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href="#Differences_between_low-dimensional_and_high-dimensional_topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Differences between low-dimensional and high-dimensional topology</span> </div> </a> <ul id="toc-Differences_between_low-dimensional_and_high-dimensional_topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Important_tools_in_geometric_topology" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Important_tools_in_geometric_topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Important tools in geometric topology</span> </div> </a> <button aria-controls="toc-Important_tools_in_geometric_topology-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 Important tools in geometric topology subsection</span> </button> <ul id="toc-Important_tools_in_geometric_topology-sublist" class="vector-toc-list"> <li id="toc-Fundamental_group" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fundamental_group"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Fundamental group</span> </div> </a> <ul id="toc-Fundamental_group-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orientability" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orientability"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Orientability</span> </div> </a> <ul id="toc-Orientability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Handle_decompositions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Handle_decompositions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Handle decompositions</span> </div> </a> <ul id="toc-Handle_decompositions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Local_flatness" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Local_flatness"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Local flatness</span> </div> </a> <ul id="toc-Local_flatness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Schönflies_theorems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Schönflies_theorems"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Schönflies theorems</span> </div> </a> <ul id="toc-Schönflies_theorems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Branches_of_geometric_topology" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Branches_of_geometric_topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Branches of geometric topology</span> </div> </a> <button aria-controls="toc-Branches_of_geometric_topology-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 Branches of geometric topology subsection</span> </button> <ul id="toc-Branches_of_geometric_topology-sublist" class="vector-toc-list"> <li id="toc-Low-dimensional_topology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Low-dimensional_topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Low-dimensional topology</span> </div> </a> <ul id="toc-Low-dimensional_topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Knot_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Knot_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Knot theory</span> </div> </a> <ul id="toc-Knot_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-High-dimensional_geometric_topology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#High-dimensional_geometric_topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>High-dimensional geometric topology</span> </div> </a> <ul id="toc-High-dimensional_geometric_topology-sublist" class="vector-toc-list"> </ul> </li> </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">5</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">6</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 id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon 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class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Topologia_geom%C3%A8trica" title="Topologia geomètrica – Catalan" lang="ca" hreflang="ca" data-title="Topologia geomètrica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Geometrisk_topologi" title="Geometrisk topologi – Danish" lang="da" hreflang="da" data-title="Geometrisk topologi" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Geometrische_Topologie" title="Geometrische Topologie – German" lang="de" hreflang="de" data-title="Geometrische Topologie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%93%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%B9%CE%BA%CE%AE_%CF%84%CE%BF%CF%80%CE%BF%CE%BB%CE%BF%CE%B3%CE%AF%CE%B1" title="Γεωμετρική τοπολογία – Greek" lang="el" hreflang="el" data-title="Γεωμετρική τοπολογία" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Topolog%C3%ADa_geom%C3%A9trica" title="Topología geométrica – Spanish" lang="es" hreflang="es" data-title="Topología geométrica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Topologie_g%C3%A9om%C3%A9trique" title="Topologie géométrique – French" lang="fr" hreflang="fr" data-title="Topologie géométrique" 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/%EA%B8%B0%ED%95%98%ED%95%99%EC%A0%81_%EC%9C%84%EC%83%81%EC%88%98%ED%95%99" title="기하학적 위상수학 – Korean" lang="ko" hreflang="ko" data-title="기하학적 위상수학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Meetkundige_topologie" title="Meetkundige topologie – Dutch" lang="nl" hreflang="nl" data-title="Meetkundige topologie" 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/%E5%B9%BE%E4%BD%95%E5%AD%A6%E7%9A%84%E3%83%88%E3%83%9D%E3%83%AD%E3%82%B8%E3%83%BC" title="幾何学的トポロジー – Japanese" lang="ja" hreflang="ja" data-title="幾何学的トポロジー" 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-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B0%E0%A9%87%E0%A8%96%E0%A8%BE%E0%A8%97%E0%A8%A3%E0%A8%BF%E0%A8%A4%E0%A8%BF%E0%A8%95_%E0%A8%9F%E0%A9%8C%E0%A8%AA%E0%A9%8C%E0%A8%B2%E0%A9%8C%E0%A8%9C%E0%A9%80" title="ਰੇਖਾਗਣਿਤਿਕ ਟੌਪੌਲੌਜੀ – Punjabi" lang="pa" hreflang="pa" data-title="ਰੇਖਾਗਣਿਤਿਕ ਟੌਪੌਲੌਜੀ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Topologie_geometric%C4%83" title="Topologie geometrică – Romanian" lang="ro" 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<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">Branch of mathematics studying (smooth) functions of manifolds</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For the mathematical object, see <a href="/wiki/Geometric_topology_(object)" title="Geometric topology (object)">Geometric topology (object)</a>.</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Borromean_Seifert_surface.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Borromean_Seifert_surface.png/220px-Borromean_Seifert_surface.png" decoding="async" width="220" height="277" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/e/e4/Borromean_Seifert_surface.png 1.5x" data-file-width="317" data-file-height="399" /></a><figcaption>A <a href="/wiki/Seifert_surface" title="Seifert surface">Seifert surface</a> bounded by a set of <a href="/wiki/Borromean_rings" title="Borromean rings">Borromean rings</a>; these surfaces can be used as tools in geometric topology</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>geometric topology</b> is the study of <a href="/wiki/Manifold" title="Manifold">manifolds</a> and <a href="/wiki/Map_(mathematics)#Maps_as_functions" title="Map (mathematics)">maps</a> between them, particularly <a href="/wiki/Embedding" title="Embedding">embeddings</a> of one manifold into another. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometric_topology&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Geometric topology as an area distinct from <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a> may be said to have originated in the 1935 classification of <a href="/wiki/Lens_space" title="Lens space">lens spaces</a> by <a href="/wiki/Reidemeister_torsion" class="mw-redirect" title="Reidemeister torsion">Reidemeister torsion</a>, which required distinguishing spaces that are <a href="/wiki/Homotopy_equivalent" class="mw-redirect" title="Homotopy equivalent">homotopy equivalent</a> but not <a href="/wiki/Homeomorphic" class="mw-redirect" title="Homeomorphic">homeomorphic</a>. This was the origin of <a href="/wiki/Simple_homotopy" class="mw-redirect" title="Simple homotopy"><i>simple</i> homotopy</a> theory. The use of the term geometric topology to describe these seems to have originated rather recently.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Differences_between_low-dimensional_and_high-dimensional_topology">Differences between low-dimensional and high-dimensional topology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometric_topology&action=edit&section=2" title="Edit section: Differences between low-dimensional and high-dimensional topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Manifolds differ radically in behavior in high and low dimension. </p><p>High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in <a href="/wiki/Codimension" title="Codimension">codimension</a> 3 and above. <a href="/wiki/Low-dimensional_topology" title="Low-dimensional topology">Low-dimensional topology</a> is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2. </p><p>Dimension 4 is special, in that in some respects (topologically), dimension 4 is high-dimensional, while in other respects (differentiably), dimension 4 is low-dimensional; this overlap yields phenomena exceptional to dimension 4, such as <a href="/wiki/Exotic_R4" title="Exotic R4">exotic differentiable structures on <b>R</b><sup>4</sup></a>. Thus the topological classification of 4-manifolds is in principle tractable, and the key questions are: does a topological manifold admit a differentiable structure, and if so, how many? Notably, the smooth case of dimension 4 is the last open case of the <a href="/wiki/Generalized_Poincar%C3%A9_conjecture" title="Generalized Poincaré conjecture">generalized Poincaré conjecture</a>; see <a href="/wiki/Gluck_twist" class="mw-redirect" title="Gluck twist">Gluck twists</a>. </p><p>The distinction is because <a href="/wiki/Surgery_theory" title="Surgery theory">surgery theory</a> works in dimension 5 and above (in fact, in many cases, it works topologically in dimension 4, though this is very involved to prove), and thus the behavior of manifolds in dimension 5 and above may be studied using the surgery theory program. In dimension 4 and below (topologically, in dimension 3 and below), surgery theory does not work. Indeed, one approach to discussing low-dimensional manifolds is to ask "what would surgery theory predict to be true, were it to work?" – and then understand low-dimensional phenomena as deviations from this. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Whitneytrickstep2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Whitneytrickstep2.svg/220px-Whitneytrickstep2.svg.png" decoding="async" width="220" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Whitneytrickstep2.svg/330px-Whitneytrickstep2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Whitneytrickstep2.svg/440px-Whitneytrickstep2.svg.png 2x" data-file-width="1450" data-file-height="200" /></a><figcaption>The <a href="/wiki/Whitney_trick" class="mw-redirect" title="Whitney trick">Whitney trick</a> requires 2+1 dimensions, hence surgery theory requires 5 dimensions.</figcaption></figure> <p>The precise reason for the difference at dimension 5 is because the <a href="/wiki/Whitney_embedding_theorem" title="Whitney embedding theorem">Whitney embedding theorem</a>, the key technical trick which underlies surgery theory, requires 2+1 dimensions. Roughly, the Whitney trick allows one to "unknot" knotted spheres – more precisely, remove self-intersections of immersions; it does this via a <a href="/wiki/Homotopy" title="Homotopy">homotopy</a> of a disk – the disk has 2 dimensions, and the homotopy adds 1 more – and thus in codimension greater than 2, this can be done without intersecting itself; hence embeddings in codimension greater than 2 can be understood by surgery. In surgery theory, the key step is in the middle dimension, and thus when the middle dimension has codimension more than 2 (loosely, 2½ is enough, hence total dimension 5 is enough), the Whitney trick works. The key consequence of this is Smale's <a href="/wiki/H-cobordism_theorem" class="mw-redirect" title="H-cobordism theorem"><i>h</i>-cobordism theorem</a>, which works in dimension 5 and above, and forms the basis for surgery theory. </p><p>A modification of the Whitney trick can work in 4 dimensions, and is called <a href="/wiki/Casson_handle" title="Casson handle">Casson handles</a> – because there are not enough dimensions, a Whitney disk introduces new kinks, which can be resolved by another Whitney disk, leading to a sequence ("tower") of disks. The limit of this tower yields a topological but not differentiable map, hence surgery works topologically but not differentiably in dimension 4. </p> <div class="mw-heading mw-heading2"><h2 id="Important_tools_in_geometric_topology">Important tools in geometric topology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometric_topology&action=edit&section=3" title="Edit section: Important tools in geometric topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/List_of_geometric_topology_topics" title="List of geometric topology topics">List of geometric topology topics</a></div> <div class="mw-heading mw-heading3"><h3 id="Fundamental_group">Fundamental group</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometric_topology&action=edit&section=4" title="Edit section: Fundamental group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Fundamental_group" title="Fundamental group">Fundamental group</a></div> <p>In all dimensions, the <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> of a manifold is a very important invariant, and determines much of the structure; in dimensions 1, 2 and 3, the possible fundamental groups are restricted, while in dimension 4 and above every <a href="/wiki/Finitely_presented_group" class="mw-redirect" title="Finitely presented group">finitely presented group</a> is the fundamental group of a manifold (note that it is sufficient to show this for 4- and 5-dimensional manifolds, and then to take products with spheres to get higher ones). </p> <div class="mw-heading mw-heading3"><h3 id="Orientability">Orientability</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometric_topology&action=edit&section=5" title="Edit section: Orientability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Orientability" title="Orientability">Orientability</a></div> <p>A manifold is orientable if it has a consistent choice of <a href="/wiki/Orientation_(mathematics)" class="mw-redirect" title="Orientation (mathematics)">orientation</a>, and a <a href="/wiki/Connected_space" title="Connected space">connected</a> orientable manifold has exactly two different possible orientations. In this setting, various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds often employ methods of <a href="/wiki/Homology_theory" class="mw-redirect" title="Homology theory">homology theory</a>, whereas for <a href="/wiki/Differentiable_manifolds" class="mw-redirect" title="Differentiable manifolds">differentiable manifolds</a> more structure is present, allowing a formulation in terms of <a href="/wiki/Differential_form" title="Differential form">differential forms</a>. An important generalization of the notion of orientability of a space is that of orientability of a family of spaces parameterized by some other space (a <a href="/wiki/Fiber_bundle" title="Fiber bundle">fiber bundle</a>) for which an orientation must be selected in each of the spaces which varies continuously with respect to changes in the parameter values. </p> <div class="mw-heading mw-heading3"><h3 id="Handle_decompositions">Handle decompositions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometric_topology&action=edit&section=6" title="Edit section: Handle decompositions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Handle_decomposition" title="Handle decomposition">Handle decomposition</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Sphere_with_three_handles.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Sphere_with_three_handles.png/220px-Sphere_with_three_handles.png" decoding="async" width="220" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Sphere_with_three_handles.png/330px-Sphere_with_three_handles.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Sphere_with_three_handles.png/440px-Sphere_with_three_handles.png 2x" data-file-width="1308" data-file-height="1004" /></a><figcaption>A 3-ball with three 1-handles attached.</figcaption></figure> <p>A <a href="/wiki/Handle_decomposition" title="Handle decomposition">handle decomposition</a> of an <i>m</i>-<a href="/wiki/Manifold" title="Manifold">manifold</a> <i>M</i> is a union </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 \emptyset =M_{-1}\subset M_{0}\subset M_{1}\subset M_{2}\subset \dots \subset M_{m-1}\subset M_{m}=M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∅</mi> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>⊂</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⊂</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊂</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⊂</mo> <mo>⋯</mo> <mo>⊂</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>⊂</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \emptyset =M_{-1}\subset M_{0}\subset M_{1}\subset M_{2}\subset \dots \subset M_{m-1}\subset M_{m}=M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b9d1239661d5e74e7969b3ed80484c452d8b69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:55.586ex; height:2.676ex;" alt="{\displaystyle \emptyset =M_{-1}\subset M_{0}\subset M_{1}\subset M_{2}\subset \dots \subset M_{m-1}\subset M_{m}=M}"></span></dd></dl> <p>where each <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 M_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eda8fd06f1cd5de22ed07385a0f8aa19773b2de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.054ex; height:2.509ex;" alt="{\displaystyle M_{i}}"></span> is obtained from <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 M_{i-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{i-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7f34b876026cca5f06e9a487af7cd3620d63328" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.154ex; height:2.509ex;" alt="{\displaystyle M_{i-1}}"></span> by the attaching 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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-<b>handles</b>. A handle decomposition is to a manifold what a <a href="/wiki/CW_complex" title="CW complex">CW-decomposition</a> is to a topological space—in many regards the purpose of a handle decomposition is to have a language analogous to CW-complexes, but adapted to the world of <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifolds</a>. Thus an <i>i</i>-handle is the smooth analogue of an <i>i</i>-cell. Handle decompositions of manifolds arise naturally via <a href="/wiki/Morse_theory" title="Morse theory">Morse theory</a>. The modification of handle structures is closely linked to <a href="/wiki/Cerf_theory" title="Cerf theory">Cerf theory</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Local_flatness">Local flatness</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometric_topology&action=edit&section=7" title="Edit section: Local flatness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Local_flatness" title="Local flatness">Local flatness</a></div> <p><a href="/wiki/Local_flatness" title="Local flatness">Local flatness</a> is a property of a <a href="/wiki/Submanifold" title="Submanifold">submanifold</a> in a <a href="/wiki/Topological_manifold" title="Topological manifold">topological manifold</a> of larger <a href="/wiki/Dimension" title="Dimension">dimension</a>. In the <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> of topological manifolds, locally flat submanifolds play a role similar to that of <a href="/wiki/Submanifold#Embedded_submanifolds" title="Submanifold">embedded submanifolds</a> in the category of <a href="/wiki/Smooth_manifolds" class="mw-redirect" title="Smooth manifolds">smooth manifolds</a>. </p><p>Suppose a <i>d</i> dimensional manifold <i>N</i> is embedded into an <i>n</i> dimensional manifold <i>M</i> (where <i>d</i> < <i>n</i>). 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 x\in N,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>N</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in N,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3c8d7fd864ef9386a936b3bf5ea31b82c595bb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.881ex; height:2.509ex;" alt="{\displaystyle x\in N,}"></span> we say <i>N</i> is <b>locally flat</b> at <i>x</i> if there is a neighborhood <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 U\subset M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>⊂</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\subset M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edd7b08ced14b1958599bba595488f7c314e59da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.323ex; height:2.176ex;" alt="{\displaystyle U\subset M}"></span> of <i>x</i> such that the <a href="/wiki/Topological_pair" title="Topological pair">topological pair</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 (U,U\cap N)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>U</mi> <mo>,</mo> <mi>U</mi> <mo>∩</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (U,U\cap N)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c56023f77636e15055a1b66540d0b3d8a3fc0552" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.055ex; height:2.843ex;" alt="{\displaystyle (U,U\cap N)}"></span> is <a href="/wiki/Homeomorphic" class="mw-redirect" title="Homeomorphic">homeomorphic</a> to the pair <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 {R} ^{n},\mathbb {R} ^{d})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {R} ^{n},\mathbb {R} ^{d})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e63ccf95fb4bae4e27d90ec31535fa6cc479c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.51ex; height:3.176ex;" alt="{\displaystyle (\mathbb {R} ^{n},\mathbb {R} ^{d})}"></span>, with a standard inclusion 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 \mathbb {R} ^{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a713426956296f1668fce772df3c60b9dde8a685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.77ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{d}}"></span> as a subspace 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 \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>. That is, there exists a homeomorphism <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 U\to R^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">→</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\to R^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/623ffcffad0076760568fa2a9b7313415e37176e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.379ex; height:2.343ex;" alt="{\displaystyle U\to R^{n}}"></span> such that the <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a> 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 U\cap N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>∩</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\cap N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56643414a96610663ee313767b4408826852931a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.429ex; height:2.176ex;" alt="{\displaystyle U\cap N}"></span> coincides with <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 {R} ^{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a713426956296f1668fce772df3c60b9dde8a685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.77ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{d}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Schönflies_theorems"><span id="Sch.C3.B6nflies_theorems"></span>Schönflies theorems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometric_topology&action=edit&section=8" title="Edit section: Schönflies theorems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Jordan-Sch%C3%B6nflies_theorem" class="mw-redirect" title="Jordan-Schönflies theorem">Jordan-Schönflies theorem</a></div> <p>The generalized <a href="/wiki/Schoenflies_theorem" class="mw-redirect" title="Schoenflies theorem">Schoenflies theorem</a> states that, if an (<i>n</i> − 1)-dimensional <a href="/wiki/Sphere" title="Sphere">sphere</a> <i>S</i> is embedded into the <i>n</i>-dimensional sphere <i>S<sup>n</sup></i> in a <a href="/wiki/Locally_flat" class="mw-redirect" title="Locally flat">locally flat</a> way (that is, the embedding extends to that of a thickened sphere), then the pair (<i>S<sup>n</sup></i>, <i>S</i>) is homeomorphic to the pair (<i>S<sup>n</sup></i>, <i>S</i><sup><i>n</i>−1</sup>), where <i>S</i><sup><i>n</i>−1</sup> is the equator of the <i>n</i>-sphere. Brown and Mazur received the <a href="/wiki/Veblen_Prize" class="mw-redirect" title="Veblen Prize">Veblen Prize</a> for their independent proofs<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><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> of this theorem. </p> <div class="mw-heading mw-heading2"><h2 id="Branches_of_geometric_topology">Branches of geometric topology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometric_topology&action=edit&section=9" title="Edit section: Branches of geometric topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Low-dimensional_topology">Low-dimensional topology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometric_topology&action=edit&section=10" title="Edit section: Low-dimensional topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Low-dimensional_topology" title="Low-dimensional topology">Low-dimensional topology</a></div> <p><a href="/wiki/Low-dimensional_topology" title="Low-dimensional topology">Low-dimensional topology</a> includes: </p> <ul><li><a href="/wiki/Surface_(topology)" title="Surface (topology)">Surfaces</a> (2-manifolds)</li> <li><a href="/wiki/3-manifold" title="3-manifold">3-manifolds</a></li> <li><a href="/wiki/4-manifold" title="4-manifold">4-manifolds</a></li></ul> <p>each have their own theory, where there are some connections. </p><p>Low-dimensional topology is strongly geometric, as reflected in the <a href="/wiki/Uniformization_theorem" title="Uniformization theorem">uniformization theorem</a> in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curvature/spherical, zero curvature/flat, negative curvature/hyperbolic – and the <a href="/wiki/Geometrization_conjecture" title="Geometrization conjecture">geometrization conjecture</a> (now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of 8 possible geometries. </p><p>2-dimensional topology can be studied as <a href="/wiki/Complex_geometry" title="Complex geometry">complex geometry</a> in one variable (<a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a> are complex curves) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure. </p> <div class="mw-heading mw-heading3"><h3 id="Knot_theory">Knot theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometric_topology&action=edit&section=11" title="Edit section: Knot theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Knot_theory" title="Knot theory">Knot theory</a></div> <p><a href="/wiki/Knot_theory" title="Knot theory">Knot theory</a> is the study of <a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">mathematical knots</a>. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In mathematical language, a knot is an <a href="/wiki/Embedding" title="Embedding">embedding</a> of a <a href="/wiki/Circle" title="Circle">circle</a> in 3-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, <b>R</b><sup>3</sup> (since we're using topology, a circle isn't bound to the classical geometric concept, but to all of its <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphisms</a>). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of <b>R</b><sup>3</sup> upon itself (known as an <a href="/wiki/Ambient_isotopy" title="Ambient isotopy">ambient isotopy</a>); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself. </p><p>To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other <a href="/wiki/3-manifold" title="3-manifold">three-dimensional spaces</a> and objects other than circles can be used; see <i><a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">knot (mathematics)</a></i>. Higher-dimensional knots are <a href="/wiki/N-sphere" title="N-sphere"><i>n</i>-dimensional spheres</a> in <i>m</i>-dimensional Euclidean space. </p> <div class="mw-heading mw-heading3"><h3 id="High-dimensional_geometric_topology">High-dimensional geometric topology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometric_topology&action=edit&section=12" title="Edit section: High-dimensional geometric topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In high-dimensional topology, <a href="/wiki/Characteristic_classes" class="mw-redirect" title="Characteristic classes">characteristic classes</a> are a basic invariant, and <a href="/wiki/Surgery_theory" title="Surgery theory">surgery theory</a> is a key theory. </p><p>A <b><a href="/wiki/Characteristic_class" title="Characteristic class">characteristic class</a></b> is a way of associating to each <a href="/wiki/Principal_bundle" title="Principal bundle">principal bundle</a> on a <a href="/wiki/Topological_space" title="Topological space">topological space</a> <i>X</i> a <a href="/wiki/Cohomology" title="Cohomology">cohomology</a> class of <i>X</i>. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses <a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">sections</a> or not. In other words, characteristic classes are global <a href="/wiki/Topological_invariant" class="mw-redirect" title="Topological invariant">invariants</a> which measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>, <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a> and <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>. </p><p><b><a href="/wiki/Surgery_theory" title="Surgery theory">Surgery theory</a></b> is a collection of techniques used to produce one <a href="/wiki/Manifold" title="Manifold">manifold</a> from another in a 'controlled' way, introduced by <a href="/wiki/John_Milnor" title="John Milnor">Milnor</a> (<a href="#CITEREFMilnor1961">1961</a>). Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with, <a href="/wiki/Handle_decomposition" title="Handle decomposition">handlebody decompositions</a>. It is a major tool in the study and classification of manifolds of dimension greater than 3. </p><p>More technically, the idea is to start with a well-understood manifold <i>M</i> and perform surgery on it to produce a manifold <i>M </i>′ having some desired property, in such a way that the effects on the <a href="/wiki/Homology_(mathematics)" title="Homology (mathematics)">homology</a>, <a href="/wiki/Homotopy_group" title="Homotopy group">homotopy groups</a>, or other interesting invariants of the manifold are known. </p><p>The classification of <a href="/wiki/Exotic_sphere" title="Exotic sphere">exotic spheres</a> by <a href="/wiki/Michel_Kervaire" title="Michel Kervaire">Kervaire</a> and <a href="/wiki/John_Milnor" title="John Milnor"> Milnor</a> (<a href="#CITEREFKervaireMilnor1963">1963</a>) led to the emergence of surgery theory as a major tool in high-dimensional topology. </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=Geometric_topology&action=edit&section=13" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Category:Maps_of_manifolds" title="Category:Maps of manifolds">Category:Maps of manifolds</a></li> <li><a href="/wiki/List_of_geometric_topology_topics" title="List of geometric topology topics">List of geometric topology topics</a></li> <li><a href="/wiki/Plumbing_(mathematics)" title="Plumbing (mathematics)">Plumbing (mathematics)</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=Geometric_topology&action=edit&section=14" 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 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.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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.meta.stackexchange.com/questions/2840/what-is-geometric-topology">"What is geometric topology?"</a>. <i>math.meta.stackexchange.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">May 30,</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=math.meta.stackexchange.com&rft.atitle=What+is+geometric+topology%3F&rft_id=https%3A%2F%2Fmath.meta.stackexchange.com%2Fquestions%2F2840%2Fwhat-is-geometric-topology&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometric+topology" 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"><a href="/wiki/Morton_Brown" title="Morton Brown">Brown, Morton</a> (1960), A proof of the generalized Schoenflies theorem. <i>Bull. Amer. Math. Soc.</i>, vol. 66, pp. 74–76. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0117695">0117695</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Mazur, Barry, On embeddings of spheres., <i>Bull. Amer. Math. Soc.</i> 65 1959 59–65. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0117693">0117693</a></span> </li> </ol></div></div> <ul><li>R. B. Sher and <a href="/wiki/R._J._Daverman" class="mw-redirect" title="R. J. Daverman">R. J. Daverman</a> (2002), <i>Handbook of Geometric Topology</i>, North-Holland. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-82432-4" title="Special:BookSources/0-444-82432-4">0-444-82432-4</a>.</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 .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol 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.navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Topology" title="Template:Topology"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Topology" title="Template talk:Topology"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Topology" title="Special:EditPage/Template:Topology"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Topology" style="font-size:114%;margin:0 4em"><a href="/wiki/Topology" title="Topology">Topology</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Fields</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_topology" title="General topology">General (point-set)</a></li> <li><a href="/wiki/Algebraic_topology" title="Algebraic topology">Algebraic</a></li> <li><a href="/wiki/Combinatorial_topology" title="Combinatorial topology">Combinatorial</a></li> <li><a href="/wiki/Continuum_(topology)" title="Continuum (topology)">Continuum</a></li> <li><a href="/wiki/Differential_topology" title="Differential topology">Differential</a></li> <li><a class="mw-selflink selflink">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" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Kleinsche_Flasche.png/90px-Kleinsche_Flasche.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Kleinsche_Flasche.png/120px-Kleinsche_Flasche.png 2x" data-file-width="1171" data-file-height="1561" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Key concepts</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/Open_set" title="Open set">Open set</a> / <a href="/wiki/Closed_set" title="Closed set">Closed set</a></li> <li><a href="/wiki/Interior_(topology)" title="Interior (topology)">Interior</a></li> <li><a href="/wiki/Continuity_(topology)" class="mw-redirect" title="Continuity (topology)">Continuity</a></li> <li><a href="/wiki/Topological_space" title="Topological space">Space</a> <ul><li><a href="/wiki/Compact_space" title="Compact space">compact</a></li> <li><a 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 href="/wiki/Second-countable_space" title="Second-countable space">Second-countable space</a></li> <li><a href="/wiki/Cobordism" title="Cobordism">Cobordism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Metrics and properties</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a></li> <li><a href="/wiki/Betti_number" title="Betti number">Betti number</a></li> <li><a href="/wiki/Winding_number" title="Winding number">Winding number</a></li> <li><a href="/wiki/Chern_class" title="Chern class">Chern number</a></li> <li><a href="/wiki/Orientability" title="Orientability">Orientability</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;"><a href="/wiki/Category:Theorems_in_topology" title="Category:Theorems in topology">Key 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="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" 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<li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikiversity_logo_2017.svg" class="mw-file-description" title="Wikiversity page"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/16px-Wikiversity_logo_2017.svg.png" decoding="async" width="16" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/24px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/32px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></a></span> <a href="https://en.wikiversity.org/wiki/Topology" class="extiw" title="wikiversity:Topology">Wikiversity</a></li> <li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" 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