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For well behaved topological spaces the two notions agree and are jointly known as computing the <em><a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a></em> 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>.</p> <h2 id="definition">Definition</h2> <p>Given 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> with an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒰</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.8em" minsize="1.8em">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mover><mo>↪</mo><mi>open</mi></mover><mi>X</mi><msub><mo maxsize="1.8em" minsize="1.8em">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \mathcal{U} \;=\; \Big\{ U_i \xhookrightarrow{open} X \Big\}_{i \in I} </annotation></semantics></math></div> <p>we write</p> <div class="maruku-equation" id="eq:CechNerve"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>𝒰</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>π</mi> <mn>0</mn></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>U</mi> <mrow><msubsup><mo>×</mo> <mi>X</mi> <mo>•</mo></msubsup></mrow></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>sSet</mi></mrow><annotation encoding="application/x-tex"> C(X,\mathcal{U}) \;\coloneqq\; \pi_0 \big( U^{\times^\bullet_X} \big) \;\in\; sSet </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> which is its <a href="Čech+nerve#FromACover">Čech+nerve</a>: whose <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/simplex">simplices</a> are the <a class="existingWikiWord" href="/nlab/show/inhabited+set">inhabited</a> <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>-fold <a class="existingWikiWord" href="/nlab/show/intersections">intersections</a> of the <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒰</mi></mrow><annotation encoding="application/x-tex">\mathcal{U}</annotation></semantics></math>.</p> <p> <div class='num_remark'> <h6>Remark</h6> <p>For sufficiently well-behaved topological spaces <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 class="existingWikiWord" href="/nlab/show/paracompact+spaces">paracompact spaces</a>) and <em><a class="existingWikiWord" href="/nlab/show/good+open+cover">good</a></em> open covers, the Čech nerve <a class="maruku-eqref" href="#eq:CechNerve">(1)</a> is <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+equivalence">simplicially homotopy equivalent</a> to the <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> – this is the statement of the <em><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></em>.</p> <p>However, in general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>𝒰</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(X,\mathcal{U})</annotation></semantics></math> may differ from the <a class="existingWikiWord" href="/nlab/show/weak+homotopy+type">weak homotopy type</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (even in the <a class="existingWikiWord" href="/nlab/show/limit">limit</a> below) in which case Čech homology may <em>differ</em> from the <a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. (See for instance the discussion at <em><a class="existingWikiWord" href="/nlab/show/well+group">well group</a></em>.)</p> </div> </p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒰</mi></mrow><annotation encoding="application/x-tex">\mathcal{U}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/refinement+of+an+open+cover">refinement</a> open cover, i.e. such that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>′</mo><mo>∈</mo><mi>𝒰</mi></mrow><annotation encoding="application/x-tex">U' \in \mathcal{U}</annotation></semantics></math>, there is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>𝒰</mi></mrow><annotation encoding="application/x-tex">U \in \mathcal{U}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>′</mo><mo>⊆</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">U' \subseteq U</annotation></semantics></math>, then these inclusions induce a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>𝒰</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>𝒰</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> C(X,\mathcal{U}) \longrightarrow C(X,\mathcal{U}') </annotation></semantics></math></div> <p>This yields an <a class="existingWikiWord" href="/nlab/show/inverse+system">inverse system</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a>.</p> <p> <div class='num_defn'> <h6>Definition</h6> <p>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>th <strong>Čech homology group</strong> of the space <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 the <a class="existingWikiWord" href="/nlab/show/limit">limit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mi>H</mi><mo stretchy="false">ˇ</mo></mover> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><munder><mo>⟵</mo><mi>𝒰</mi></munder></munder><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>𝒰</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \check{H}_n(X) \;\coloneqq\; \underset{ \underset{\mathcal{U}}{\longleftarrow} }{lim} H_n(X,\,\mathcal{U}) </annotation></semantics></math></div> <p>over the <a class="existingWikiWord" href="/nlab/show/inverse+system">inverse system</a> of <a class="existingWikiWord" href="/nlab/show/open+covers">open covers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>, of the <a class="existingWikiWord" href="/nlab/show/simplicial+homology">simplicial homology</a> <a class="existingWikiWord" href="/nlab/show/homology+group">groups</a> of the <a href="Čech+nerve#FromACover">Čech nerve</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(X,\alpha)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>𝒰</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>H</mi> <mi>n</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>𝒰</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H_n(X,\,\mathcal{U}) \;\coloneqq\; H_n\big(C(X,\,\mathcal{U})\big) \,. </annotation></semantics></math></div> <p></p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>It is to be noted that these groups do not constitute a <a class="existingWikiWord" href="/nlab/show/homology+theory">homology theory</a> in the sense of the <a class="existingWikiWord" href="/nlab/show/Eilenberg-Steenrod+axioms">Eilenberg-Steenrod axioms</a> as the <em>exactness axiom</em> fails in general. There is a “corrected” theory known under the name <a class="existingWikiWord" href="/nlab/show/strong+homology">strong homology</a>.</p> </div> </p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C4%8Cech+methods">Čech methods</a>, <a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a>, <a class="existingWikiWord" href="/nlab/show/cellular+homology">cellular homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+homotopy+type">étale homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/well+group">well group</a></p> </li> </ul> <h2 id="references">References</h2> <p>Exposition:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/David+H.+Fremlin">David H. Fremlin</a>, Section 2 of: <em>Singular homology for amateurs</em> (2016) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://www1.essex.ac.uk/maths/people/fremlin/n16703.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/FremlinSIngularHomology.pdf" title="pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/closed+covers">closed covers</a> in Čech homology:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Edwin+E.+Floyd">Edwin E. Floyd</a>, <em>Closed coverings in Čech homology theory</em>, Trans. Amer. Math. Soc. <strong>84</strong> (1957) 319-337 [<a href="https://doi.org/10.1090/S0002-9947-1957-0087100-2">doi:10.1090/S0002-9947-1957-0087100-2</a>, <a href="http://www.ams.org/journals/tran/1957-084-02/S0002-9947-1957-0087100-2/S0002-9947-1957-0087100-2.pdf">pdf</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 17, 2023 at 22:17:06. 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