CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> 3-manifold in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> 3-manifold </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/6952/#Item_4" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="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='#triangulability_and_smoothing'>Triangulability and smoothing</a></li> <li><a href='#poincar_conjecture'>Poincaré conjecture</a></li> <li><a href='#geometrization_conjecture'>Geometrization conjecture</a></li> <li><a href='#virtually_fibered_conjecture'>Virtually fibered conjecture</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> <li><a href='#general'>General</a></li> <li><a href='#hyperbolic_3manifolds'>Hyperbolic 3-manifolds</a></li> <li><a href='#vafawitten_theory'>Vafa-Witten theory</a></li> <li><a href='#quantum_invariants'>Quantum invariants</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>3-manifold</em> is a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> 3. (Our default meaning of “manifold” is <a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a>, unless a qualifier is added, e.g., <em>smooth</em> manifold.)</p> <h2 id="properties">Properties</h2> <h3 id="triangulability_and_smoothing">Triangulability and smoothing</h3> <p>The following is taken from <a href="#Hatcher">Hatcher</a>:</p> <blockquote> <p>A pleasant feature of 3-manifolds, in contrast to higher dimensions, is that there is no essential difference between smooth, piecewise linear, and topological manifolds. It was shown by Bing and <a href="#Moise52">Moise</a> in the 1950s that <a class="existingWikiWord" href="/nlab/show/triangulation+theorem">every topological 3-manifold can be triangulated</a> as a simplicial complex whose combinatorial type is unique up to subdivision. And every triangulation of a 3-manifold can be taken to be a smooth triangulation in some differential structure on the manifold, unique up to diffeomorphism. Thus every topological 3-manifold has a unique smooth structure, and the classifications up to diffeomorphism and homeomorphism coincide.</p> </blockquote> <p>Thus it makes no essential difference if we consider 3-manifolds as mere topological manifolds, or as <a class="existingWikiWord" href="/nlab/show/piecewise-linear+manifolds">piecewise-linear manifolds</a> or <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a>. It’s often technically convenient to work in the smooth category.</p> <h3 id="poincar_conjecture">Poincaré conjecture</h3> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+conjecture">Poincaré conjecture</a>)</strong></p> <p>Every <a class="existingWikiWord" href="/nlab/show/simply+connected">simply connected</a> <a class="existingWikiWord" href="/nlab/show/compact+space">compact</a> 3-manifold without boundary is <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphic</a> to the 3-sphere.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>A proof strategy was given by <a class="existingWikiWord" href="/nlab/show/Richard+Hamilton">Richard Hamilton</a>: imagine the manifold is equipped with a <a class="existingWikiWord" href="/nlab/show/metric">metric</a>. Follow the <a class="existingWikiWord" href="/nlab/show/Ricci+flow">Ricci flow</a> of that metric through the space of metrics. As the flow proceeds along parameter time, it will from time to time pass through metrics that describe singular geometries where the compact metric manifold pinches off into separate manifolds. Follow the flow through these singularities and then continue the flow on each of the resulting components. If this process terminates in finite parameter time with the metric on each component stabilizing to that of the round 3-sphere, then the original manifold was a 3-sphere.</p> <p>The hard technical part of this program is to show that the passage through the singularities can be controlled. This was finally shown by <a class="existingWikiWord" href="/nlab/show/Grigori+Perelman">Grigori Perelman</a>.</p> </div> <h3 id="geometrization_conjecture">Geometrization conjecture</h3> <p>The <em><a class="existingWikiWord" href="/nlab/show/geometrization+conjecture">geometrization conjecture</a></em> says that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure.</p> <h3 id="virtually_fibered_conjecture">Virtually fibered conjecture</h3> <p>The <em><a class="existingWikiWord" href="/nlab/show/virtually+fibered+conjecture">virtually fibered conjecture</a></em> says that every closed, irreducible, atoroidal 3-manifold with infinite fundamental group has a finite cover which is a surface bundle over the circle.</p> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/3-sphere">3-sphere</a>/<a class="existingWikiWord" href="/nlab/show/SU%282%29">SU(2)</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(3)</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/SO%283%29">SO(3)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lens+space">lens space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hantzsche-Wendt+manifold">Hantzsche-Wendt manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Seifert+manifold">Seifert manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperbolic+3-manifold">hyperbolic 3-manifold</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/atoroidal+3-manifold">atoroidal 3-manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Seifert+3-manifold">Seifert 3-manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperbolic+3-manifold">hyperbolic 3-manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dehn+surgery">Dehn surgery</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kirby+calculus">Kirby calculus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/low+dimensional+topology">low dimensional topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/knot+theory">knot theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+topology">arithmetic topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+2-framing">Atiyah 2-framing</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lens+space">lens space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associative+submanifold">associative submanifold</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> <h3 id="general">General</h3> <p>Review:</p> <ul> <li id="Thurston92"> <p><a class="existingWikiWord" href="/nlab/show/William+Thurston">William Thurston</a>: <em>Three-dimensional geometry and topology</em>, preliminary draft, University of Minnesota (1992) [1979: <a href="https://archive.org/details/ThurstonTheGeometryAndTopologyOfThreeManifolds/mode/2up">ark:/13960/t3714t34v</a>, 1991: <a class="existingWikiWord" href="/nlab/files/Thurston-3dGeometry-1991.pdf" title="pdf">pdf</a>, 2002: <a href="https://www.math.unl.edu/~jkettinger2/thurston.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Thurston-3dGeometry-2002.pdf" title="pdf">pdf</a>]</p> <p>the first three chapters of which are published in expanded form as:</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/William+Thurston">William Thurston</a>: <em>The Geometry and Topology of Three-Manifolds</em>, Princeton University Press (1997) [<a href="https://press.princeton.edu/books/hardcover/9780691083049/three-dimensional-geometry-and-topology-volume-1">ISBN:9780691083049</a>, <a href="https://en.wikipedia.org/wiki/The_geometry_and_topology_of_three-manifolds">Wikipedia page</a>]</p> <p>in particular orbifolds are discussed in <a href="http://library.msri.org/books/gt3m/PDF/13.pdf">chapter 13</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Vladimir+Turaev">Vladimir Turaev</a>: <em>Quantum invariants of knots and 3-manifolds</em>, de Gruyter Studies in Mathematics <strong>18</strong>, de Gruyter & Co. (1994) [<a href="https://doi.org/10.1515/9783110435221">doi:10.1515/9783110435221</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/turaev5.pdf">pdf</a>]</p> </li> <li id="Hatcher"> <p><a class="existingWikiWord" href="/nlab/show/Allen+Hatcher">Allen Hatcher</a>, <em>The classification of 3-manifolds – a brief overview</em>, (<a href="https://www.math.cornell.edu/~hatcher/Papers/3Msurvey.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tomotada+Ohtsuki">Tomotada Ohtsuki</a>: <em>Quantum Invariants – A Study of Knots, 3-Manifolds, and Their Sets</em>, World Scientific (2001) [<a href="https://doi.org/10.1142/4746">doi:10.1142/4746</a>]</p> </li> <li id="Martelli16"> <p>Bruno Martelli, <em>An Introduction to Geometric Topology</em> (<a href="https://arxiv.org/abs/1610.02592">arXiv:1610.02592</a>)</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/triangulation+theorem">triangulation theorem</a> for <a class="existingWikiWord" href="/nlab/show/3-manifolds">3-manifolds</a>:</p> <ul> <li id="Moise52"><a class="existingWikiWord" href="/nlab/show/Edwin+E.+Moise">Edwin E. Moise</a>, <em>Affine Structures in 3-Manifolds: V. The Triangulation Theorem and Hauptvermutung</em>, Annals of Mathematics Second Series, Vol. 56, No. 1 (Jul., 1952), pp. 96-114 (<a href="https://doi.org/10.2307/1969769">doi:10.2307/1969769</a>, <a href="https://www.jstor.org/stable/1969769">jstor:1969769</a>)</li> </ul> <p>3-manifolds as <a class="existingWikiWord" href="/nlab/show/branched+covers">branched covers</a> of the <a class="existingWikiWord" href="/nlab/show/3-sphere">3-sphere</a>:</p> <ul> <li>J. Montesinos, <em>A representation of closed orientable 3-manifolds as 3-fold branched coverings of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">S^3</annotation></semantics></math></em>, Bull. Amer. Math. Soc. 80 (1974), 845-846 (<a href="https://projecteuclid.org/euclid.bams/1183535815">Euclid:1183535815</a>)</li> </ul> <p>See also</p> <ul> <li id="BottCattaneo98"><a class="existingWikiWord" href="/nlab/show/Raoul+Bott">Raoul Bott</a>, <a class="existingWikiWord" href="/nlab/show/Alberto+Cattaneo">Alberto Cattaneo</a>, <em>Integral Invariants of 3-Manifolds</em>, J. Diff. Geom., 48 (1998) 91-133 (<a href="https://arxiv.org/abs/dg-ga/9710001">arXiv:dg-ga/9710001</a>)</li> </ul> <h3 id="hyperbolic_3manifolds">Hyperbolic 3-manifolds</h3> <p>On <a class="existingWikiWord" href="/nlab/show/hyperbolic+3-manifolds">hyperbolic 3-manifolds</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/William+Thurston">William Thurston</a>, <em>Hyperbolic Structures on 3-manifolds, I: Deformation of acylindrical manifolds</em>, Annals of Math, 124 (1986), 203–246 (<a href="https://www.jstor.org/stable/1971277">jstor:1971277</a>, <a href="https://arxiv.org/abs/math/9801019">arXiv:math/9801019</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/William+Thurston">William Thurston</a>, <em>Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle</em> (<a href="https://arxiv.org/abs/math/9801045">arXiv:math/9801045</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/William+Thurston">William Thurston</a>, <em>Three dimensional manifolds, Kleinian groups and hyperbolic geometry</em>, Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 3 (1982), 357-381 (<a href="https://projecteuclid.org/euclid.bams/1183548782">euclid.bams/1183548782</a>)</p> </li> </ul> <h3 id="vafawitten_theory">Vafa-Witten theory</h3> <p>Computations of Vafa-Witten invariants of 3-manifolds are given in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Sergei+Gukov">Sergei Gukov</a>, Artan Sheshmani, <a class="existingWikiWord" href="/nlab/show/Shing-Tung+Yau">Shing-Tung Yau</a>, <em>3-manifolds and Vafa-Witten theory</em> (<a href="https://arxiv.org/abs/2207.05775">arXiv:2207.05775</a>).</li> </ul> <h3 id="quantum_invariants">Quantum invariants</h3> <blockquote> <p>(see also at <em><a class="existingWikiWord" href="/nlab/show/knot+theory">knot theory</a>)</em></p> </blockquote> <ul> <li>Yuya Murakami: <em>A proof of a conjecture of Gukov-Pei-Putrov-Vafa</em> [<a href="https://arxiv.org/abs/2302.13526">arXiv:2302.13526</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 4, 2024 at 06:39:22. See the <a href="/nlab/history/3-manifold" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/3-manifold" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/6952/#Item_4">Discuss</a><span class="backintime"><a href="/nlab/revision/3-manifold/30" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/3-manifold" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/3-manifold" accesskey="S" class="navlink" id="history" rel="nofollow">History (30 revisions)</a> <a href="/nlab/show/3-manifold/cite" style="color: black">Cite</a> <a href="/nlab/print/3-manifold" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/3-manifold" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>