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> string topology 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> string topology </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/1045/#Item_24" 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="topology">Topology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topology">topology</a></strong> (<a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>, <a class="existingWikiWord" href="/nlab/show/point-free+topology">point-free topology</a>)</p> <p>see also <em><a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></em> and <em><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></em></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology">Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>, <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+for+the+topology">base for the topology</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood+base">neighbourhood base</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finer+topology">finer/coarser topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a>, <a class="existingWikiWord" href="/nlab/show/topological+interior">interior</a>, <a class="existingWikiWord" href="/nlab/show/topological+boundary">boundary</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separation+axiom">separation</a>, <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sobriety</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/uniformly+continuous+function">uniformly continuous function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+embedding">embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+map">open map</a>, <a class="existingWikiWord" href="/nlab/show/closed+map">closed map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a>, <a class="existingWikiWord" href="/nlab/show/sub-net">sub-net</a>, <a class="existingWikiWord" href="/nlab/show/filter">filter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/convergence">convergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a><a class="existingWikiWord" href="/nlab/show/Top">Top</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a></li> </ul> </li> </ul> <p><strong><a href="Top#UniversalConstructions">Universal constructions</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a>, <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>, <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>,</p> </li> <li> <p>fiber space, <a class="existingWikiWord" href="/nlab/show/space+attachment">space attachment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product+space">product space</a>, <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+telescope">mapping telescope</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/colimits+of+normal+spaces">colimits of normal spaces</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nice+topological+space">nice topological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>, <a class="existingWikiWord" href="/nlab/show/metrisable+space">metrisable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+space">Kolmogorov space</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>, <a class="existingWikiWord" href="/nlab/show/regular+space">regular space</a>, <a class="existingWikiWord" href="/nlab/show/normal+space">normal space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sober+space">sober space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a>, <a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a></p> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+topological+space">sequentially compact</a>, <a class="existingWikiWord" href="/nlab/show/countably+compact+topological+space">countably compact</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a>, <a class="existingWikiWord" href="/nlab/show/sigma-compact+topological+space">sigma-compact</a>, <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a>, <a class="existingWikiWord" href="/nlab/show/countably+paracompact+topological+space">countably paracompact</a>, <a class="existingWikiWord" href="/nlab/show/strongly+compact+topological+space">strongly compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+space">second-countable space</a>, <a class="existingWikiWord" href="/nlab/show/first-countable+space">first-countable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a>, <a class="existingWikiWord" href="/nlab/show/locally+contractible+space">locally contractible space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+space">connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+connected+space">locally connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simply-connected+space">simply-connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+simply-connected+space">locally simply-connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a>, <a class="existingWikiWord" href="/nlab/show/point+space">point space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+space">codiscrete space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/order+topology">order topology</a>, <a class="existingWikiWord" href="/nlab/show/specialization+topology">specialization topology</a>, <a class="existingWikiWord" href="/nlab/show/Scott+topology">Scott topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/real+line">real line</a>, <a class="existingWikiWord" href="/nlab/show/plane">plane</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a>, <a class="existingWikiWord" href="/nlab/show/cone">cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>, <a class="existingWikiWord" href="/nlab/show/ball">ball</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle">circle</a>, <a class="existingWikiWord" href="/nlab/show/torus">torus</a>, <a class="existingWikiWord" href="/nlab/show/annulus">annulus</a>, <a class="existingWikiWord" href="/nlab/show/Moebius+strip">Moebius strip</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polytope">polytope</a>, <a class="existingWikiWord" href="/nlab/show/polyhedron">polyhedron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+space">projective space</a> (<a class="existingWikiWord" href="/nlab/show/real+projective+space">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space+%28mathematics%29">configuration space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path">path</a>, <a class="existingWikiWord" href="/nlab/show/loop">loop</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a>: <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>, <a class="existingWikiWord" href="/nlab/show/topology+of+uniform+convergence">topology of uniform convergence</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/path+space">path space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cantor+space">Cantor space</a>, <a class="existingWikiWord" href="/nlab/show/Mandelbrot+space">Mandelbrot space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+curve">Peano curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+with+two+origins">line with two origins</a>, <a class="existingWikiWord" href="/nlab/show/long+line">long line</a>, <a class="existingWikiWord" href="/nlab/show/Sorgenfrey+line">Sorgenfrey line</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-topology">K-topology</a>, <a class="existingWikiWord" href="/nlab/show/Dowker+space">Dowker space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Warsaw+circle">Warsaw circle</a>, <a class="existingWikiWord" href="/nlab/show/Hawaiian+earring+space">Hawaiian earring space</a></p> </li> </ul> <p><strong>Basic statements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hausdorff+spaces+are+sober">Hausdorff spaces are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schemes+are+sober">schemes are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+images+of+compact+spaces+are+compact">continuous images of compact spaces are compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces">closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact">open subspaces of compact Hausdorff spaces are locally compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff">quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lebesgue+number+lemma">Lebesgue number lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces">sequentially compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+totally+bounded">sequentially compact metric spaces are totally bounded</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous">continuous metric space valued function on compact metric space is uniformly continuous</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+are+normal">paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity">paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+injections+are+embeddings">closed injections are embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+maps+to+locally+compact+spaces+are+closed">proper maps to locally compact spaces are closed</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings">injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+sigma-compact+spaces+are+paracompact">locally compact and sigma-compact spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+second-countable+spaces+are+sigma-compact">locally compact and second-countable spaces are sigma-compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+regular+spaces+are+paracompact">second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CW-complexes+are+paracompact+Hausdorff+spaces">CW-complexes are paracompact Hausdorff spaces</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Urysohn%27s+lemma">Urysohn's lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tietze+extension+theorem">Tietze extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tychonoff+theorem">Tychonoff theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tube+lemma">tube lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael%27s+theorem">Michael's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brouwer%27s+fixed+point+theorem">Brouwer's fixed point theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jordan+curve+theorem">Jordan curve theorem</a></p> </li> </ul> <p><strong>Analysis Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/intermediate+value+theorem">intermediate value theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extreme+value+theorem">extreme value theorem</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological homotopy theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>, <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+extension+property">homotopy extension property</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibration">Hurewicz cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+cofiber+sequence">cofiber sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+category">Strøm model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></p> </li> </ul> </div></div> <h4 id="higher_algebra">Higher algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> <h2 id="algebraic_theories">Algebraic theories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#the_string_operations'>The string operations</a></li> <ul> <li><a href='#the_string_product'>The string product</a></li> <li><a href='#the_bvoperator'>The BV-operator</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#InTermsOfTQFTs'>As a TQFT</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In <strong>string topology</strong> one studies the <a class="existingWikiWord" href="/nlab/show/BV-algebra">BV-algebra</a>-structure on the <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a> of the <a class="existingWikiWord" href="/nlab/show/free+loop+space">free loop space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">X^{S^1}</annotation></semantics></math> of an <a class="existingWikiWord" href="/nlab/show/oriented">oriented</a> <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, or more generally the <a class="existingWikiWord" href="/nlab/show/framed+little+2-disk+operad">framed little 2-disk algebra</a>-structure on the singular <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>. This is a special case of the general algebraic structure on higher order <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, as discussed there.</p> <p>The study of <em>string topology</em> was initated by <a class="existingWikiWord" href="/nlab/show/Moira+Chas">Moira Chas</a> and <a class="existingWikiWord" href="/nlab/show/Dennis+Sullivan">Dennis Sullivan</a>.</p> <h2 id="the_string_operations">The string operations</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">L X</annotation></semantics></math> for its <a class="existingWikiWord" href="/nlab/show/free+loop+space">free loop space</a> (for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> regarded as a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>) and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>L</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_\bullet(L X)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a> of this space (with coefficients in the <a class="existingWikiWord" href="/nlab/show/integer">integer</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>).</p> <h3 id="the_string_product">The string product</h3> <div class="num_defn"> <h6 id="definition">Definition</h6> <p>The <strong>string product</strong> is a morphism of <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>s</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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>L</mi><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>L</mi><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>H</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>dim</mi><mi>X</mi></mrow></msub><mo stretchy="false">(</mo><mi>L</mi><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (-)\cdot(-) : H_\bullet(L X) \otimes H_\bullet(L X) \to H_{\bullet - dim X}(L X) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">dim X</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, defined as follows:</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ev</mi> <mo>*</mo></msub><mo>:</mo><mi>L</mi><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">ev_* : L X \to X</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/evaluation+map">evaluation map</a> at the basepoint of the loops.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>α</mi><mo stretchy="false">]</mo><mo>∈</mo><msub><mi>H</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>L</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[\alpha] \in H_i(L X)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>β</mi><mo stretchy="false">]</mo><mo>∈</mo><msub><mi>H</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><mi>L</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[\beta] \in H_j(L X)</annotation></semantics></math> we can find representatives <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ev</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ev(\alpha)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ev</mi><mo stretchy="false">(</mo><mi>β</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ev(\beta)</annotation></semantics></math> intersect <a class="existingWikiWord" href="/nlab/show/transversal+map">transversally</a>. There is then an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>i</mi><mo>+</mo><mi>j</mi><mo stretchy="false">)</mo><mo>−</mo><mi>dim</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">((i+j)-dim X)</annotation></semantics></math>-chain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>⋅</mo><mi>β</mi></mrow><annotation encoding="application/x-tex">\alpha \cdot \beta</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ev</mi><mo stretchy="false">(</mo><mi>α</mi><mo>⋅</mo><mi>β</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ev(\alpha \cdot \beta)</annotation></semantics></math> is the chain given by that intersection: above <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ev</mi><mo stretchy="false">(</mo><mi>α</mi><mo>⋅</mo><mi>β</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \in ev(\alpha \cdot \beta)</annotation></semantics></math> this is the loop obtained by concatenating <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>β</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">\beta_x</annotation></semantics></math> at their common basepoint. The <em>string product</em> is then defined using such representatives by</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><mi>α</mi><mo stretchy="false">]</mo><mo>⋅</mo><mo stretchy="false">[</mo><mi>β</mi><mo stretchy="false">]</mo><mo>:</mo><mo>=</mo><mo stretchy="false">[</mo><mi>α</mi><mo>⋅</mo><mi>β</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\alpha] \cdot [\beta] := [\alpha \cdot \beta] \,. </annotation></semantics></math></div></div> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>The string product is <a class="existingWikiWord" href="/nlab/show/associativity">associative</a> and graded-commutative.</p> </div> <p>This is due to (<a href="#ChasSullivan">ChasSullivan</a>). There is is a more elegant way to capture this, due to (<a href="#CohenJones">CohenJones</a>):</p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>→</mo><mn>8</mn><mo>←</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex"> S^1 \coprod S^1 \to 8 \leftarrow S^1 </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/cospan">cospan</a> that exhibts the inner and the outer circle of the figure “8” topological space. By forming <a class="existingWikiWord" href="/nlab/show/hom+space">hom space</a>s this induces the <a class="existingWikiWord" href="/nlab/show/span">span</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>X</mi> <mn>8</mn></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>in</mi></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>out</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>L</mi><mi>X</mi><mo>×</mo><mi>L</mi><mi>X</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>L</mi><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && X^8 \\ & {}^{\mathllap{in}}\swarrow && \searrow^{\mathrlap{out}} \\ L X \times L X &&&& L X } \,. </annotation></semantics></math></div> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>in</mi> <mo>!</mo></msup></mrow><annotation encoding="application/x-tex">in^!</annotation></semantics></math> for the “pullback” in <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>in</mi></mrow><annotation encoding="application/x-tex">in</annotation></semantics></math> (the dual <a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>) and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>out</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">out_*</annotation></semantics></math> for the ordinary pushforward.</p> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p>The string product is the pull-push operation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>out</mi> <mo>*</mo></msub><mo>∘</mo><msup><mi>in</mi> <mo>!</mo></msup><mo>:</mo><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>L</mi><mi>X</mi><mo>×</mo><mi>L</mi><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>L</mi><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>L</mi><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>H</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>dim</mi><mi>X</mi></mrow></msub><mo stretchy="false">(</mo><mi>L</mi><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> out_* \circ in^! : H_\bullet(L X \times L X) \simeq H_\bullet(L X) \otimes H_\bullet(L X) \to H_{\bullet - dim X}(L X) \,. </annotation></semantics></math></div></div> <p>This is due to (<a href="#CohenJones">CohenJones</a>).</p> <h3 id="the_bvoperator">The BV-operator</h3> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>Define a morphism of <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>:</mo><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>L</mi><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>H</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>L</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Delta : H_\bullet(L X) \to H_{\bullet + 1}(L X) </annotation></semantics></math></div> <p>as follows. Consider first the rotation map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><mi>L</mi><mi>X</mi><mo>→</mo><mi>L</mi><mi>X</mi></mrow><annotation encoding="application/x-tex"> \rho : S^1 \times L X \to L X </annotation></semantics></math></div> <p>that sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>γ</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>t</mi><mo>↦</mo><mi>γ</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>+</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\theta, \gamma) \mapsto (t \mapsto \gamma(\theta + t))</annotation></semantics></math>. Then take</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>:</mo><mi>a</mi><mo>↦</mo><msub><mi>ρ</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">]</mo><mo>×</mo><mi>a</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \Delta : a \mapsto \rho_* ([S^1] \times a) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">]</mo><mo>∈</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[S^1] \in H_1(S^1)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/fundamental+class">fundamental class</a> of the <a class="existingWikiWord" href="/nlab/show/circle">circle</a>.</p> </div> <p>This is called the <strong>BV-operator</strong> for string topology.</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/Goldman+bracket">Goldman bracket</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>L</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_0(L X)</annotation></semantics></math> is equivalent to the string product applied to the image of the BV-operator</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><mo stretchy="false">[</mo><msub><mi>γ</mi> <mn>1</mn></msub><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><msub><mi>γ</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo stretchy="false">}</mo><mo>=</mo><mi>Δ</mi><mo stretchy="false">[</mo><msub><mi>Γ</mi> <mn>1</mn></msub><mo stretchy="false">]</mo><mo>⋅</mo><mi>Δ</mi><mo stretchy="false">[</mo><msub><mi>Γ</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \{[\gamma_1], [\gamma_2]\} = \Delta[\Gamma_1] \cdot \Delta[\Gamma_2] \,. </annotation></semantics></math></div></div> <p>This is due to (<a href="#ChasSullivan">ChasSullivan</a>).</p> <h2 id="properties">Properties</h2> <h3 id="InTermsOfTQFTs">As a TQFT</h3> <p>The structures studied in the <em>string topology</em> of a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> may be understood as being essentially the data of a 2-dimensional <a class="existingWikiWord" href="/nlab/show/topological+field+theory">topological field theory</a> <a class="existingWikiWord" href="/nlab/show/sigma+model">sigma model</a> with <a class="existingWikiWord" href="/nlab/show/target+space">target space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, or rather its linearization to an <a class="existingWikiWord" href="/nlab/show/HQFT">HQFT</a> (with due care on some technical subtleties).</p> <p>The idea is that the <a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a> of a closed or open <a class="existingWikiWord" href="/nlab/show/string">string</a>-<a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> propagating on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a> or path space of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, respectively. The space of <a class="existingWikiWord" href="/nlab/show/state">state</a>s of the string is some space of sections over this configuration space, to which the (co)homology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>L</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_\bullet(L X)</annotation></semantics></math> is an approximation. The string topology operations are then the <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>-representation with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in the <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>Bord</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Ch</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> H_\bullet(Bord_2) \to Ch_\bullet </annotation></semantics></math></div> <p>given by the <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> corresponding to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-modelon these state spaces, acting on these state spaces.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex">\,,</annotation></semantics></math></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/orientation">oriented</a> <a class="existingWikiWord" href="/nlab/show/compact+space">compact</a> <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi><mo>=</mo><mo stretchy="false">{</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>⋯</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\mathcal{B} = \{A, B , \cdots\}</annotation></semantics></math> a collection of oriented compact submanifolds write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P_X(A,B)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/path+space">path space</a> of paths in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> that start in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \subset X</annotation></semantics></math> and end in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">B \subset X</annotation></semantics></math>.</p> <div class="num_theorem"> <h6 id="theorem_3">Theorem</h6> <p>The tuple <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>L</mi><mi>M</mi><mo>,</mo><mi>ℚ</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">{</mo><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>P</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>,</mo><mi>ℚ</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">}</mo> <mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><mi>ℬ</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(H_\bullet(L M, \mathbb{Q}), \{H_\bullet(P_X(A,B), \mathbb{Q})\}_{A,B \in \mathcal{B}})</annotation></semantics></math> carries the structure of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/HCFT">HCFT</a> with <em>positive boundary</em> and set of <a class="existingWikiWord" href="/nlab/show/branes">branes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">\mathcal{B}</annotation></semantics></math>, such that the correlators in the closed sector are the standard string topology operation.</p> </div> <p>For <a class="existingWikiWord" href="/nlab/show/closed+strings">closed strings</a> this is discussed in (<a href="#CohenGodin03">Cohen-Godin 03</a>, <a href="#Tamanoi07">Tamanoi 07</a>). For <a class="existingWikiWord" href="/nlab/show/open+strings">open strings</a> on a single <a class="existingWikiWord" href="/nlab/show/brane">brane</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi><mo>=</mo><mo stretchy="false">{</mo><mo>*</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\mathcal{B} = \{*\}</annotation></semantics></math> this was shown in (<a href="#Godin">Godin 07</a>), where the general statement for arbitrary branes is conjectured. A detailed proof of this general statement is in (<a href="#Kupers">Kupers 11</a>).</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>These constructions work by regarding the <a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a> from 2-dimensional <a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a> with maps to the base space as <a class="existingWikiWord" href="/nlab/show/correspondences">correspondences</a> and then applying pull-push (pullback followed by <a class="existingWikiWord" href="/nlab/show/push-forward+in+generalized+cohomology">push-forward in cohomology</a>/<a class="existingWikiWord" href="/nlab/show/Umkehr+maps">Umkehr maps</a>) to these. Hence these quantum field theory realizations of string topology may be thought of as arising from a <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> process of the form <em><a class="existingWikiWord" href="/nlab/show/path+integral+as+a+pull-push+transform">path integral as a pull-push transform</a>/<a class="existingWikiWord" href="/nlab/show/motivic+quantization">motivic quantization</a></em>.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Goldman+bracket">Goldman bracket</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+chord+diagram">Sullivan chord diagram</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+integral+as+a+pull-push+transform">path integral as a pull-push transform</a></p> </li> </ul> <h2 id="references">References</h2> <p>The original references include the following:</p> <ul> <li id="ChasSullivan"> <p><a class="existingWikiWord" href="/nlab/show/Moira+Chas">Moira Chas</a>, <a class="existingWikiWord" href="/nlab/show/Dennis+Sullivan">Dennis Sullivan</a>, <em>String topology</em>, Ann. Math. <a href="http://arxiv.org/abs/math/9911159">math.GT/9911159</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ralph+Cohen">Ralph Cohen</a>, John R. Klein, <a class="existingWikiWord" href="/nlab/show/Dennis+Sullivan">Dennis Sullivan</a>, <em>The homotopy invariance of the string topology loop product and string bracket</em>, J. of Topology 2008 <strong>1</strong>(2):391-408; <a href="http://dx.doi.org/10.1112/jtopol/jtn001">doi</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ralph+Cohen">Ralph Cohen</a>, <em>Homotopy and geometric perspectives on string topology</em>, <a href="http://math.stanford.edu/~ralph/skyesummary.pdf">pdf</a></p> </li> </ul> <p>In</p> <ul> <li id="CohenJones"><a class="existingWikiWord" href="/nlab/show/Ralph+Cohen">Ralph Cohen</a>, <a class="existingWikiWord" href="/nlab/show/John+David+Stuart+Jones">John David Stuart Jones</a>, <em>A homotopy theoretic realization of string topology</em> , Math. Ann. 324 <p>(2002), no. 4, (<a href="http://arxiv.org/abs/math/0107187">arXiv:0107187</a>)</p> </li> </ul> <p>the string product was realized as genuine pull-push (in terms of dual <a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a> via <a class="existingWikiWord" href="/nlab/show/Thom+isomorphism">Thom isomorphism</a>).</p> <p>The interpretation of closed string topology as an <a class="existingWikiWord" href="/nlab/show/HQFT">HQFT</a> is discussed in</p> <ul> <li id="CohenGodin03"> <p><a class="existingWikiWord" href="/nlab/show/Ralph+Cohen">Ralph Cohen</a>, <a class="existingWikiWord" href="/nlab/show/Veronique+Godin">Veronique Godin</a>, <em><a class="existingWikiWord" href="/nlab/show/A+Polarized+View+of+String+Topology">A Polarized View of String Topology</a></em> (<a href="http://arxiv.org/abs/math/0303003">arXiv:math/0303003</a>)</p> </li> <li id="Tamanoi07"> <p>Hirotaka Tamanoi, <em>Loop coproducts in string topology and triviality of higher genus TQFT operations</em> (2007) (<a href="http://arxiv.org/abs/0706.1276">arXiv</a>)</p> </li> </ul> <p>A detailed discussion and generalization to the open-closed <a class="existingWikiWord" href="/nlab/show/HQFT">HQFT</a> in the presence of a single space-filling <a class="existingWikiWord" href="/nlab/show/brane">brane</a> is in</p> <ul> <li id="Godin"><a class="existingWikiWord" href="/nlab/show/Veronique+Godin">Veronique Godin</a>, <em>Higher string topology operations</em> (2007)(<a href="http://arxiv.org/abs/0711.4859">arXiv:0711.4859</a>)</li> </ul> <p>The generalization to multiple <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> is discussed in</p> <ul> <li id="Kupers"><a class="existingWikiWord" href="/nlab/show/Sander+Kupers">Sander Kupers</a>, <em>String topology operations</em> MS thesis (2011) (<a href="http://math.stanford.edu/~kupers/thesis7thjune2011.pdf">pdf</a>)</li> </ul> <p>Exposition of the perspective of regarding <a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a>-operations as the <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> of a <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a> <a class="existingWikiWord" href="/nlab/show/sigma+model">sigma model</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ralph+Cohen">Ralph Cohen</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Voronov">Alexander Voronov</a>, <em>Notes on string topology</em>, in: <a class="existingWikiWord" href="/nlab/show/Ralph+Cohen">Ralph Cohen</a>, <a class="existingWikiWord" href="/nlab/show/Kathryn+Hess">Kathryn Hess</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Voronov">Alexander Voronov</a>, <em>String topology and cyclic homology</em>, Advanced courses in mathematics CRM Barcelona, Birkhäuser 2006 (<a href="http://arxiv.org/abs/math/0503625">math.GT/05036259</a>, <a href="https://doi.org/10.1007/3-7643-7388-1">doi:10.1007/3-7643-7388-1</a>, <a href="http://gen.lib.rus.ec/get?md5=adde9464705ede0fea6b435edb58fbe7">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dennis+Sullivan">Dennis Sullivan</a>, <em>Sigma models and string topology</em>, in: Mikhail Lyubich, Leon Takhtajan (eds.), <em>Graphs and Patterns in Mathematics and Theoretical Physics</em>, Proc. Symp. Pure Math. 73 (2005) (<a href="https://doi.org/10.1090/pspum/073">doi:10.1090/pspum/073</a>, <a href="http://inspirehep.net/record/1697823">spire:1697823</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dennis+Sullivan">Dennis Sullivan</a>, <em>Open and closed string field theory interpreted in classical algebraic topology</em>, chapter 11 in: <a class="existingWikiWord" href="/nlab/show/Ulrike+Tillmann">Ulrike Tillmann</a> (ed.) <em>Topology, Geometry and Quantum Field Theory</em>, Cambridge University Press (2005) (<a href="https://doi.org/10.1017/CBO9780511526398.014">doi:10.1017/CBO9780511526398.014</a>)</p> </li> </ul> <p>For target space a <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> of a <a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a> or <a class="existingWikiWord" href="/nlab/show/compact+space">compact</a> <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> this is discussed in</p> <ul> <li>David Chataur, <a class="existingWikiWord" href="/nlab/show/Luc+Menichi">Luc Menichi</a>, <em>String topology of classifying spaces</em> (<a href="http://math.univ-angers.fr/perso/lmenichi/String_Classifiant09.pdf">pdf</a>)</li> </ul> <p>Arguments that this string-topology <a class="existingWikiWord" href="/nlab/show/HQFT">HQFT</a> should refine to a chain-level theory – a <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> – were given in</p> <ul> <li id="Costello"><a class="existingWikiWord" href="/nlab/show/Kevin+Costello">Kevin Costello</a>, <em>Topological conformal field theories and Calabi-Yau <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories</em> (2004) , (<a href="https://arxiv.org/abs/math/0412149">arXiv:0412149</a>)</li> </ul> <p>and</p> <ul id="Lurie"> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">On the Classification of Topological Field Theories</a></em></li> </ul> <p>(see example 4.2.16, remark 4.2.17).</p> <p>For the string product and the BV-operator this extension has been known early on, it yields a <a class="existingWikiWord" href="/nlab/show/homotopy+BV+algebra">homotopy BV algebra</a> considered in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Scott+Wilson">Scott Wilson</a>, around page 101 of: <em>On the Algebra and Geometry of a Manifold’s Chains and Cochains</em> (2005) (<a href="http://qcpages.qc.cuny.edu/~swilson/main.pdf">pdf</a>)</li> </ul> <p>Evidence for the existence of the <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> version by exhibiting a <a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a> that looks like it ought to be the dg-category of string-topology <a class="existingWikiWord" href="/nlab/show/branes">branes</a> (hence ought to correspond to the TCFT under the suitable version of the <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a>-version of the <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>) is discussed in</p> <ul> <li id="BlumbergCoheneTeleman09"><a class="existingWikiWord" href="/nlab/show/Andrew+Blumberg">Andrew Blumberg</a>, <a class="existingWikiWord" href="/nlab/show/Ralph+Cohen">Ralph Cohen</a>, <a class="existingWikiWord" href="/nlab/show/Constantin+Teleman">Constantin Teleman</a>, <em>Open-closed field theories, string topology, and Hochschild homology</em>, pp. 53–56 in Alpine Perspectives on Algebraic Topology, Contemp. Math. 504, Amer. Math. Soc. 2009 <p>(<a href="https://arxiv.org/abs/0906.5198">arXiv:0906.5198</a>)</p> </li> </ul> <p>Refinements of string topology from <a class="existingWikiWord" href="/nlab/show/homology+groups">homology groups</a> to the full <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a>-<a class="existingWikiWord" href="/nlab/show/spectra">spectra</a> is discussed in (<a href="#BlumbergCoheneTeleman09">Blumberg-Cohen-Teleman 09</a>) and in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ralph+Cohen">Ralph Cohen</a>, <a class="existingWikiWord" href="/nlab/show/John+Jones">John Jones</a>, <em>A homotopy theoretic realization of string topology</em>, Mathematische Annalen (<a href="https://arxiv.org/abs/math/0107187">arXiv:math/0107187</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ralph+Cohen">Ralph Cohen</a>, <a class="existingWikiWord" href="/nlab/show/John+Jones">John Jones</a>, <em>Gauge theory and string topology</em> (<a href="https://arxiv.org/abs/1304.0613">arXiv:1304.0613</a>)</p> </li> <li> <p>Kate Gruher, Paolo Salvatore, <em>Generalized string topology operations</em>, Proc. London Math. Soc. <strong>96</strong>:1 (2008) 78–106 <a href="https://doi.org/10.1112/plms/pdm030">doi</a> <a href="https://arxiv.org/abs/math/0602210">math.AT/0602210</a></p> </li> </ul> <p>Further generalization to target spaces that are more generally <a class="existingWikiWord" href="/nlab/show/differentiable+stacks">differentiable stacks</a>/<a class="existingWikiWord" href="/nlab/show/Lie+groupoids">Lie groupoids</a> is discussed in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Kai+Behrend">Kai Behrend</a>, <a class="existingWikiWord" href="/nlab/show/Gregory+Ginot">Gregory Ginot</a>, <a class="existingWikiWord" href="/nlab/show/Behrang+Noohi">Behrang Noohi</a>, <a class="existingWikiWord" href="/nlab/show/Ping+Xu">Ping Xu</a>, <em>String topology for stacks</em>, (89 pages) <a href="http://arxiv.org/abs/0712.3857">arxiv/0712.3857</a>; <em>String topology for loop stacks</em>, C. R. Math. Acad. Sci. Paris, <strong>344</strong> (2007), no. 4, 247–252, (6 pages, <a href="">pdf</a>)</p> </li> <li> <p>Po Hu, <em>Higher string topology on general spaces</em>, Proc. London Math. Soc. <strong>93</strong> (2006) 515-544, <a href="http://dx.doi.org/10.1112/S0024611506015838">doi</a>, <a href="http://www.math.wayne.edu/~po/koszul04.ps">ps</a></p> </li> </ul> <p>The relation between string topology and <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> in dimenion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>></mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\gt 1</annotation></semantics></math> is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dmitry+Vaintrob">Dmitry Vaintrob</a>, <em>The String topology BV algebra, Hochschild cohomology and the Goldman bracket on surfaces</em> (<a href="http://arxiv.org/abs/math/0702859">arXiv:0702859</a>)</li> </ul> <p>More developments are in</p> <ul> <li>Eric Malm, <em>String topology and the based loop space</em>, 2009 (<a href="http://arxiv.org/abs/1103.6198">arXiv:1103.6198</a>, <a href="http://math.ucr.edu/~jbergner/ucr-st-present.pdf">slides</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 13, 2023 at 02:56:14. See the <a href="/nlab/history/string+topology" 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/string+topology" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/1045/#Item_24">Discuss</a><span class="backintime"><a href="/nlab/revision/string+topology/29" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/string+topology" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/string+topology" accesskey="S" class="navlink" id="history" rel="nofollow">History (29 revisions)</a> <a href="/nlab/show/string+topology/cite" style="color: black">Cite</a> <a href="/nlab/print/string+topology" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/string+topology" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>