X)</annotation></semantics></math> the structure of a <a class="existingWikiWord" href="/nlab/show/filtered+chain+complex">filtered chain complex</a>:</p> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>↪</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>↪</mo><mi>⋯</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X_0 \hookrightarrow X_1 \hookrightarrow \cdots \hookrightarrow X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/CW+complex">CW complex</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">p \in \mathbb{N}</annotation></semantics></math>, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>p</mi></msub><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>p</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> F_p C_\bullet(X) \coloneqq C_\bullet(X_p) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/singular+chain+complex">singular chain complex</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>p</mi></msub><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X_p \hookrightarrow X</annotation></semantics></math>. The given <a class="existingWikiWord" href="/nlab/show/topological+subspace">topological subspace</a> inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>p</mi></msub><mo>↪</mo><msub><mi>X</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_p \hookrightarrow X_{p+1}</annotation></semantics></math> induce <a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a> inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>p</mi></msub><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>F</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_p C_\bullet(X) \hookrightarrow F_{p+1} C_\bullet(X)</annotation></semantics></math> and these equip the singular chain complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_\bullet(X)</annotation></semantics></math> 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 the structure of a bounded <a class="existingWikiWord" href="/nlab/show/filtered+chain+complex">filtered chain complex</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>↪</mo><msub><mi>F</mi> <mn>0</mn></msub><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>F</mi> <mn>1</mn></msub><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>F</mi> <mn>2</mn></msub><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>⋯</mi><mo>↪</mo><msub><mi>F</mi> <mn>∞</mn></msub><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 0 \hookrightarrow F_0 C_\bullet(X) \hookrightarrow F_1 C_\bullet(X) \hookrightarrow F_2 C_\bullet(X) \hookrightarrow \cdots \hookrightarrow F_\infty C_\bullet(X) \coloneqq C_\bullet(X) \,. </annotation></semantics></math></div> <p>(If <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 of finite <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">dim X</annotation></semantics></math> then this is a bounded filtration.)</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mi>r</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{E^r_{p,q}(X)\}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a> corresponding to this filtering.</p> </div> <p>We identify various of the pages of this spectral sequences with structures in singular homology theory.</p> <div class="num_prop" id="PagesInTheSpectralSequenceOfTheFilteredSingularComplex"> <h6 id="proposition_3">Proposition</h6> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">r = 0</annotation></semantics></math> – <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mn>0</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>C</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>p</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mi>C</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E^0_{p,q}(X) \simeq C_{p+q}(X_p)/C_{p+q}(X_{p-1})</annotation></semantics></math> is the group of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_{p-1}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/relative+homology">relative (p+q)-chains</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">X_p</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">r = 1</annotation></semantics></math> – <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mn>1</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>p</mi></msub><mo>,</mo><msub><mi>X</mi> <mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E^1_{p,q}(X) \simeq H_{p+q}(X_p, X_{p-1})</annotation></semantics></math> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_{p-1}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/relative+singular+homology">relative singular homology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">X_p</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">r = 2</annotation></semantics></math> – <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msubsup><mi>H</mi> <mi>p</mi> <mi>CW</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mi>for</mi><mspace width="thickmathspace"></mspace><mi>q</mi><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mi>otherwise</mi></mtd></mtr></mtable></mrow></mrow></mrow><annotation encoding="application/x-tex">E^2_{p,q}(X) \simeq \left\{ \array{ H_p^{CW}(X) & for\; q = 0 \\ 0 & otherwise } \right.</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>=</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">r = \infty</annotation></semantics></math> – <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>F</mi> <mi>p</mi></msub><msub><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mi>F</mi> <mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E^\infty_{p,q}(X) \simeq F_p H_{p+q}(X) / F_{p-1} H_{p+q}(X) </annotation></semantics></math>.</p> </li> </ul> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>(…)</p> </div> <p>This now directly implies the isomorphism between the cellular homology and the singular homology of a CW-complex <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>:</p> <div class="num_cor" id="CelluarEquivalentToSingularFromSpectralSequence"> <h6 id="corollary">Corollary</h6> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mo>•</mo> <mi>CW</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^{CW}_\bullet(X) \simeq H_\bullet(X) </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>By the third item of prop. <a class="maruku-ref" href="#PagesInTheSpectralSequenceOfTheFilteredSingularComplex"></a> the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>=</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(r = 2)</annotation></semantics></math>-page of the spectral sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mi>r</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{E^r_{p,q}(X)\}</annotation></semantics></math> is concentrated in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>q</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(q = 0)</annotation></semantics></math>-row. This implies that all <a class="existingWikiWord" href="/nlab/show/differentials">differentials</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>></mo><mn>2</mn></mrow><annotation encoding="application/x-tex">r \gt 2</annotation></semantics></math> vanish, since their domain and codomain groups necessarily have different values of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>. Accordingly we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E^\infty_{p,q}(X) \simeq E^2_{p,q}(X) </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p,q</annotation></semantics></math>. By the third and fourth item of prop. <a class="maruku-ref" href="#PagesInTheSpectralSequenceOfTheFilteredSingularComplex"></a> this is equivalently</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>p</mi></msub><msub><mi>H</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msubsup><mi>H</mi> <mi>p</mi> <mi>CW</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> G_p H_{p}(X) \simeq H^{CW}_p(X) \,. </annotation></semantics></math></div> <p>Finally observe that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>p</mi></msub><msub><mi>H</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>H</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G_p H_p(X) \simeq H_p(X)</annotation></semantics></math> by the definition of the filtering on the homology as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>p</mi></msub><msub><mi>H</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo lspace="mediummathspace" rspace="mediummathspace">∶−</mo><mi>image</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>p</mi></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mi>H</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_p H_p(X) \coloneq image(H_p(X_p) \to H_p(X))</annotation></semantics></math> and by standard properties of singular homology of <a class="existingWikiWord" href="/nlab/show/CW+complexes">CW complexes</a> discusses at <em><a href="CW+complex#SingularHomology">CW complex – singular homology</a></em>.</p> <h2 id="software">Software</h2> <p>There are convenient software implementations for large-scale computations of cellular homology: one may use <a href="http://www.linalg.org">LinBox</a>, <a href="http://chomp.rutgers.edu">CHomP</a> or <a href="http://www.sas.upenn.edu/~vnanda/perseus">Perseus</a>.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a>, <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+homology">Čech homology</a></p> </li> </ul> <h2 id="references">References</h2> <div> <h3 id="early_references_on_cohomology">Early references on (co)homology</h3> <p>The original references on <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a>/<a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> and <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> in the form of <a class="existingWikiWord" href="/nlab/show/cellular+cohomology">cellular cohomology</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andrei+Kolmogoroff">Andrei Kolmogoroff</a>, <em>Über die Dualität im Aufbau der kombinatorischen Topologie</em>, Recueil Mathématique 1(43) (1936), 97–102. (<a href="http://mi.mathnet.ru/msb5361">mathnet</a>)</li> </ul> <p>A footnote on the first page reads as follows, giving attribution to <a href="#Alexander35a">Alexander 35a</a>, <a href="#Alexander35a">35b</a>:</p> <blockquote> <p>Die Resultate dieser Arbeit wurden für den Fall gewöhnlicher Komplexe vom Verfasser im Frühling und im Sommer 1934 erhalten und teilweise an der Internationalen Konferenz für Tensoranalysis (Moskau) im Mai 1934 vorgetragen. Die hier dargestellte allgemeinere Theorie bildete den Gegenstand eines Vortrages, den der Verfasser an der Internationalen Topologischen Konferenz (Moskau, September 1935) hielt; bei letzterer Gelegenheit erfuhr er, dass ein grosser Teil dieser Resultate im Falle von Komplexen indessen von Herrn Alexander erhalten worden ist. Vgl. die inzwischen erschienenen Noten von Herrn <a class="existingWikiWord" href="/nlab/show/J.+W.+Alexander">Alexander</a> in den «Proceedings of the National Academy of Sciences U.S.A.», 21, (1935), 509—512. Herr Alexander trug über seine Resultate ebenfalls an der Moskauer Topologischen Konferenz vor. Verallgemeinerungen für abgeschlossene Mengen und die Konstruktion eines Homologieringes für Komplexe und abgeschlossene Mengen, über welche der Verfasser ebenso an der Tensorkonferenz 1934 vorgetragen hat, werden in einer weiteren Publikation dargestellt. Diese weitere Begriffsbildungen sind übrigens ebenfalls von Herrn Alexander gefunden und teilweise in den erwähnten Noten publiziert.</p> </blockquote> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Andrei+Kolmogoroff">Andrei Kolmogoroff</a>, <em>Homologiering des Komplexes und des lokal-bicompakten Raumes</em>, Recueil Mathématique 1(43) (1936), 701–705. <a href="http://mi.mathnet.ru/msb5475">mathnet</a>.</p> </li> <li id="Alexander35a"> <p><a class="existingWikiWord" href="/nlab/show/J.+W.+Alexander">J. W. Alexander</a>, <em>On the chains of a complex and their duals</em>, Proc. Nat. Acad. Sei. USA, 21(1935), 509–511 (<a href="https://doi.org/10.1073/pnas.21.8.50">doi:10.1073/pnas.21.8.50</a>)</p> </li> <li id="Alexander35b"> <p><a class="existingWikiWord" href="/nlab/show/J.+W.+Alexander">J. W. Alexander</a>, <em>On the ring of a compact metric space</em>, Proc. Nat. Acad. Sci. USA, 21 (1935), 511–512 (<a href="https://doi.org/10.1073/pnas.21.8.511">doi:10.1073/pnas.21.8.511</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/J.+W.+Alexander">J. W. Alexander</a>, <em>On the connectivity ring of an abstract space</em>, Ann. of Math., 37 (1936), 698–708 (<a href="https://doi.org/10.2307/1968484">doi:10.2307/1968484</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/alexcon.pdf">pdf</a>)</p> </li> </ul> <p>The term “cohomology” was introduced by <a class="existingWikiWord" href="/nlab/show/Hassler+Whitney">Hassler Whitney</a> in</p> <ul> <li id="Whitney37"><a class="existingWikiWord" href="/nlab/show/Hassler+Whitney">Hassler Whitney</a>, <em>On matrices of integers and combinatorial topology</em>. Duke Mathematical Journal 3:1 (1937), 35–45 (<a href="https://projecteuclid.org/journals/duke-mathematical-journal/volume-3/issue-1/On-matrices-of-integers-and-combinatorial-topology/10.1215/S0012-7094-37-00304-1.short">doi:10.1215/S0012-7094-37-00304-1</a>)</li> </ul> <p>See also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hassler+Whitney">Hassler Whitney</a>, <em>On products in a complex</em>, Annals of Math. 39 (1938) 397–432 (<a href="https://doi.org/10.2307/1968795">doi:10.2307/1968795</a>)</li> </ul> <p>The notion of <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a> is due to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, <em>Singular homology theory</em>, Annals of Mathematics 45:3 (1944) (<a href="https://doi.org/10.2307/1969185">doi:10.2307/1969185</a>)</li> </ul> <p>The notion of <a class="existingWikiWord" href="/nlab/show/monadic+cohomology">monadic cohomology</a> via <a class="existingWikiWord" href="/nlab/show/canonical+resolutions">canonical resolutions</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Barr">Michael Barr</a>, <a class="existingWikiWord" href="/nlab/show/Jon+Beck">Jon Beck</a>, <em>Homology and Standard Constructions</em>, in: <em><a class="existingWikiWord" href="/nlab/show/Seminar+on+Triples+and+Categorical+Homology+Theory">Seminar on Triples and Categorical Homology Theory</a></em>, Lecture Notes in Maths. <strong>80</strong>, Springer (1969), Reprints in Theory and Applications of Categories <strong>18</strong> (2008) 186-248 [<a href="http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html">tac:18</a>, <a href="http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Barr">Michael Barr</a>, <em>Cartan-Eilenberg cohomology and triples</em>, J. Pure Applied Algebra <strong>112</strong> 3 (1996) 219–238 [<a href="https://doi.org/10.1016/0022-4049(95)00138-7">doi:10.1016/0022-4049(95)00138-7</a>, <a href="https://www.math.mcgill.ca/barr/papers/coho.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Barr-CECohomology.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Barr">Michael Barr</a>, <em>Algebraic cohomology: the early days</em>, in <em>Galois Theory, Hopf Algebras, and Semiabelian Categories</em>, Fields Institute Communications <strong>43</strong> (2004) 1–26 [<a href="https://doi.org/10.1090/fic/043">doi:10.1090/fic/043</a>, <a href="https://www.math.mcgill.ca/barr/papers/algcohom.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Barr-AlgebraicCohomology.pdf" title="pdf">pdf</a>]</p> </li> </ul> <p>The general abstract perspective on <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> (subsuming <a class="existingWikiWord" href="/nlab/show/sheaf+cohomology">sheaf cohomology</a>, <a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a>, <a class="existingWikiWord" href="/nlab/show/non-abelian+cohomology">non-abelian cohomology</a> and indications of <a class="existingWikiWord" href="/nlab/show/Whitehead-generalized+cohomology">Whitehead-generalized cohomology</a>) was essentially established in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kenneth+Brown">Kenneth Brown</a>, <em><a class="existingWikiWord" href="/nlab/show/BrownAHT">Abstract homotopy theory and generalized sheaf cohomology</a></em> (1973)</li> </ul> </div> <h3 id="general">General</h3> <p>A standard textbook account is from p. 139 on in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Allen+Hatcher">Allen Hatcher</a>, <em><a href="http://www.math.cornell.edu/~hatcher/AT/ATpage.html">Algebraic topology</a></em>, Cambridge Univ. Press 2002,</li> </ul> <p>Lecture notes include</p> <ul> <li>Lisa Jeffrey, <em>Homology of CW-complexes and Cellular homology</em> (<a href="http://www.math.toronto.edu/~mat1300/oldnotes/cellular-homology.pdf">pdf</a>)</li> </ul> <p>Formulation in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ulrik+Buchholtz">Ulrik Buchholtz</a>, <a class="existingWikiWord" href="/nlab/show/Favonia">Kuen-Bang Hou (Favonia)</a>, <em><a class="existingWikiWord" href="/nlab/show/Cellular+Cohomology+in+Homotopy+Type+Theory">Cellular Cohomology in Homotopy Type Theory</a></em>, Logical Methods in Computer Science, <strong>16</strong> 2 (2020) (<a href="https://arxiv.org/abs/1802.02191">arXiv:1802.02191</a>, <a href="https://lmcs.episciences.org/6518">lmcs:6518</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Axel+Ljungstr%C3%B6m">Axel Ljungström</a>: <em>More cellular (co)homology in HoTT</em>, <a href="CQTS#LjungströmApr2024">talk at</a> <em><a href="CQTS#RunningHoTT2024">Running HoTT 2024</a></em>, <a class="existingWikiWord" href="/nlab/show/CQTS">CQTS</a>@NYUAD (April 2024) [video: <a href="https://cdnapisec.kaltura.com/html5/html5lib/v2.73.2/mwEmbedFrame.php/p/1674401/uiconf_id/23435151/entry_id/1_r5x4tca9?wid=_1674401&iframeembed=true&playerId=kaltura_player&entry_id=1_r5x4tca9">kt</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 13, 2024 at 07:58:14. 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