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float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="manifolds_and_cobordisms">Manifolds and cobordisms</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a></strong> and <strong><a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/cobordism+theory">cobordism theory</a>, <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Cobordism+and+Complex+Oriented+Cohomology">Introduction</a></em></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+Euclidean+space">locally Euclidean space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+transformation">coordinate transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atlas">atlas</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold">manifold</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, ,<a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinite+dimensional+manifold">infinite dimensional manifold</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Banach+manifold">Banach manifold</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+manifold">Hilbert manifold</a>, <a class="existingWikiWord" href="/nlab/show/ILH+manifold">ILH manifold</a>, <a class="existingWikiWord" href="/nlab/show/Frechet+manifold">Frechet manifold</a>, <a class="existingWikiWord" href="/nlab/show/convenient+manifold">convenient manifold</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>, <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/B-bordism">B-bordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/genus">genus</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Genera and invariants</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/signature+genus">signature genus</a>, <a class="existingWikiWord" href="/nlab/show/Kervaire+invariant">Kervaire invariant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-hat+genus">A-hat genus</a>, <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a></p> </li> </ul> <p><strong>Classification</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-manifolds">2-manifolds</a>/<a class="existingWikiWord" href="/nlab/show/surfaces">surfaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus of a surface</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3-manifolds">3-manifolds</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kirby+calculus">Kirby calculus</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4-manifolds">4-manifolds</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dehn+surgery">Dehn surgery</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom%27s+transversality+theorem">Thom's transversality theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+construction">Pontrjagin-Thom construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galatius-Tillmann-Madsen-Weiss+theorem">Galatius-Tillmann-Madsen-Weiss theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometrization+conjecture">geometrization conjecture</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+conjecture">Poincaré conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptization+conjecture">elliptization conjecture</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#Classification'>Classification</a></li> <li><a href='#Homotopy'>Homotopy</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <strong>surface</strong> is a <a class="existingWikiWord" href="/nlab/show/space">space</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> 2, usually understood to be <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected</a>.</p> <p>In <a class="existingWikiWord" href="/nlab/show/differential+topology">differential</a> <a class="existingWikiWord" href="/nlab/show/topology">topology</a>/<a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> this means a 2-<a class="existingWikiWord" href="/nlab/show/dimension+of+a+manifold">dimensional</a> (<a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable</a>/<a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth</a>) <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a>.</p> <p>In <a class="existingWikiWord" href="/nlab/show/complex+analytic+geometry">complex analytic geometry</a> this usually means a <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a> of <em>complex</em> dimension 2 (hence real dimension 4).</p> <p>Similarly and more generally, in <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a> an <em><a class="existingWikiWord" href="/nlab/show/algebraic+surface">algebraic surface</a></em> is a <a class="existingWikiWord" href="/nlab/show/variety">variety</a> of algebraic dimension 2.</p> <h2 id="properties">Properties</h2> <h3 id="Classification">Classification</h3> <p>The <a class="existingWikiWord" href="/nlab/show/orientation">oriented</a> <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed</a> <a class="existingWikiWord" href="/nlab/show/real+manifold">real</a> surfaces are classified, up to <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, by a <a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">g \in \mathbb{N}</annotation></semantics></math> called the <em><a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus</a></em> (intuitively its “number of holes”, in a sense):</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus</a></th><th><a class="existingWikiWord" href="/nlab/show/orientation">oriented</a> <a class="existingWikiWord" href="/nlab/show/surface">surface</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-sphere">2-sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-torus">2-torus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝕋</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{T}^2</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/double+torus">double torus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Σ</mi> <mn>2</mn> <mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\Sigma^2_2</annotation></semantics></math></td></tr> </tbody></table> <p>(cf. <a href="#Kinsey91">Kinsey 1991 Thm. 4.14</a>, <a href="#GallierXu13">Gallier & Xu 2013 Thm 6.3</a>, <a href="#BBR21">Thm. 4.14</a>)</p> <p id="FundamentalPolygons"> Here the closed oriented surface <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Σ</mi> <mi>g</mi> <mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\Sigma^2_g</annotation></semantics></math> of genus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> may be obtained from the <a class="existingWikiWord" href="/nlab/show/regular+polygon">regular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>4</mn> <mi>g</mi> </mrow> <annotation encoding="application/x-tex">4g</annotation> </semantics> </math>-gon</a> by identifying</p> <ul> <li>all the <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> <a class="existingWikiWord" href="/nlab/show/vertices">vertices</a> with a single point,</li> </ul> <p>and, going clockwise for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>g</mi><mo>−</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">k \in \{0, \cdots, g-1\}</annotation></semantics></math>,</p> <ul> <li> <p>the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>4</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">4k+1</annotation></semantics></math>st boundary <a class="existingWikiWord" href="/nlab/show/edge">edge</a> with the reverse of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>4</mn><mi>k</mi><mo>+</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">4k +3</annotation></semantics></math>rd,</p> </li> <li> <p>the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>4</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">4k+2</annotation></semantics></math>nd edge with the reverse of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>4</mn><mi>k</mi><mo>+</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">4k+4</annotation></semantics></math>th.</p> </li> </ul> <center> <img src="/nlab/files/FundamPolygonsForOrntdClosedSurfaces.jpg" width="420" /> </center> <p>Similarly, the <a class="existingWikiWord" href="/nlab/show/orientation">non-orientable</a> <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed</a> <a class="existingWikiWord" href="/nlab/show/real+manifold">real</a> <a class="existingWikiWord" href="/nlab/show/surfaces">surfaces</a> are classified by a <a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>∈</mo><msub><mi>ℕ</mi> <mrow><mo>≥</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">h \in \mathbb{N}_{\geq 1}</annotation></semantics></math>, also called the (non-orientable) <em>genus</em> or the <em>number of crosscaps</em></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/genus+of+a+surface"># crosscaps</a></th><th><a class="existingWikiWord" href="/nlab/show/orientation">non-orientable</a> <a class="existingWikiWord" href="/nlab/show/surface">surface</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/real+projective+plane">projective plane</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><msup><mi>P</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}P^2</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Klein+bottle">Klein 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href="/nlab/show/regular+polygon">regular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>2</mn> <mi>h</mi> </mrow> <annotation encoding="application/x-tex">2h</annotation> </semantics> </math>-gon</a> by identifying</p> <ul> <li>all the <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> <a class="existingWikiWord" href="/nlab/show/vertices">vertices</a> with a single point,</li> </ul> <p>and, going clockwise for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>h</mi><mo>−</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">k \in \{0, \cdots, h-1\}</annotation></semantics></math>,</p> <ul> <li>the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">2k+1</annotation></semantics></math>st boundary edge with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">2k + 2</annotation></semantics></math>nd.</li> </ul> <center> <img src="/nlab/files/FundamenPolygonsForNonOrntdClosedSurfaces.jpg" width="310" /> </center> <blockquote> <p>(This is a conveniently concise but not the most intuitively visualized choice of fundamental polygons – other choices are possible and often discussed, cf. also <a href="https://mathoverflow.net/q/172784/381">MO:q/172784</a>.)</p> </blockquote> <h3 id="Homotopy">Homotopy</h3> <p>The <a href="#Classification">above</a> classification may be restated in <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebro-topological 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b_i)_{i=1}^g</annotation></semantics></math>, and the <a class="existingWikiWord" href="/nlab/show/attaching+map">attaching map</a> is a representative in this free group of the composition of <a class="existingWikiWord" href="/nlab/show/group+commutators">group commutators</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>a</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>b</mi> <mi>i</mi></msub><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>a</mi> <mi>i</mi></msub><mo>⋅</mo><msub><mi>b</mi> <mi>i</mi></msub><mo>⋅</mo><msubsup><mi>a</mi> <mi>i</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>⋅</mo><msubsup><mi>b</mi> <mi>i</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> [a_i, b_i] \;\coloneqq\; a_i \cdot b_i \cdot a_i^{-1} \cdot b_i^{-1} \,, </annotation></semantics></math></div> <p>as indicated.</p> <p>Similarly for (the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of) the non-orientable surface <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Σ</mi> <mover><mi>h</mi><mo>¯</mo></mover> <mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\Sigma^2_{\overline{h}}</annotation></semantics></math> of crosscap number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math>:</p> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="163.21pt" height="104.587pt" viewBox="0 0 163.21 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style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -50.019255 -34.603644 L 27.957929 -34.603644 " transform="matrix(0.995183,0,0,-0.995183,81.211903,54.434138)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.488027 2.868687 C -2.032708 1.149468 -1.020018 0.333035 0.000523519 -0.000603557 C -1.020018 -0.334242 -2.032708 -1.14675 -2.488027 -2.869894 " transform="matrix(0.995183,0,0,-0.995183,109.272917,88.870493)"></path> </g> </svg> <p>It follows that:</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p></p> <p>The <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> of the <a class="existingWikiWord" href="/nlab/show/orientation">oriented</a> <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed</a> <a class="existingWikiWord" href="/nlab/show/real+manifold">real</a> surface <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Σ</mi> <mi>g</mi> <mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\Sigma^2_g</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">g \in \mathbb{N}</annotation></semantics></math> (see <a href="#Classification">above</a>) is the <a class="existingWikiWord" href="/nlab/show/quotient+group">quotient group</a> of the <a class="existingWikiWord" href="/nlab/show/free+group">free group</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>g</mi></mrow><annotation encoding="application/x-tex">2g</annotation></semantics></math> generators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>g</mi></msub><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>b</mi> <mi>g</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a_1, \cdots, a_g, \, b_1, \cdots, b_g)</annotation></semantics></math> by the <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a> generated by the group product of the <a class="existingWikiWord" href="/nlab/show/group+commutators">group commutators</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>a</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>b</mi> <mi>i</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a_i, b_i]</annotation></semantics></math> of the sequence of <a class="existingWikiWord" href="/nlab/show/pairs">pairs</a> of generators:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><msubsup><mi>Σ</mi> <mi>g</mi> <mn>2</mn></msubsup><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">⟨</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>g</mi></msub><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>b</mi> <mi>g</mi></msub><mo maxsize="1.2em" minsize="1.2em">⟩</mo><mo maxsize="1.2em" minsize="1.2em">/</mo><mo maxsize="1.8em" minsize="1.8em">(</mo><mstyle displaystyle="false"><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>i</mi></msub></mrow><mo stretchy="false">[</mo><msub><mi>a</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>b</mi> <mi>i</mi></msub><mo stretchy="false">]</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mstyle></mrow><annotation encoding="application/x-tex"> \pi_1\big( \Sigma^2_g \big) \;\simeq\; \big\langle a_1, \cdots, a_g ,\, b_1, \cdots, b_g \big\rangle \big/ \Big( \textstyle{\prod_i} [a_i, b_i] \Big) \,. </annotation></semantics></math></div> <p>Similarly, the <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> of the <a class="existingWikiWord" href="/nlab/show/orientation">non-orientable</a> <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed</a> <a class="existingWikiWord" href="/nlab/show/real+manifold">real</a> surface <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Σ</mi> <mover><mi>h</mi><mo>¯</mo></mover> <mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\Sigma^2_{\overline{h}}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>∈</mo><msub><mi>ℕ</mi> <mrow><mo>≥</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">h \in \mathbb{N}_{\geq 1}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">crosscaps</a> is the <a class="existingWikiWord" href="/nlab/show/quotient+group">quotient group</a> of the <a class="existingWikiWord" href="/nlab/show/free+group">free group</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> generators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>h</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a_1, \cdots, a_h)</annotation></semantics></math> by the <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a> generated by the group product of the squares <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>a</mi> <mspace width="thinmathspace"></mspace> <mn>2</mn></msubsup><mo>≔</mo><mspace width="thinmathspace"></mspace><msub><mi>a</mi> <mi>i</mi></msub><mo>⋅</mo><msub><mi>a</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">a^2_ \,\coloneqq\, a_i \cdot a_i</annotation></semantics></math> of the sequence of <a class="existingWikiWord" href="/nlab/show/pairs">pairs</a> of generators:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><msubsup><mi>Σ</mi> <mover><mi>h</mi><mo>¯</mo></mover> <mn>2</mn></msubsup><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">⟨</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>h</mi></msub><mo maxsize="1.2em" minsize="1.2em">⟩</mo><mo maxsize="1.2em" minsize="1.2em">/</mo><mo maxsize="1.8em" minsize="1.8em">(</mo><mstyle displaystyle="false"><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>i</mi></msub></mrow><msubsup><mi>a</mi> <mi>i</mi> <mn>2</mn></msubsup><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mstyle></mrow><annotation encoding="application/x-tex"> \pi_1\big( \Sigma^2_{\overline{h}} \big) \;\simeq\; \big\langle a_1, \cdots, a_h \big\rangle \big/ \Big( \textstyle{\prod_i} a_i^2 \Big) \,. </annotation></semantics></math></div> <p></p> </div> </p> <p>(cf. <a href="#GallierXu13">Gallier & Xu 2013 p 100</a>, <a href="#Actipes13">Actipes 2013 Thm. 6.3</a>).</p> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/plane">plane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere">sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/torus">torus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Klein+bottle">Klein bottle</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/punctured">punctured</a> surface</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry+of+curves+and+surfaces">differential geometry of curves and surfaces</a></p> </li> <li> <p>analog for dimension 1: <a class="existingWikiWord" href="/nlab/show/curve">curve</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+curve">algebraic curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus of a surface</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/area">area</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+space+of+curves">moduli space of curves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+surface">complex surface</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/surface+knot">surface knot</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypersurface">hypersurface</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/minimal+surface">minimal surface</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a> in <a class="existingWikiWord" href="/nlab/show/low-dimensional+topology">low</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a></strong>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-manifolds">2-manifolds</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3-manifolds">3-manifolds</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4-manifolds">4-manifolds</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/5-manifolds">5-manifolds</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/8-manifolds">8-manifolds</a></p> </li> </ul> </div> <h2 id="references">References</h2> <p>Monographs:</p> <ul> <li id="Kinsey91"> <p><a class="existingWikiWord" href="/nlab/show/L.+Christine+Kinsey">L. Christine Kinsey</a>: <em>Topology of Surfaces</em>, Spinger (1991) [<a href="https://doi.org/10.1007/978-1-4612-0899-0">doi:10.1007/978-1-4612-0899-0</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/kinsey.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Richard+E.+Schwartz">Richard E. Schwartz</a>: <em>Mostly Surfaces</em>, American Mathematical Society, Student Mathematical Library <strong>60</strong> (2011) [draft: <a href="http://www.math.brown.edu/reschwar/Papers/surfacebook.pdf">pdf</a>, endmatter:<a href="https://www.ams.org/books/stml/060/stml060-endmatter.pdf">pdf</a>]</p> </li> <li id="GallierXu13"> <p><a class="existingWikiWord" href="/nlab/show/Jean+Gallier">Jean Gallier</a>, <a class="existingWikiWord" href="/nlab/show/Dianna+Xu">Dianna Xu</a>: <em>A Guide to the Classification Theorem for Compact Surfaces</em>, Springer (2013) [<a href="https://doi.org/10.1007/978-3-642-34364-3">doi:10.1007/978-3-642-34364-3</a>, <a href="https://en.wikipedia.org/wiki/A_Guide_to_the_Classification_Theorem_for_Compact_Surfaces">Wikipedia entry</a>]</p> </li> <li id="BBR21"> <p>Clark Bray, Adrian Butcher, Simon Rubinstein-Salzedo, chapters 2 & 4 of: <em>Algebraic Topology</em>, Springer (2021) [<a href="https://doi.org/10.1007/978-3-030-70608-1">doi:10.1007/978-3-030-70608-1</a>, <a href="https://link.springer.com/content/pdf/10.1007/978-3-030-70608-1.pdf">pdf</a>]</p> </li> </ul> <p>See also:</p> <ul> <li> <p>Wikipedia: <em><a href="https://en.wikipedia.org/wiki/Surface_(topology)">Surface (topology)</a></em></p> </li> <li> <p>Wikipedia: <em><a href="https://en.wikipedia.org/wiki/Genus_g_surface">Genus g surface</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Manifold+Atlas">Manifold Atlas</a>, <em><a href="www.map.mpim-bonn.mpg.de/2-manifolds">2-manifolds</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+Andrews">Peter Andrews</a>: <em>The Classification of Surfaces</em>, The American Mathematical Monthly <strong>95</strong> 9 (1988) 861-867 [<a href="https://doi.org/10.2307/2322906">doi:10.2307/2322906</a>, <a href="https://www.jstor.org/stable/2322906">jstor:2322906</a>]</p> </li> </ul> <p>Review and exposition of the classification of surfaces by fundamental <a class="existingWikiWord" href="/nlab/show/polygons">polygons</a>:</p> <ul> <li> <p>E. C. Zeeman: <em>An Introduction to Topology – The Classification theorem for Surfaces</em> [<a href="https://www.maths.ed.ac.uk/~v1ranick/surgery/zeeman.pdf">pdf</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/surgery/ecztop.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Zeeman-ClassificationOfSurfaces.pdf" title="pdf">pdf</a>]</p> </li> <li> <p>Chen Hui George Teo: <em>Classification of Surfaces</em>, REU notes (2011) [<a href="https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Teo.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/CHGT-ClassificationOfSurfaces.pdf" title="pdf">pdf</a>]</p> </li> <li> <p>Thomas George: <em>The Classification of Surfaces with Boundary</em>, REU notes (2001) [<a href="https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/George.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/George-SurfacesWithBoundary.pdf" title="pdf">pdf</a>]</p> </li> <li> <p>Eugene Gorsky: <em>Classification of Surfaces</em>, lecture notes (2021) [<a href="https://www.math.ucdavis.edu/~egorskiy/MAT215C-s21/Sosinsky.pdf">pdf</a>]</p> </li> <li> <p>Ana da Silva Rodrigues: <em>Classification of Surfaces</em>, BSc thesis, ETH (2023) [<a href="https://people.math.ethz.ch/~acannas/Student_Papers/BSc_Theses/2023_bsc_da_silva_classification_of_surfaces.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Rodrigues-ClassificationOfSurfaces.pdf" title="pdf">pdf</a>]</p> </li> </ul> <p>See also:</p> <ul> <li>Wikipedia: <em><a href="https://en.wikipedia.org/wiki/Fundamental_polygon">Fundamental polygon</a></em></li> </ul> <p>Review of the computation of the <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> of surfaces:</p> <ul> <li id="Actipes13">Matthew Actipes: <em>On the fundamental group of surfaces</em>, REU notes (2013) [<a href="http://math.uchicago.edu/~may/REU2013/REUPapers/Actipes.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Actipes-FundamentalGroupOfSurfaces.pdf" title="pdf">pdf</a>]</li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a> of surfaces:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/William+Fulton">William Fulton</a>, §18 of: <em>Algebraic Topology – A First Course</em>, Graduate Texts in Mathematics <strong>153</strong>, Springer (1995) [<a href="https://doi.org/10.1007/978-1-4612-4180-5">doi:10.1007/978-1-4612-4180-5</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 15, 2025 at 12:09:09. 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