Betti number - Wikipedia

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href="#Formal_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Formal definition</span> </div> </a> <ul id="toc-Formal_definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Poincaré_polynomial" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Poincaré_polynomial"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Poincaré polynomial</span> </div> </a> <ul id="toc-Poincaré_polynomial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Betti_numbers_of_a_graph" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Betti_numbers_of_a_graph"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Betti numbers of a graph</span> </div> </a> <ul id="toc-Betti_numbers_of_a_graph-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Betti_numbers_of_a_simplicial_complex" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Betti_numbers_of_a_simplicial_complex"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Betti numbers of a simplicial complex</span> </div> </a> <ul id="toc-Betti_numbers_of_a_simplicial_complex-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Betti_numbers_of_the_projective_plane" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Betti_numbers_of_the_projective_plane"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Betti numbers of the projective plane</span> </div> </a> <ul id="toc-Betti_numbers_of_the_projective_plane-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Properties</span> </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Euler_characteristic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Euler_characteristic"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Euler characteristic</span> </div> </a> <ul id="toc-Euler_characteristic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cartesian_product" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cartesian_product"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Cartesian product</span> </div> </a> <ul id="toc-Cartesian_product-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symmetry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Symmetry"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Symmetry</span> </div> </a> <ul id="toc-Symmetry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Different_coefficients" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Different_coefficients"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Different coefficients</span> </div> </a> <ul id="toc-Different_coefficients-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-More_examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#More_examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>More examples</span> </div> </a> <ul id="toc-More_examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationship_with_dimensions_of_spaces_of_differential_forms" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relationship_with_dimensions_of_spaces_of_differential_forms"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Relationship with dimensions of spaces of differential forms</span> </div> </a> <ul id="toc-Relationship_with_dimensions_of_spaces_of_differential_forms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div 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class="mw-page-title-main">Betti number</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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href="https://de.wikipedia.org/wiki/Betti-Zahl" title="Betti-Zahl – German" lang="de" hreflang="de" data-title="Betti-Zahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_de_Betti" title="Número de Betti – Spanish" lang="es" hreflang="es" data-title="Número de Betti" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%A8%D8%AA%DB%8C" title="عدد بتی – Persian" lang="fa" hreflang="fa" data-title="عدد بتی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_de_Betti" title="Nombre de Betti – French" lang="fr" hreflang="fr" data-title="Nombre de Betti" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B2%A0%ED%8B%B0_%EC%88%98" title="베티 수 – Korean" lang="ko" hreflang="ko" data-title="베티 수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_di_Betti" title="Numero di Betti – Italian" lang="it" hreflang="it" data-title="Numero di Betti" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Betti-getal" title="Betti-getal – Dutch" lang="nl" hreflang="nl" data-title="Betti-getal" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%99%E3%83%83%E3%83%81%E6%95%B0" title="ベッチ数 – Japanese" lang="ja" hreflang="ja" data-title="ベッチ数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%91%D0%B5%D1%82%D1%82%D0%B8" title="Число Бетти – Russian" lang="ru" hreflang="ru" data-title="Число Бетти" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk 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data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Roughly, the number of k-dimensional holes on a topological surface</div> <p>In <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>, the <b>Betti numbers</b> are used to distinguish <a href="/wiki/Topological_space" title="Topological space">topological spaces</a> based on the connectivity of <i>n</i>-dimensional <a href="/wiki/Simplicial_complex" title="Simplicial complex">simplicial complexes</a>. For the most reasonable finite-dimensional <a href="/wiki/Topological_space" title="Topological space">spaces</a> (such as <a href="/wiki/Compact_manifold" class="mw-redirect" title="Compact manifold">compact manifolds</a>, finite <a href="/wiki/Simplicial_complexes" class="mw-redirect" title="Simplicial complexes">simplicial complexes</a> or <a href="/wiki/CW_complexes" class="mw-redirect" title="CW complexes">CW complexes</a>), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they are all finite. </p><p>The <i>n</i><sup>th</sup> Betti number represents the <a href="/wiki/Rank_of_a_group" title="Rank of a group">rank</a> of the <i>n</i><sup>th</sup> <a href="/wiki/Homology_group" class="mw-redirect" title="Homology group">homology group</a>, denoted <i>H</i><sub><i>n</i></sub>, which tells us the maximum number of cuts that can be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> For example, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}(X)\cong 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>≅</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n}(X)\cong 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/859e8f13866d3a53fa6c81a7f7484d2fd44e99c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.2ex; height:2.843ex;" alt="{\displaystyle H_{n}(X)\cong 0}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{n}(X)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}(X)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6cddca019b14fe932b060127d2cbfeffd3718c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.266ex; height:2.843ex;" alt="{\displaystyle b_{n}(X)=0}"></span>, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}(X)\cong \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>≅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n}(X)\cong \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b8ef9768281c98a79a828e3d10770d08368011b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.588ex; height:2.843ex;" alt="{\displaystyle H_{n}(X)\cong \mathbb {Z} }"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{n}(X)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}(X)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db1481910b8fab4d43c4e06535aa81930683e36f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.266ex; height:2.843ex;" alt="{\displaystyle b_{n}(X)=1}"></span>, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}(X)\cong \mathbb {Z} \oplus \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>≅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>⊕</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n}(X)\cong \mathbb {Z} \oplus \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63563c3749aa3993e5dc518186a4ef60dd391528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.979ex; height:2.843ex;" alt="{\displaystyle H_{n}(X)\cong \mathbb {Z} \oplus \mathbb {Z} }"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{n}(X)=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}(X)=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/533ddc802776a1225c3cf3112e7d803b04272841" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.266ex; height:2.843ex;" alt="{\displaystyle b_{n}(X)=2}"></span>, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}(X)\cong \mathbb {Z} \oplus \mathbb {Z} \oplus \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>≅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>⊕</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>⊕</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n}(X)\cong \mathbb {Z} \oplus \mathbb {Z} \oplus \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa31803495188b625e5c5f30cd8bff8d5a5f5e63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.369ex; height:2.843ex;" alt="{\displaystyle H_{n}(X)\cong \mathbb {Z} \oplus \mathbb {Z} \oplus \mathbb {Z} }"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{n}(X)=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}(X)=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1634f39cd6d76bf3ad3844ac69fe6dc2c87dc9d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.266ex; height:2.843ex;" alt="{\displaystyle b_{n}(X)=3}"></span>, etc. Note that only the ranks of infinite groups are considered, so for example if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}(X)\cong \mathbb {Z} ^{k}\oplus \mathbb {Z} /(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>≅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>⊕</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n}(X)\cong \mathbb {Z} ^{k}\oplus \mathbb {Z} /(2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fbfcc36a9031383420745166af9888f7b2bc14a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.201ex; height:3.176ex;" alt="{\displaystyle H_{n}(X)\cong \mathbb {Z} ^{k}\oplus \mathbb {Z} /(2)}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /(2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bb59df265fa1a42d2f5c60d0f4706bc8619791b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.685ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /(2)}"></span> is the <a href="/wiki/Finite_cyclic_group" class="mw-redirect" title="Finite cyclic group">finite cyclic group</a> of order 2, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{n}(X)=k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}(X)=k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21175cdb9ce342a083595b04bf545376d76b9563" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.315ex; height:2.843ex;" alt="{\displaystyle b_{n}(X)=k}"></span>. These finite components of the homology groups are their <a href="/wiki/Torsion_subgroup" title="Torsion subgroup">torsion subgroups</a>, and they are denoted by <b>torsion coefficients</b>. </p><p>The term "Betti number" was coined by <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a> after <a href="/wiki/Enrico_Betti" title="Enrico Betti">Enrico Betti</a>. The modern formulation is due to <a href="/wiki/Emmy_Noether#Second_epoch_(1920–1926):_Contributions_to_topology" title="Emmy Noether">Emmy Noether</a>. Betti numbers are used today in fields such as <a href="/wiki/Simplicial_homology" title="Simplicial homology">simplicial homology</a>, <a href="/wiki/Computer_science" title="Computer science">computer science</a> and <a href="/wiki/Digital_images" class="mw-redirect" title="Digital images">digital images</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Geometric_interpretation">Geometric interpretation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Betti_number&action=edit&section=1" title="Edit section: Geometric interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Torus_cycles.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Torus_cycles.png/220px-Torus_cycles.png" decoding="async" width="220" height="299" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/5/54/Torus_cycles.png 1.5x" data-file-width="229" data-file-height="311" /></a><figcaption>For a torus, the first Betti number is <i>b</i><sub>1</sub> = 2 , which can be intuitively thought of as the number of circular "holes"</figcaption></figure> <p>Informally, the <i>k</i>th Betti number refers to the number of <i>k</i>-dimensional <i>holes</i> on a topological surface. A "<i>k</i>-dimensional <i>hole</i>" is a <i>k</i>-dimensional cycle that is not a boundary of a (<i>k</i>+1)-dimensional object. </p><p>The first few Betti numbers have the following definitions for 0-dimensional, 1-dimensional, and 2-dimensional <a href="/wiki/Simplicial_complex" title="Simplicial complex">simplicial complexes</a>: </p> <ul><li><i>b</i><sub>0</sub> is the number of connected components;</li> <li><i>b</i><sub>1</sub> is the number of one-dimensional or "circular" holes;</li> <li><i>b</i><sub>2</sub> is the number of two-dimensional "voids" or "cavities".</li></ul> <p>Thus, for example, a torus has one connected surface component so <i>b</i><sub>0</sub> = 1, two "circular" holes (one equatorial and one <a href="/wiki/Zonal_and_meridional" class="mw-redirect" title="Zonal and meridional">meridional</a>) so <i>b</i><sub>1</sub> = 2, and a single cavity enclosed within the surface so <i>b</i><sub>2</sub> = 1. </p><p>Another interpretation of <i>b</i><sub>k</sub> is the maximum number of <i>k</i>-dimensional curves that can be removed while the object remains connected. For example, the torus remains connected after removing two 1-dimensional curves (equatorial and meridional) so <i>b</i><sub>1</sub> = 2.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The two-dimensional Betti numbers are easier to understand because we can see the world in 0, 1, 2, and 3-dimensions. </p> <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Betti_number&action=edit&section=2" title="Edit section: Formal definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a non-negative <a href="/wiki/Integer" title="Integer">integer</a> <i>k</i>, the <i>k</i>th Betti number <i>b</i><sub><i>k</i></sub>(<i>X</i>) of the space <i>X</i> is defined as the <a href="/wiki/Rank_of_an_abelian_group" title="Rank of an abelian group">rank</a> (number of linearly independent generators) of the <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> <i>H</i><sub><i>k</i></sub>(<i>X</i>), the <i>k</i>th <a href="/wiki/Homology_group" class="mw-redirect" title="Homology group">homology group</a> of <i>X</i>. The <i>k</i>th homology group is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{k}=\ker \delta _{k}/\operatorname {Im} \delta _{k+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mi>ker</mi> <mo>⁡</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>Im</mi> <mo>⁡</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{k}=\ker \delta _{k}/\operatorname {Im} \delta _{k+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6820f0a88e5f80f34a8f895c4da6c85ca5aa4aaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.732ex; height:2.843ex;" alt="{\displaystyle H_{k}=\ker \delta _{k}/\operatorname {Im} \delta _{k+1}}"></span>, the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74fee48740841008fe3a4585440d650dbea735ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.121ex; height:2.676ex;" alt="{\displaystyle \delta _{k}}"></span>s are the boundary maps of the <a href="/wiki/Simplicial_complex" title="Simplicial complex">simplicial complex</a> and the rank of H<sub>k</sub> is the <i>k</i>th Betti number. Equivalently, one can define it as the <a href="/wiki/Vector_space_dimension" class="mw-redirect" title="Vector space dimension">vector space dimension</a> of <i>H</i><sub><i>k</i></sub>(<i>X</i>; <b>Q</b>) since the homology group in this case is a vector space over <b>Q</b>. The <a href="/wiki/Universal_coefficient_theorem" title="Universal coefficient theorem">universal coefficient theorem</a>, in a very simple torsion-free case, shows that these definitions are the same. </p><p>More generally, given a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> <i>F</i> one can define <i>b</i><sub><i>k</i></sub>(<i>X</i>, <i>F</i>), the <i>k</i>th Betti number with coefficients in <i>F</i>, as the vector space dimension of <i>H</i><sub><i>k</i></sub>(<i>X</i>, <i>F</i>). </p> <div class="mw-heading mw-heading2"><h2 id="Poincaré_polynomial"><span id="Poincar.C3.A9_polynomial"></span>Poincaré polynomial</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Betti_number&action=edit&section=3" title="Edit section: Poincaré polynomial"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>Poincaré polynomial</b> of a surface is defined to be the <a href="/wiki/Generating_function" title="Generating function">generating function</a> of its Betti numbers. For example, the Betti numbers of the torus are 1, 2, and 1; thus its Poincaré polynomial is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+2x+x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+2x+x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6e17ac0c9d473449e475ddac2f6ad61cb8db39a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.719ex; height:2.843ex;" alt="{\displaystyle 1+2x+x^{2}}"></span>. The same definition applies to any topological space which has a finitely generated homology. </p><p>Given a topological space which has a finitely generated homology, the Poincaré polynomial is defined as the generating function of its Betti numbers, via the polynomial where the coefficient of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/150d38e238991bc4d0689ffc9d2a852547d2658d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.548ex; height:2.343ex;" alt="{\displaystyle x^{n}}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28e2d72f6dd9375c8f1f59f1effd9b4e5492ac97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.509ex;" alt="{\displaystyle b_{n}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Betti_number&action=edit&section=4" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Betti_numbers_of_a_graph">Betti numbers of a graph</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Betti_number&action=edit&section=5" title="Edit section: Betti numbers of a graph"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider a <a href="/wiki/Topological_graph_theory" title="Topological graph theory">topological graph</a> <i>G</i> in which the set of vertices is <i>V</i>, the set of edges is <i>E</i>, and the set of connected components is <i>C</i>. As explained in the page on <a href="/wiki/Graph_homology" title="Graph homology">graph homology</a>, its homology groups are given by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{k}(G)={\begin{cases}\mathbb {Z} ^{|C|}&k=0\\\mathbb {Z} ^{|E|+|C|-|V|}&k=1\\\{0\}&{\text{otherwise}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> </mtd> <mtd> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> </mtd> <mtd> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{k}(G)={\begin{cases}\mathbb {Z} ^{|C|}&k=0\\\mathbb {Z} ^{|E|+|C|-|V|}&k=1\\\{0\}&{\text{otherwise}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e21fd5a00d53dbea9ef41aedafa0ad30f71fa409" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:35.324ex; height:9.176ex;" alt="{\displaystyle H_{k}(G)={\begin{cases}\mathbb {Z} ^{|C|}&k=0\\\mathbb {Z} ^{|E|+|C|-|V|}&k=1\\\{0\}&{\text{otherwise}}\end{cases}}}"></span></dd></dl> <p>This may be proved straightforwardly by <a href="/wiki/Mathematical_induction" title="Mathematical induction">mathematical induction</a> on the number of edges. A new edge either increments the number of 1-cycles or decrements the number of connected components. </p><p>Therefore, the "zero-th" Betti number <i>b</i><sub>0</sub>(<i>G</i>) equals |<i>C</i>|, which is simply the number of connected components.<sup id="cite_ref-Hage1996_3-0" class="reference"><a href="#cite_note-Hage1996-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The first Betti number <i>b</i><sub>1</sub>(<i>G</i>) equals |<i>E</i>| + |<i>C</i>| - |<i>V</i>|. It is also called the <a href="/wiki/Cyclomatic_number" class="mw-redirect" title="Cyclomatic number">cyclomatic number</a>—a term introduced by <a href="/wiki/Gustav_Kirchhoff" title="Gustav Kirchhoff">Gustav Kirchhoff</a> before Betti's paper.<sup id="cite_ref-Kotiuga2010_4-0" class="reference"><a href="#cite_note-Kotiuga2010-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> See <a href="/wiki/Cyclomatic_complexity" title="Cyclomatic complexity">cyclomatic complexity</a> for an application to <a href="/wiki/Software_engineering" title="Software engineering">software engineering</a>. </p><p>All other Betti numbers are 0. </p> <div class="mw-heading mw-heading3"><h3 id="Betti_numbers_of_a_simplicial_complex">Betti numbers of a simplicial complex</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Betti_number&action=edit&section=6" title="Edit section: Betti numbers of a simplicial complex"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Simplicialexample.png" class="mw-file-description"><img alt="Example" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Simplicialexample.png/160px-Simplicialexample.png" decoding="async" width="160" height="292" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/8/87/Simplicialexample.png 1.5x" data-file-width="213" data-file-height="389" /></a><figcaption></figcaption></figure> <p>Consider a <a href="/wiki/Simplicial_complex" title="Simplicial complex">simplicial complex</a> with 0-simplices: a, b, c, and d, 1-simplices: E, F, G, H and I, and the only 2-simplex is J, which is the shaded region in the figure. There is one connected component in this figure (<i>b</i><sub>0</sub>); one hole, which is the unshaded region (<i>b</i><sub>1</sub>); and no "voids" or "cavities" (<i>b</i><sub>2</sub>). </p><p>This means that the rank of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43910602a221b7a4c373791f94793e3008622070" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{0}}"></span> is 1, the rank of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d4d9a872a55b209f2eb7cc23a71e5e1541bd1f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{1}}"></span> is 1 and the rank of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fa4324515cc7343ee952e3840a1bb1aa8c7f74c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{2}}"></span> is 0. </p><p>The Betti number sequence for this figure is 1, 1, 0, 0, ...; the Poincaré polynomial is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43e308c59c1f1efa99ad225ef13c66e413123aff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.72ex; height:2.343ex;" alt="{\displaystyle 1+x\,}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Betti_numbers_of_the_projective_plane">Betti numbers of the projective plane</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Betti_number&action=edit&section=7" title="Edit section: Betti numbers of the projective plane"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The homology groups of the <a href="/wiki/Projective_plane" title="Projective plane">projective plane</a> <i>P</i> are:<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{k}(P)={\begin{cases}\mathbb {Z} &k=0\\\mathbb {Z} _{2}&k=1\\\{0\}&{\text{otherwise}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mtd> <mtd> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{k}(P)={\begin{cases}\mathbb {Z} &k=0\\\mathbb {Z} _{2}&k=1\\\{0\}&{\text{otherwise}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2826df19ed5ba6ed8afc40e4fd50bdfbd8efe11a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:27.878ex; height:8.509ex;" alt="{\displaystyle H_{k}(P)={\begin{cases}\mathbb {Z} &k=0\\\mathbb {Z} _{2}&k=1\\\{0\}&{\text{otherwise}}\end{cases}}}"></span></dd></dl> <p>Here, <b>Z</b><sub>2</sub> is the <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a> of order 2. The 0-th Betti number is again 1. However, the 1-st Betti number is 0. This is because <i>H</i><sub>1</sub>(<i>P</i>) is a finite group - it does not have any infinite component. The finite component of the group is called the <b>torsion coefficient</b> of <i>P</i>. The (rational) Betti numbers <i>b</i><sub><i>k</i></sub>(<i>X</i>) do not take into account any <a href="/wiki/Torsion_subgroup" title="Torsion subgroup">torsion</a> in the homology groups, but they are very useful basic topological invariants. In the most intuitive terms, they allow one to count the number of <i>holes</i> of different dimensions. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Betti_number&action=edit&section=8" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Euler_characteristic">Euler characteristic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Betti_number&action=edit&section=9" title="Edit section: Euler characteristic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a finite CW-complex <i>K</i> we have </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi (K)=\sum _{i=0}^{\infty }(-1)^{i}b_{i}(K,F),\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi (K)=\sum _{i=0}^{\infty }(-1)^{i}b_{i}(K,F),\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b57ab6d1ab8a1f9ebf70f1edc2610358ae05894" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.844ex; height:6.843ex;" alt="{\displaystyle \chi (K)=\sum _{i=0}^{\infty }(-1)^{i}b_{i}(K,F),\,}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi (K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi (K)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c09215a6b003dce27b2e3d3aaed8aed927caf1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.33ex; height:2.843ex;" alt="{\displaystyle \chi (K)}"></span> denotes <a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a> of <i>K</i> and any field <i>F</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Cartesian_product">Cartesian product</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Betti_number&action=edit&section=10" title="Edit section: Cartesian product"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For any two spaces <i>X</i> and <i>Y</i> we have </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{X\times Y}=P_{X}P_{Y},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>×</mo> <mi>Y</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{X\times Y}=P_{X}P_{Y},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e917fc8ac3b26c8b4ed895fcb575d8310f8d375d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.505ex; height:2.509ex;" alt="{\displaystyle P_{X\times Y}=P_{X}P_{Y},}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8348dd8ce7e6f7f4778ee01fa5bdc7b828afd98c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.125ex; height:2.509ex;" alt="{\displaystyle P_{X}}"></span> denotes the <b>Poincaré polynomial</b> of <i>X</i>, (more generally, the <a href="/wiki/Hilbert%E2%80%93Poincar%C3%A9_series" title="Hilbert–Poincaré series">Hilbert–Poincaré series</a>, for infinite-dimensional spaces), i.e., the <a href="/wiki/Generating_function" title="Generating function">generating function</a> of the Betti numbers of <i>X</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{X}(z)=b_{0}(X)+b_{1}(X)z+b_{2}(X)z^{2}+\cdots ,\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mi>z</mi> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{X}(z)=b_{0}(X)+b_{1}(X)z+b_{2}(X)z^{2}+\cdots ,\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2be6676145cfb6ed4268b20ab0b0483bb3bb543b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:42.542ex; height:3.176ex;" alt="{\displaystyle P_{X}(z)=b_{0}(X)+b_{1}(X)z+b_{2}(X)z^{2}+\cdots ,\,\!}"></span></dd></dl> <p>see <a href="/wiki/K%C3%BCnneth_theorem" title="Künneth theorem">Künneth theorem</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Symmetry">Symmetry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Betti_number&action=edit&section=11" title="Edit section: Symmetry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <i>X</i> is <i>n</i>-dimensional manifold, there is symmetry interchanging <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b98e1d6a69bccd09a4b9b69bdf03a08c1706c8c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.446ex; height:2.343ex;" alt="{\displaystyle n-k}"></span>, for any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{k}(X)=b_{n-k}(X),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{k}(X)=b_{n-k}(X),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b22ea768bc6967f1e179831c081ab100c933ed91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.761ex; height:2.843ex;" alt="{\displaystyle b_{k}(X)=b_{n-k}(X),}"></span></dd></dl> <p>under conditions (a <i>closed</i> and <i>oriented</i> manifold); see <a href="/wiki/Poincar%C3%A9_duality" title="Poincaré duality">Poincaré duality</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Different_coefficients">Different coefficients</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Betti_number&action=edit&section=12" title="Edit section: Different coefficients"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The dependence on the field <i>F</i> is only through its <a href="/wiki/Characteristic_(field)" class="mw-redirect" title="Characteristic (field)">characteristic</a>. If the homology groups are <a href="/wiki/Torsion_(algebra)" title="Torsion (algebra)">torsion-free</a>, the Betti numbers are independent of <i>F</i>. The connection of <i>p</i>-torsion and the Betti number for <a href="/wiki/Characteristic_p" class="mw-redirect" title="Characteristic p">characteristic <i>p</i></a>, for <i>p</i> a prime number, is given in detail by the <a href="/wiki/Universal_coefficient_theorem" title="Universal coefficient theorem">universal coefficient theorem</a> (based on <a href="/wiki/Tor_functor" title="Tor functor">Tor functors</a>, but in a simple case). </p> <div class="mw-heading mw-heading2"><h2 id="More_examples">More examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Betti_number&action=edit&section=13" title="Edit section: More examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol><li>The Betti number sequence for a circle is 1, 1, 0, 0, 0, ...; <dl><dd>the Poincaré polynomial is <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43e308c59c1f1efa99ad225ef13c66e413123aff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.72ex; height:2.343ex;" alt="{\displaystyle 1+x\,}"></span>.</dd></dl></dd></dl></li> <li>The Betti number sequence for a three-<a href="/wiki/Torus" title="Torus">torus</a> is 1, 3, 3, 1, 0, 0, 0, ... . <dl><dd>the Poincaré polynomial is <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1+x)^{3}=1+3x+3x^{2}+x^{3}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+x)^{3}=1+3x+3x^{2}+x^{3}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfdab40d93b2d2adf94ff2d02579005b924f7d63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.788ex; height:3.176ex;" alt="{\displaystyle (1+x)^{3}=1+3x+3x^{2}+x^{3}\,}"></span>.</dd></dl></dd></dl></li> <li>Similarly, for an <i>n</i>-<a href="/wiki/Torus" title="Torus">torus</a>, <dl><dd>the Poincaré polynomial is <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1+x)^{n}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+x)^{n}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/375cb569620a5c398b698ae8f2e4c4b1241edba5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.747ex; height:2.843ex;" alt="{\displaystyle (1+x)^{n}\,}"></span> (by the <a href="/wiki/K%C3%BCnneth_theorem" title="Künneth theorem">Künneth theorem</a>), so the Betti numbers are the <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficients</a>.</dd></dl></dd></dl></li></ol> <p>It is possible for spaces that are infinite-dimensional in an essential way to have an infinite sequence of non-zero Betti numbers. An example is the infinite-dimensional <a href="/wiki/Complex_projective_space" title="Complex projective space">complex projective space</a>, with sequence 1, 0, 1, 0, 1, ... that is periodic, with <a href="/wiki/Period_length" class="mw-redirect" title="Period length">period length</a> 2. In this case the Poincaré function is not a polynomial but rather an infinite series </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+x^{2}+x^{4}+\dotsb }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+x^{2}+x^{4}+\dotsb }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a22da3a63955da66d311329a1de901e08db7d34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.175ex; height:2.843ex;" alt="{\displaystyle 1+x^{2}+x^{4}+\dotsb }"></span>,</dd></dl> <p>which, being a geometric series, can be expressed as the rational function </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{1-x^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{1-x^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/860d7e3de9a9388332bf9047d85e3de0ccf6f2f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.87ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{1-x^{2}}}.}"></span></dd></dl> <p>More generally, any sequence that is periodic can be expressed as a sum of geometric series, generalizing the above. For example <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c,a,b,c,\dots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c,a,b,c,\dots ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebec54a57ded78dd5c39eaba864a0668297a9364" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.429ex; height:2.509ex;" alt="{\displaystyle a,b,c,a,b,c,\dots ,}"></span> has the generating function </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(a+bx+cx^{2}\right)/\left(1-x^{3}\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(a+bx+cx^{2}\right)/\left(1-x^{3}\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df965392611b08bbc831cc0cb3cddefee16b8547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.599ex; height:3.343ex;" alt="{\displaystyle \left(a+bx+cx^{2}\right)/\left(1-x^{3}\right)\,}"></span></dd></dl> <p>and more generally <a href="/wiki/Linear_recursive_sequence" class="mw-redirect" title="Linear recursive sequence">linear recursive sequences</a> are exactly the sequences generated by <a href="/wiki/Rational_functions" class="mw-redirect" title="Rational functions">rational functions</a>; thus the Poincaré series is expressible as a rational function if and only if the sequence of Betti numbers is a linear recursive sequence. </p><p>The Poincaré polynomials of the compact simple <a href="/wiki/Lie_groups" class="mw-redirect" title="Lie groups">Lie groups</a> are: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}P_{SU(n+1)}(x)&=\left(1+x^{3}\right)\left(1+x^{5}\right)\cdots \left(1+x^{2n+1}\right)\\P_{SO(2n+1)}(x)&=\left(1+x^{3}\right)\left(1+x^{7}\right)\cdots \left(1+x^{4n-1}\right)\\P_{Sp(n)}(x)&=\left(1+x^{3}\right)\left(1+x^{7}\right)\cdots \left(1+x^{4n-1}\right)\\P_{SO(2n)}(x)&=\left(1+x^{2n-1}\right)\left(1+x^{3}\right)\left(1+x^{7}\right)\cdots \left(1+x^{4n-5}\right)\\P_{G_{2}}(x)&=\left(1+x^{3}\right)\left(1+x^{11}\right)\\P_{F_{4}}(x)&=\left(1+x^{3}\right)\left(1+x^{11}\right)\left(1+x^{15}\right)\left(1+x^{23}\right)\\P_{E_{6}}(x)&=\left(1+x^{3}\right)\left(1+x^{9}\right)\left(1+x^{11}\right)\left(1+x^{15}\right)\left(1+x^{17}\right)\left(1+x^{23}\right)\\P_{E_{7}}(x)&=\left(1+x^{3}\right)\left(1+x^{11}\right)\left(1+x^{15}\right)\left(1+x^{19}\right)\left(1+x^{23}\right)\left(1+x^{27}\right)\left(1+x^{35}\right)\\P_{E_{8}}(x)&=\left(1+x^{3}\right)\left(1+x^{15}\right)\left(1+x^{23}\right)\left(1+x^{27}\right)\left(1+x^{35}\right)\left(1+x^{39}\right)\left(1+x^{47}\right)\left(1+x^{59}\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>n</mi> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>17</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>19</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>27</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>35</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>27</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>35</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>39</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>47</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>59</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P_{SU(n+1)}(x)&=\left(1+x^{3}\right)\left(1+x^{5}\right)\cdots \left(1+x^{2n+1}\right)\\P_{SO(2n+1)}(x)&=\left(1+x^{3}\right)\left(1+x^{7}\right)\cdots \left(1+x^{4n-1}\right)\\P_{Sp(n)}(x)&=\left(1+x^{3}\right)\left(1+x^{7}\right)\cdots \left(1+x^{4n-1}\right)\\P_{SO(2n)}(x)&=\left(1+x^{2n-1}\right)\left(1+x^{3}\right)\left(1+x^{7}\right)\cdots \left(1+x^{4n-5}\right)\\P_{G_{2}}(x)&=\left(1+x^{3}\right)\left(1+x^{11}\right)\\P_{F_{4}}(x)&=\left(1+x^{3}\right)\left(1+x^{11}\right)\left(1+x^{15}\right)\left(1+x^{23}\right)\\P_{E_{6}}(x)&=\left(1+x^{3}\right)\left(1+x^{9}\right)\left(1+x^{11}\right)\left(1+x^{15}\right)\left(1+x^{17}\right)\left(1+x^{23}\right)\\P_{E_{7}}(x)&=\left(1+x^{3}\right)\left(1+x^{11}\right)\left(1+x^{15}\right)\left(1+x^{19}\right)\left(1+x^{23}\right)\left(1+x^{27}\right)\left(1+x^{35}\right)\\P_{E_{8}}(x)&=\left(1+x^{3}\right)\left(1+x^{15}\right)\left(1+x^{23}\right)\left(1+x^{27}\right)\left(1+x^{35}\right)\left(1+x^{39}\right)\left(1+x^{47}\right)\left(1+x^{59}\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad5d09dcb0fa37ddcf6afbed2efeb1c418d4575f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.505ex; width:92.812ex; height:32.176ex;" alt="{\displaystyle {\begin{aligned}P_{SU(n+1)}(x)&=\left(1+x^{3}\right)\left(1+x^{5}\right)\cdots \left(1+x^{2n+1}\right)\\P_{SO(2n+1)}(x)&=\left(1+x^{3}\right)\left(1+x^{7}\right)\cdots \left(1+x^{4n-1}\right)\\P_{Sp(n)}(x)&=\left(1+x^{3}\right)\left(1+x^{7}\right)\cdots \left(1+x^{4n-1}\right)\\P_{SO(2n)}(x)&=\left(1+x^{2n-1}\right)\left(1+x^{3}\right)\left(1+x^{7}\right)\cdots \left(1+x^{4n-5}\right)\\P_{G_{2}}(x)&=\left(1+x^{3}\right)\left(1+x^{11}\right)\\P_{F_{4}}(x)&=\left(1+x^{3}\right)\left(1+x^{11}\right)\left(1+x^{15}\right)\left(1+x^{23}\right)\\P_{E_{6}}(x)&=\left(1+x^{3}\right)\left(1+x^{9}\right)\left(1+x^{11}\right)\left(1+x^{15}\right)\left(1+x^{17}\right)\left(1+x^{23}\right)\\P_{E_{7}}(x)&=\left(1+x^{3}\right)\left(1+x^{11}\right)\left(1+x^{15}\right)\left(1+x^{19}\right)\left(1+x^{23}\right)\left(1+x^{27}\right)\left(1+x^{35}\right)\\P_{E_{8}}(x)&=\left(1+x^{3}\right)\left(1+x^{15}\right)\left(1+x^{23}\right)\left(1+x^{27}\right)\left(1+x^{35}\right)\left(1+x^{39}\right)\left(1+x^{47}\right)\left(1+x^{59}\right)\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Relationship_with_dimensions_of_spaces_of_differential_forms">Relationship with dimensions of spaces of differential forms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Betti_number&action=edit&section=14" title="Edit section: Relationship with dimensions of spaces of differential forms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In geometric situations when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is a <a href="/wiki/Closed_manifold" title="Closed manifold">closed manifold</a>, the importance of the Betti numbers may arise from a different direction, namely that they predict the dimensions of vector spaces of <a href="/wiki/Closed_differential_form" class="mw-redirect" title="Closed differential form">closed differential forms</a> <i><a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a></i> <a href="/wiki/Exact_differential_form" class="mw-redirect" title="Exact differential form">exact differential forms</a>. The connection with the definition given above is via three basic results, <a href="/wiki/De_Rham%27s_theorem" class="mw-redirect" title="De Rham's theorem">de Rham's theorem</a> and <a href="/wiki/Poincar%C3%A9_duality" title="Poincaré duality">Poincaré duality</a> (when those apply), and the <a href="/wiki/Universal_coefficient_theorem" title="Universal coefficient theorem">universal coefficient theorem</a> of <a href="/wiki/Homology_theory" class="mw-redirect" title="Homology theory">homology theory</a>. </p><p>There is an alternate reading, namely that the Betti numbers give the dimensions of spaces of <a href="/wiki/Harmonic_form" class="mw-redirect" title="Harmonic form">harmonic forms</a>. This requires the use of some of the results of <a href="/wiki/Hodge_theory" title="Hodge theory">Hodge theory</a> on the <a href="/wiki/Hodge_Laplacian" class="mw-redirect" title="Hodge Laplacian">Hodge Laplacian</a>. </p><p>In this setting, <a href="/wiki/Morse_theory" title="Morse theory">Morse theory</a> gives a set of inequalities for alternating sums of Betti numbers in terms of a corresponding alternating sum of the number of <a href="/wiki/Critical_point_(mathematics)" title="Critical point (mathematics)">critical points</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fef58cebf23adff9199f17325aefb5515fdca99d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.666ex; height:2.509ex;" alt="{\displaystyle N_{i}}"></span> of a <a href="/wiki/Morse_function" class="mw-redirect" title="Morse function">Morse function</a> of a given <a href="/wiki/Morse_theory" title="Morse theory">index</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{i}(X)-b_{i-1}(X)+\cdots \leq N_{i}-N_{i-1}+\cdots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>≤</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{i}(X)-b_{i-1}(X)+\cdots \leq N_{i}-N_{i-1}+\cdots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed22248280bc6d0bff0e68a0864fc5ec332b1fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.646ex; height:2.843ex;" alt="{\displaystyle b_{i}(X)-b_{i-1}(X)+\cdots \leq N_{i}-N_{i-1}+\cdots .}"></span></dd></dl> <p><a href="/wiki/Edward_Witten" title="Edward Witten">Edward Witten</a> gave an explanation of these inequalities by using the Morse function to modify the <a href="/wiki/Exterior_derivative" title="Exterior derivative">exterior derivative</a> in the <a href="/wiki/De_Rham_complex" class="mw-redirect" title="De Rham complex">de Rham complex</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Betti_number&action=edit&section=15" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Topological_data_analysis" title="Topological data analysis">Topological data analysis</a></li> <li><a href="/wiki/Torsion_coefficient_(topology)" class="mw-redirect" title="Torsion coefficient (topology)">Torsion coefficient</a></li> <li><a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Betti_number&action=edit&section=16" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBarile,_and_Weisstein" class="citation web cs1">Barile, and Weisstein, Margherita and Eric. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/BettiNumber.html">"Betti number"</a>. From MathWorld--A Wolfram Web Resource.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Betti+number&rft.pub=From+MathWorld--A+Wolfram+Web+Resource.&rft.aulast=Barile%2C+and+Weisstein&rft.aufirst=Margherita+and+Eric&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FBettiNumber.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABetti+number" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Archived at <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211212/XxFGokyYo6g">Ghostarchive</a> and the <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200829013025/https://www.youtube.com/watch?v=XxFGokyYo6g&gl=US&hl=en">Wayback Machine</a>: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlbin2019" class="citation web cs1">Albin, Pierre (2019). <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=XxFGokyYo6g">"History of algebraic topology"</a>. <i><a href="/wiki/YouTube" title="YouTube">YouTube</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=YouTube&rft.atitle=History+of+algebraic+topology&rft.date=2019&rft.aulast=Albin&rft.aufirst=Pierre&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DXxFGokyYo6g&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABetti+number" class="Z3988"></span></span> </li> <li id="cite_note-Hage1996-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hage1996_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPer_Hage1996" class="citation book cs1">Per Hage (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZBdLknuP0BYC&pg=PA49"><i>Island Networks: Communication, Kinship, and Classification Structures in Oceania</i></a>. Cambridge University Press. p. 49. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-55232-5" title="Special:BookSources/978-0-521-55232-5"><bdi>978-0-521-55232-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Island+Networks%3A+Communication%2C+Kinship%2C+and+Classification+Structures+in+Oceania&rft.pages=49&rft.pub=Cambridge+University+Press&rft.date=1996&rft.isbn=978-0-521-55232-5&rft.au=Per+Hage&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DZBdLknuP0BYC%26pg%3DPA49&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABetti+number" class="Z3988"></span></span> </li> <li id="cite_note-Kotiuga2010-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kotiuga2010_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeter_Robert_Kotiuga2010" class="citation book cs1">Peter Robert Kotiuga (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mqLXi0FRIZwC&pg=PA20"><i>A Celebration of the Mathematical Legacy of Raoul Bott</i></a>. American Mathematical Soc. p. 20. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-8381-5" title="Special:BookSources/978-0-8218-8381-5"><bdi>978-0-8218-8381-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Celebration+of+the+Mathematical+Legacy+of+Raoul+Bott&rft.pages=20&rft.pub=American+Mathematical+Soc.&rft.date=2010&rft.isbn=978-0-8218-8381-5&rft.au=Peter+Robert+Kotiuga&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DmqLXi0FRIZwC%26pg%3DPA20&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABetti+number" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Archived at <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211212/NgrIPPqYKjQ">Ghostarchive</a> and the <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130926094324/http://www.youtube.com/watch?v=NgrIPPqYKjQ&list=PL0F555888A4C2329B">Wayback Machine</a>: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWildberger2012" class="citation web cs1">Wildberger, Norman J. (2012). <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=NgrIPPqYKjQ&list=PL6763F57A61FE6FE8&index=41&t=0s">"Delta complexes, Betti numbers and torsion"</a>. <i><a href="/wiki/YouTube" title="YouTube">YouTube</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=YouTube&rft.atitle=Delta+complexes%2C+Betti+numbers+and+torsion&rft.date=2012&rft.aulast=Wildberger&rft.aufirst=Norman+J.&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DNgrIPPqYKjQ%26list%3DPL6763F57A61FE6FE8%26index%3D41%26t%3D0s&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABetti+number" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWitten1982" class="citation cs2"><a href="/wiki/Edward_Witten" title="Edward Witten">Witten, Edward</a> (1982), "Supersymmetry and Morse theory", <i><a href="/wiki/Journal_of_Differential_Geometry" title="Journal of Differential Geometry">Journal of Differential Geometry</a></i>, <b>17</b> (4): 661–692, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4310%2Fjdg%2F1214437492">10.4310/jdg/1214437492</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Differential+Geometry&rft.atitle=Supersymmetry+and+Morse+theory&rft.volume=17&rft.issue=4&rft.pages=661-692&rft.date=1982&rft_id=info%3Adoi%2F10.4310%2Fjdg%2F1214437492&rft.aulast=Witten&rft.aufirst=Edward&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABetti+number" class="Z3988"></span><span style="position:relative; top: -2px;"><span typeof="mw:File"><a href="/wiki/Open_access" title="open access publication – free to read"><img alt="Open access icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Open_Access_logo_PLoS_transparent.svg/9px-Open_Access_logo_PLoS_transparent.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Open_Access_logo_PLoS_transparent.svg/14px-Open_Access_logo_PLoS_transparent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Open_Access_logo_PLoS_transparent.svg/18px-Open_Access_logo_PLoS_transparent.svg.png 2x" data-file-width="640" data-file-height="1000" /></a></span></span></span> </li> </ol></div></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWarner1983" class="citation cs2">Warner, Frank Wilson (1983), <i>Foundations of differentiable manifolds and Lie groups</i>, New York: Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90894-3" title="Special:BookSources/0-387-90894-3"><bdi>0-387-90894-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+differentiable+manifolds+and+Lie+groups&rft.place=New+York&rft.pub=Springer&rft.date=1983&rft.isbn=0-387-90894-3&rft.aulast=Warner&rft.aufirst=Frank+Wilson&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABetti+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoe1998" class="citation cs2">Roe, John (1998), <i>Elliptic Operators, Topology, and Asymptotic Methods</i>, Research Notes in Mathematics Series, vol. 395 (Second ed.), Boca Raton, FL: Chapman and Hall, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-582-32502-1" title="Special:BookSources/0-582-32502-1"><bdi>0-582-32502-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elliptic+Operators%2C+Topology%2C+and+Asymptotic+Methods&rft.place=Boca+Raton%2C+FL&rft.series=Research+Notes+in+Mathematics+Series&rft.edition=Second&rft.pub=Chapman+and+Hall&rft.date=1998&rft.isbn=0-582-32502-1&rft.aulast=Roe&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABetti+number" class="Z3988"></span>.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output 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href="/wiki/Geometric_topology" title="Geometric topology">Geometric</a> <ul><li><a href="/wiki/Low-dimensional_topology" title="Low-dimensional topology">low-dimensional</a></li></ul></li> <li><a href="/wiki/Homology_(mathematics)" title="Homology (mathematics)">Homology</a> <ul><li><a href="/wiki/Cohomology" title="Cohomology">cohomology</a></li></ul></li> <li><a href="/wiki/Set-theoretic_topology" title="Set-theoretic topology">Set-theoretic</a></li> <li><a href="/wiki/Digital_topology" title="Digital topology">Digital</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="4" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/Klein_bottle" title="Klein bottle"><img alt="Computer graphics rendering of a Klein bottle" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Kleinsche_Flasche.png/60px-Kleinsche_Flasche.png" decoding="async" width="60" height="80" class="mw-file-element" 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href="/wiki/Connected_space" title="Connected space">connected</a></li> <li><a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a></li> <li><a href="/wiki/Metric_space" title="Metric space">metric</a></li> <li><a href="/wiki/Uniform_space" title="Uniform space">uniform</a></li></ul></li> <li><a href="/wiki/Homotopy" title="Homotopy">Homotopy</a> <ul><li><a href="/wiki/Homotopy_group" title="Homotopy group">homotopy group</a></li> <li><a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a></li></ul></li> <li><a href="/wiki/Simplicial_complex" title="Simplicial complex">Simplicial complex</a></li> <li><a href="/wiki/CW_complex" title="CW complex">CW complex</a></li> <li><a href="/wiki/Polyhedral_complex" title="Polyhedral complex">Polyhedral complex</a></li> <li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Bundle_(mathematics)" title="Bundle (mathematics)">Bundle (mathematics)</a></li> <li><a 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results</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_fixed-point_theorem" title="Banach fixed-point theorem">Banach fixed-point theorem</a></li> <li><a href="/wiki/De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li> <li><a href="/wiki/Invariance_of_domain" title="Invariance of domain">Invariance of domain</a></li> <li><a href="/wiki/Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a></li> <li><a href="/wiki/Tychonoff%27s_theorem" title="Tychonoff's theorem">Tychonoff's theorem</a></li> <li><a href="/wiki/Urysohn%27s_lemma" title="Urysohn's lemma">Urysohn's lemma</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" 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Betti number - Wikipedia