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> Hypothesis H 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"> <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> nLab Hypothesis H </h1> <div class="navigation"> <a href='#navEnd'>Skip the Navigation Links</a> | <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/14819/#Item_20" 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> </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> <blockquote> This page is a non-maintained mirror of <a href="https://ncatlab.org/schreiber/show/Hypothesis+H">ncatlab.org/schreiber/show/Hypothesis+H</a>. See there for the up-to-date version. </blockquote> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>… models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, … see also <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a> Introductions <ul> <li> <a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a> </li> </ul> Definitions <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> Paths and cylinders <ul> <li> <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/path+object">path object</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a> </li> </ul> </li> </ul> Homotopy groups <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> Basic facts <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> Theorems <ul> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem </li> </ul> </div></div> <h4 id="algebraic_topology">Algebraic topology</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a> – application of <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a> and <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> to the study of (<a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>) <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy</a> <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> <a class="existingWikiWord" href="/nlab/show/homology">homology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a> </li> </ul> </div></div> <h4 id="cohomology">Cohomology</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>, <a class="existingWikiWord" href="/nlab/show/coboundary">coboundary</a>, <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homology">homology</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/taf">taf</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> differential cohomology <ul> <li> <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a> <a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a> </li> </ul> <div> <a href="/nlab/edit/cohomology+-+contents">Edit this sidebar</a> </div></div></div> <h4 id="supergeometry">Super-Geometry</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a> and (<a class="existingWikiWord" href="/nlab/show/synthetic+differential+supergeometry">synthetic</a> ) <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a> <h2 id="background">Background</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/graded+object">graded object</a> </li> </ul> <h2 id="introductions">Introductions</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+superalgebra">geometry of physics – superalgebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">geometry of physics – supergeometry</a> </li> </ul> <h2 id="superalgebra">Superalgebra</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/super+commutative+monoid">super commutative monoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+ring">super ring</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/supercommutative+ring">supercommutative ring</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/exterior+ring">exterior ring</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Clifford+ring">Clifford ring</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+module">super module</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a>, <a class="existingWikiWord" href="/nlab/show/SVect">SVect</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+algebra">super algebra</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/supercommutative+algebra">supercommutative algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/exterior+algebra">exterior algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Clifford+algebra">Clifford algebra</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/superdeterminant">superdeterminant</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/chain+complex+of+super+vector+spaces">chain complex of super vector spaces</a> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes+of+super+vector+spaces">model structure</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+superalgebra">differential graded-commutative superalgebra</a> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/super+L-infinity+algebra">super L-infinity algebra</a> </li> </ul> <h2 id="supergeometry">Supergeometry</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/superpoint">superpoint</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+Cartesian+space">super Cartesian space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>, <a class="existingWikiWord" href="/nlab/show/SDiff">SDiff</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/NQ-supermanifold">NQ-supermanifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+vector+bundle">super vector bundle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Euclidean+supermanifold">Euclidean supermanifold</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+spacetime">super spacetime</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/integration+over+supermanifolds">integration over supermanifolds</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Berezin+integral">Berezin integral</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/super+Lie+group">super translation group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+line+2-bundle">super line 2-bundle</a> </li> </ul> <h2 id="supersymmetry">Supersymmetry</h2> <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/division+algebra+and+supersymmetry">division algebra and supersymmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supermultiplet">supermultiplet</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BPS+state">BPS state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/M-theory+super+Lie+algebra">M-theory super Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/type+II+super+Lie+algebra">type II super Lie algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a>, <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a> </li> </ul> <h2 id="supersymmetric_field_theory">Supersymmetric field theory</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/superfield">superfield</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supersymmetric+quantum+mechanics">supersymmetric quantum mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/adinkra">adinkra</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/gauged+supergravity">gauged supergravity</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/superstring+theory">superstring theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a> </li> </ul> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+model+for+elliptic+cohomology">geometric model for elliptic cohomology</a></li> </ul> <div> <a href="/nlab/edit/supergeometry+-+contents">Edit this sidebar</a> </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='#TheOpenProblemOfFormulatingMTheory'>The open problem of formulating M-Theory</a></li> <li><a href='#hypothesis_h'>Hypothesis H</a></li> <ul> <li><a href='#StatementOfHypothesisH'>Statement</a></li> <li><a href='#MotivationFromHomotopyTheoreticReanalysisOfBraneScan'>Motivation from homotopy theoretic re-analysis of the brane-scan</a></li> </ul> <li><a href='#Implications'>Implications</a></li> <ul> <li><a href='#anomaly_cancellation'>Anomaly cancellation</a></li> <ul> <li><a href='#OnCurvedButNonSingularManifolds'>On curved but non-singular 8-manifolds</a></li> <li><a href='#on_flat_but_singular_orbifolds'>On flat but singular orbifolds:</a></li> </ul> <li><a href='#microscopic_brane_physics'>Microscopic brane physics</a></li> </ul> <li><a href='#see_also'>See also</a></li> <li><a href='#FAQ'>FAQ</a></li> <ul> <li><a href='#references_for_the_faq'>References for the <a href="#FAQ">FAQ</a></a></li> </ul> <li><a href='#References'>References</a></li> <ul> <li><a href='#ReferencesInRational'>Motivation by results in rational Cohomotopy</a></li> <li><a href='#ReferencesFormulationInFullCohomotopyTheory'>Formulation in full Cohomotopy theory</a></li> <li><a href='#ExpositionAndTalkNotes'>Exposition and talk notes</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> Hypothesis H is a hypothesis about the mathematical formulation of the theory of fundamental physics known as <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>. The core statement of the hypothesis is that quantum <a class="existingWikiWord" href="/nlab/show/charge+quantization">charge quantization</a> in <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> happens in the <a class="existingWikiWord" href="/nlab/show/non-abelian+cohomology">non-abelian cohomology</a> theory called <a class="existingWikiWord" href="/nlab/show/cohomotopy">cohomotopy</a> (in more detail: <a class="existingWikiWord" href="/nlab/show/twisted+cohomotopy">twisted cohomotopy</a>). This originates with an observation due to <a href="#Sati13">Sati 2013</a> (<a href="https://arxiv.org/pdf/1310.1060.pdf#page=17">Sec. 2.5</a>) and has become the theme of a research program developed by <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a> and <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, see the list of references <a href="#References">below</a>. (In a play on how the “M” in “M-theory” has been <a href="https://ncatlab.org/nlab/show/M-theory#MagicMysteryQuote">said</a> to be either for “Membrane” or for “Mystery”, so the “H” in “Hypothesis H” may be read as being either for “Homotopy” or for “Hisham”.) <h2 id="TheOpenProblemOfFormulatingMTheory">The open problem of formulating M-Theory</h2> It is an open problem to mathematically formulate <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>: Supposedly a single coherent <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory of physics</a> whose various <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limiting</a> cases reproduce the zoo of <a class="existingWikiWord" href="/nlab/show/perturbative+string+theory">perturbative string theories</a> (<a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">HET</a>/<a class="existingWikiWord" href="/nlab/show/type+I+string+theory">I</a>/<a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">IIA</a>/<a class="existingWikiWord" href="/nlab/show/type+IIB+string+theory">IIB</a>&<a class="existingWikiWord" href="/nlab/show/F-theory">F</a>/<a class="existingWikiWord" href="/nlab/show/D%3D11+supergravity">11d SuGra</a>) and the expected <a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">dualities</a> relating them; but which also makes mathematical sense of <a class="existingWikiWord" href="/nlab/show/non-perturbative+effect">non-perturbative</a> <a class="existingWikiWord" href="/nlab/show/D-brane">D-</a>/<a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a> physics, and hence solves, via <a class="existingWikiWord" href="/nlab/show/intersecting+D-brane+models">intersecting D-brane models</a> of <a class="existingWikiWord" href="/nlab/show/confinement">confined</a> <a class="existingWikiWord" href="/nlab/show/quantization+of+Yang-Mills+theory">quantum</a> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> (<a class="existingWikiWord" href="/nlab/show/holographic+QCD">holographic QCD</a>), the <a class="existingWikiWord" href="/nlab/show/mass+gap+problem">last of the Millennium Problems</a>: <center> <a href="https://ncatlab.org/schreiber/files/Schreiber-MTheoryMathematics2020-v200126.pdf#page=8"> <img src="https://ncatlab.org/schreiber/files/ProblemQCDToProblemM230120.jpg" width="670" /> </a> </center> In the <a class="existingWikiWord" href="/nlab/show/physics">physics</a>-minded <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> community, parts of the literature tends to forget or downplay the <a class="existingWikiWord" href="/nlab/show/conjecture">conjectural</a> nature of many assumptions and leaps of faiths that are being made when it comes to discussion of <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a>/<a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a> dynamics and generally of <a class="existingWikiWord" href="/nlab/show/non-perturbative+effects">non-perturbative effects</a> in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>. More widely recognized is the fact that the theory of <a class="existingWikiWord" href="/nlab/show/intersecting+branes">coincident</a> <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a> is missing, together with the formulation of its expected decoupling limit to the <a class="existingWikiWord" href="/nlab/show/D%3D6+N%3D%282%2C0%29+SCFT">D=6 N=(2,0) SCFT</a> – but of course these major open problems are but one aspect of full <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>. For string-theoretic references that do highlight the open problem of formulating <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> see at <a href="https://ncatlab.org/nlab/show/M-theory#TheOpenProblem">M-theory – The open problem</a>. In <a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a>, it is rare but not unfamiliar that an open problem is not the proof of a theorem within an established theory, but is the establishing of a theory in the first place. A famous example of such a situation is the search for a theory of “absolute geometry” over the “<a class="existingWikiWord" href="/nlab/show/field+with+one+element">field with one element</a>”. In this <a class="existingWikiWord" href="/nlab/show/analogy">analogy</a>, the various <a class="existingWikiWord" href="/nlab/show/perturbative+string+theories">perturbative string theories</a> (<a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">HET</a>, <a class="existingWikiWord" href="/nlab/show/type+I+string+theory">I</a>, <a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">IIA</a>, <a class="existingWikiWord" href="/nlab/show/type+IIB+string+theory">IIB</a> and their <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactified</a> <a class="existingWikiWord" href="/nlab/show/perturbative+string+theory+vacua">perturbative string theory vacua</a>) correspond to <a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometries</a> over <a class="existingWikiWord" href="/nlab/show/ground+field">base</a> <a class="existingWikiWord" href="/nlab/show/prime+fields">prime fields</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_p</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">p \geq 2</annotation></semantics></math>, and the would-be <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> corresponds to a theory of a “<a class="existingWikiWord" href="/nlab/show/field+with+one+element">field with one element</a>” that unifies and completes all this, by describing it at a deeper level (literally: a deeper base). Hence the task here is to conjure a mathematical theory <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">\mathcal{X}</annotation></semantics></math>, hypothesize that and explain how <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">\mathcal{X}</annotation></semantics></math> is the putative <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>, and then rigorously work out the mathematical implications 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">\mathcal{X}</annotation></semantics></math> in order to check to which extent they include the required design criterion “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒳</mi><mo>⇒</mo><mi>ST</mi></mrow><annotation encoding="application/x-tex">\mathcal{X} \Rightarrow ST</annotation></semantics></math>”. To the extent that <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">\mathcal{X}</annotation></semantics></math> implies known or expected phenomena in <a class="existingWikiWord" href="/nlab/show/perturbative+string+theory">perturbative string theory</a> the hypothesis that <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">\mathcal{X}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> finds support, to the extent that it doesn’t <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">\mathcal{X}</annotation></semantics></math> needs to be modified to or be replaced by some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒳</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\mathcal{X}'</annotation></semantics></math>, and the process re-started. <h2 id="hypothesis_h">Hypothesis H</h2> <h3 id="StatementOfHypothesisH">Statement</h3> In the large-<a class="existingWikiWord" href="/nlab/show/scale">scale</a>/low-<a class="existingWikiWord" href="/nlab/show/energy">energy</a> <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit</a> of <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> given by <a class="existingWikiWord" href="/nlab/show/D%3D11+supergravity">D=11 supergravity</a>, the <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a> must consist of <a class="existingWikiWord" href="/nlab/show/torsion+constraints+in+supergravity">super-torsion free</a> <a class="existingWikiWord" href="/nlab/show/super+Cartan+geometry">super orbi</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mn>10</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mn>32</mn></mstyle></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{10,1\vert \mathbf{32}}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/V-manifold">-folds</a> equipped with a suitable <a class="existingWikiWord" href="/nlab/show/higher+gauge+field">higher gauge field</a>: the <a class="existingWikiWord" href="/nlab/show/C-field">C-field</a>. <center> <a href="https://ncatlab.org/schreiber/files/Schreiber-MTheoryMathematics2020-v200126.pdf#page=12"> <img src="https://ncatlab.org/schreiber/files/MTheoryInSupergravityLimit200123.jpg" width="670" /> </a> </center> The first ingredient of a <a class="existingWikiWord" href="/nlab/show/non-perturbative+effect">non-perturbative</a> <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantization</a> of this <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> must be a choice of <a class="existingWikiWord" href="/nlab/show/Dirac+charge+quantization">Dirac charge quantization</a>-condition for the <a class="existingWikiWord" href="/nlab/show/C-field">C-field</a>. <center> <a href="https://ncatlab.org/schreiber/files/Schreiber-MTheoryMathematics2020-v200126.pdf#page=14"> <img src="https://ncatlab.org/schreiber/files/HypothesisH200123.jpg" width="670" /> </a> </center> Hypothesis H: (<a href="#FSS19b">FSS 19b</a>,<a href="#FSS19c">FSS 19c</a>) The <a class="existingWikiWord" href="/nlab/show/C-field">C-field</a> is <a class="existingWikiWord" href="/nlab/show/Dirac+charge+quantization">charge quantized</a> in <a class="existingWikiWord" href="/nlab/show/J-homomorphism">J-</a><a class="existingWikiWord" href="/nlab/show/twisted+cohomotopy">twisted</a> <a class="existingWikiWord" href="/nlab/show/cohomotopy+cohomology+theory">cohomotopy cohomology theory</a>. <center> <a href="https://ncatlab.org/schreiber/files/Schreiber-MTheoryMathematics2020-v200126.pdf#page=21"> <img src="https://ncatlab.org/schreiber/files/HypothesisH.jpg" width="640" /> </a> </center><div style="float:right;margin:0 10px 10px 0;"> <a href="https://arxiv.org/pdf/1903.02834.pdf#page=14"> <img src="https://ncatlab.org/schreiber/files/TheRational4SphereAppears.jpg" width="380" /> </a> </div> <h3 id="MotivationFromHomotopyTheoreticReanalysisOfBraneScan">Motivation from homotopy theoretic re-analysis of the brane-scan</h3> The <a href="#HypothesisH">Hypothesis H</a> (<a href="#StatementOfHypothesisH">above</a>) is motivated from re-analysis (based on <a href="#Sati13">Sati 13, Sec. 2.5</a> see <a href="#FSS19a">FSS 19a</a> for comprehensive review) of <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+fundamental+super+p-branes">super p-brane WZ terms</a> in <a class="existingWikiWord" href="/nlab/show/super+homotopy+theory">super homotopy theory</a>. The resulting <a class="existingWikiWord" href="/schreiber/show/brane+bouquet">brane bouquet</a> proves that – in the approximation of <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a> – <a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a> charge is in <a class="existingWikiWord" href="/nlab/show/rational+cohomotopy">rational cohomotopy</a> in exactly the same way that <a class="existingWikiWord" href="/nlab/show/D-brane+charge+quantization+in+K-theory">D-brane charge is in twisted K-theory</a>. <a href="https://arxiv.org/pdf/1606.03206.pdf#page=20"> <img src="https://ncatlab.org/schreiber/files/SupercocyclesRationalTwistedKTheory.jpg" width="440" /> </a> Detailed lecture note on this point are at <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+fundamental+super+p-branes">geometry of physics – fundamental super p-branes</a>, in the section <a href="https://ncatlab.org/nlab/show/geometry+of+physics+--+fundamental+super+p-branes#MFluxFields">M-Flux fields</a>. <hr /><center> <a href="https://ncatlab.org/schreiber/files/EquivariantSuperHomotopyTheory120314a.pdf#page=74"> <img src="https://ncatlab.org/schreiber/files/BraneBouquetAnimated.gif" width="450 />" /></a> </center> <h2 id="Implications">Implications</h2> Assuming <a href="#HypothesisH">Hypothesis H</a>, we have a well-defined <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a> and may rigorously derive properties of the resulting <a class="existingWikiWord" href="/nlab/show/states">states</a> and <a class="existingWikiWord" href="/nlab/show/observables">observables</a>. These are the implications of <a href="#HypothesisH">Hypothesis H</a>. To the extent that these implications match known facts of <a class="existingWikiWord" href="/nlab/show/perturbative+string+theory">perturbative string theory</a> and to the extent that they match <a class="existingWikiWord" href="/nlab/show/folklore">folklore</a> about <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>, <a href="#HypothesisH">Hypothesis H</a> finds support, to the extend that they do not <a href="#HypothesisH">Hypothesis H</a> may need to be adjusted or abandoned. <h3 id="anomaly_cancellation">Anomaly cancellation</h3> What physicists call <a class="existingWikiWord" href="/nlab/show/anomalies">anomalies</a> are what mathematician call <a class="existingWikiWord" href="/nlab/show/obstructions">obstructions</a> to the construction of a <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">physical theory</a>. Hence <a class="existingWikiWord" href="/nlab/show/anomaly+cancellation">cancellation</a> of all <a class="existingWikiWord" href="/nlab/show/anomalies">anomalies</a> in every <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> that <a class="existingWikiWord" href="/nlab/show/folklore">folklore</a> expects to be a <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limiting case</a> of <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> is strictly necessary for <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> to exist: Any remaining anomaly is an <a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a> against the existence of the theory that it is an anomaly of. A long list of anomaly cancellation conditions in M-theory has been argued for in the literature, obtained from a variety of different arguments and under various assumptions, but supported by a web of consistency checks. Theorem (<a href="#FSS19b">FSS 19b</a>, <a href="#FSS19c">FSS 19c</a>, <a href="#SS19a">SS 19a</a>) <a href="##HypothesisH">Hypothesis H</a> implies the following list of <a class="existingWikiWord" href="/nlab/show/anomaly+cancellation">anomaly cancellation</a> consistency conditions expected in the <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> <a class="existingWikiWord" href="/nlab/show/folklore">folklore</a>: <center> <a href="https://arxiv.org/pdf/1904.10207.pdf#page=2"> <img src="https://ncatlab.org/schreiber/files/HypothesisHImplications.jpg" width="740" /> </a> </center> <h4 id="OnCurvedButNonSingularManifolds">On curved but non-singular 8-manifolds</h4> Here is a quick survey of the implications of <a href="#HypothesisH">Hypothesis H</a> on expected M-theory anomaly cancellation conditions for <a class="existingWikiWord" href="/nlab/show/M-theory+on+8-manifolds">M-theory on 8-manifolds</a>: download: <a class="existingWikiWord" href="/schreiber/files/HypothesisHIntroductionAndSurvey191125.pdf" title="pdf">pdf</a> <center> <a href="https://arxiv.org/pdf/1904.10207.pdf#page=2"> <img src="https://ncatlab.org/schreiber/files/AnmlCnclltnInMThryOn8MfdsViaHypothesisH.jpg" width="740" /> </a> </center> <h4 id="on_flat_but_singular_orbifolds">On flat but singular orbifolds:</h4> <div style="float:right;margin:0 10px 10px 0;"> <a href="https://arxiv.org/pdf/1909.12277.pdf#page=5"> <img src="https://ncatlab.org/schreiber/files/AbsoluteDBraneOPlaneCharge.jpg" width="500" /> </a> </div> <a href="#HypothesisH">Hypothesis H</a> implies <a class="existingWikiWord" href="/nlab/show/RR-field+tadpole+cancellation">RR-field tadpole cancellation</a> on <a class="existingWikiWord" href="/nlab/show/flat+orbifolds">flat orbifolds</a> with <a class="existingWikiWord" href="/nlab/show/ADE-singularities">ADE-singularities</a>, including the absolute <a class="existingWikiWord" href="/nlab/show/O-plane+charge">O-plane charge</a> and hence the <a class="existingWikiWord" href="/nlab/show/SemiSpin%2832%29">SemiSpin(32)</a>-<a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> of <a class="existingWikiWord" href="/nlab/show/type+I%27+string+theory">type I' string theory</a>, <a class="existingWikiWord" href="/nlab/show/duality+between+type+I+and+heterotic+string+theory">dually</a> of <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a>. This is the heart of what came to be known as the “first string theory revolution” (<a href="https://ncatlab.org/nlab/show/string+theory#Schwarz07">Schwarz 07</a>). From <a href="#SS19a">SS 19a</a>: <center> <a href="https://arxiv.org/pdf/1909.12277.pdf#page=2"> <img src="https://ncatlab.org/schreiber/files/RRFieldTadpoleCancellationViaHypothesisH.jpg" width="740" /> </a> </center> <h3 id="microscopic_brane_physics">Microscopic brane physics</h3> The core aspect of <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a> physics is the <a class="existingWikiWord" href="/nlab/show/folklore">folklore</a> (see <a href="#BMSS19">BMSS 19, p. 3</a>) that, at low <a class="existingWikiWord" href="/nlab/show/energy">energy</a>, <a class="existingWikiWord" href="/nlab/show/coincident+branes">coincident</a> <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> carry <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills</a> <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> on their <a class="existingWikiWord" href="/nlab/show/worldvolumes">worldvolumes</a>. This is the starting point of <a class="existingWikiWord" href="/nlab/show/geometric+engineering+of+quantum+field+theory">geometric engineering of quantum field theory</a> in <a class="existingWikiWord" href="/nlab/show/intersecting+D-brane+models">intersecting D-brane models</a> and without this phenomenon, string theory would be irrelevant for <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a>. Moreover, <a class="existingWikiWord" href="/nlab/show/folklore">folklore</a> has it (but see the issues with the <a class="existingWikiWord" href="/nlab/show/nonabelian+DBI+action">nonabelian DBI action</a> <a href="https://ncatlab.org/nlab/show/M-theory#OpenProblemNonabelianDBI">here</a>) that at small-<a class="existingWikiWord" href="/nlab/show/scale">scale</a>/high-<a class="existingWikiWord" href="/nlab/show/energy">energy</a> <a class="existingWikiWord" href="/nlab/show/intersecting+brane">intersecting</a> <a class="existingWikiWord" href="/nlab/show/branes">branes</a> show the deep quantum phenomena associated with <a class="existingWikiWord" href="/nlab/show/monopoles+in+Yang-Mills+theory">monopoles in Yang-Mills theory</a>, which ultimately lead to the partial formulation of <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> via the <a class="existingWikiWord" href="/nlab/show/BFSS+matrix+model">BFSS matrix model</a> and the <a class="existingWikiWord" href="/nlab/show/BMN+matrix+model">BMN matrix model</a>. <a href="#HypothesisH">Hypothesis H</a> implies that the <a class="existingWikiWord" href="/nlab/show/states">states</a> and <a class="existingWikiWord" href="/nlab/show/observables">observables</a> of <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> reflect the following expected phenomena for <a class="existingWikiWord" href="/nlab/show/Dp-D%28p%2B2%29-brane+intersections">Dp-D(p+2)-brane intersections</a>: From <a href="#SS19c">SS 19c</a>: <a href="https://ncatlab.org/schreiber/files/DifferentialCohomotopyIntersectingBranes200116.pdf#page=3"> <img src="https://ncatlab.org/schreiber/files/EmergenceOfIntersectinBraneObservablesII.jpg" width="800" /> </a> <h2 id="see_also">See also</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+cohomotopy">twisted cohomotopy</a></li> <li><a class="existingWikiWord" href="/nlab/show/equivariant+cohomotopy">equivariant cohomotopy</a></li> <li><a class="existingWikiWord" href="/nlab/show/differential+cohomotopy">differential cohomotopy</a></li> </ul> <h2 id="FAQ">FAQ</h2> <ul> <li> <a href="https://mathoverflow.net/q/377071/381">MO-question:377071/381</a>: <blockquote> “In M-theory, what can hypothesis H tell us that quantization in ordinary cohomology cannot?” </blockquote> answer at: <a href="https://mathoverflow.net/a/377154/381">MO-answer:377154/381</a> </li> </ul> <h3 id="references_for_the_faq">References for the <a href="#FAQ">FAQ</a></h3> <ul> <li id="DMW00"> E. Diaconescu, G. Moore, E. Witten, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math> Gauge Theory, and a Derivation of K-Theory from M-Theory, Adv. Theor. Math. Phys. 6:1031-1134, 2003 (<a href="http://arxiv.org/abs/hep-th/0005090">arXiv:hep-th/0005090</a>), summarised in: A Derivation of K-Theory from M-Theory (<a href="http://arxiv.org/abs/hep-th/0005091">arXiv:hep-th/0005091</a>) </li> <li id="HS02"> M Hopkins, I. Singer, <a class="existingWikiWord" href="/nlab/show/Quadratic+Functions+in+Geometry%2C+Topology%2C+and+M-Theory">Quadratic Functions in Geometry, Topology, and M-Theory</a>, J. Differential Geom. Volume 70, Number 3 (2005), 329-452 (<a href="http://arxiv.org/abs/math.AT/0211216">arXiv:math.AT/0211216</a>, <a href="https://projecteuclid.org/euclid.jdg/1143642908">euclid:1143642908</a>) </li> <li id="DFM03"> E. Diaconescu, D. Freed, G. Moore, The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>-theory 3-form and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math>-gauge theory, chapter in: Elliptic Cohomology Geometry, Applications, and Higher Chromatic Analogues, Cambridge University Press 2007 (<a href="http://arxiv.org/abs/hep-th/0312069">arXiv:hep-th/0312069</a>, <a href="https:
//doi.org/10.1017/CBO9780511721489">doi:10.1017/CBO9780511721489</a>) </li> <li id="MSa03"> V. Mathai, H. Sati, Some Relations between Twisted K-theory and E8 Gauge Theory, JHEP 0403:016, 2004 (<a href="https://arxiv.org/abs/hep-th/0312033">arXiv:hep-th/0312033</a>, <a href="https://iopscience.iop.org/article/10.1088/1126-6708/2004/03/016">doi:10.1088/1126-6708/2004/03/016</a>) </li> <li id="Sa13"> H. Sati, Framed M-branes, corners, and topological invariants, J. Math. Phys. 59 (2018), 062304 (<a href="http://arxiv.org/abs/1310.1060">arXiv:1310.1060</a>) </li> <li id="FSS15"> D. Fiorenza, H. Sati, U. Schreiber, The WZW term of the M5-brane and differential cohomotopy, J. Math. Phys. 56, 102301 (2015) (<a href="http://arxiv.org/abs/1506.07557">arXiv:1506.07557</a>, <a href="http://scitation.aip.org/content/aip/journal/jmp/56/10/10.1063/1.4932618">doi:10.1063/1.4932618</a>) </li> <li id="FSS16"> D. Fiorenza, H. Sati, U. Schreiber, Rational sphere valued supercocycles in M-theory and type IIA string theory, J. Geom. Phys., Vol 114 (2017) (<a href="http://arxiv.org/abs/1606.03206">arXiv:1606.03206</a>, <a href="http://dx.doi.org/10.1016/j.geomphys.2016.11.024">doi:10.1016/j.geomphys.2016.11.024</a>) </li> <li id="BMSS18"> V. Braunack-Mayer, H. Sati, U. Schreiber, Gauge enhancement of super M-branes via parametrized stable homotopy theory, Comm. Math. Phys. 371: 197 (2019) (<a href="https://arxiv.org/abs/1806.01115">arXiv:1806.01115</a>, <a href="https://doi.org/10.1007/s00220-019-03441-4">doi:10.1007/s00220-019-03441-4</a>) </li> <li id="BSS18"> S. Burton, H. Sati, U. Schreiber, Lift of fractional D-brane charge to equivariant Cohomotopy theory, J. Geom. Phys., 2020 (in print) (<a href="https://arxiv.org/abs/1812.09679">arXiv:1812.09679</a>) </li> <li id="FSS19a"> D. Fiorenza, H. Sati, U. Schreiber, The rational higher structure of M-theory, in: Proceedings of Higher Structures in M-Theory 2018, Fortsch. Phys. 2019 (<a href="https://arxiv.org/abs/1903.02834">arXiv:1903.02834</a>, <a href="https://doi.org/10.1002/prop.201910017">doi:10.1002/prop.201910017</a>) </li> <li id="FSS19b"> D. Fiorenza, H. Sati, U. Schreiber, Twisted Cohomotopy implies M-Theory anomaly cancellation on 8-manifolds, Comm. Math. Phys. 377(3), 1961-2025 (2020) (<a href="https://arxiv.org/abs/1904.10207">arXiv:1904.10207</a>, <a href="https://doi.org/10.1007/s00220-020-03707-2">doi:10.1007/s00220-020-03707-2</a>) </li> <li id="FSS19c"> D. Fiorenza, H. Sati, U. Schreiber, Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino-term of the M5-brane, Comm. Math. Phys. 2020 (in print) (<a href="https://arxiv.org/abs/1906.07417">arXiv:1906.07417</a>) </li> <li id="FSS19d"> D. Fiorenza, H. Sati, U. Schreiber, Twistorial Cohomotopy Implies Green-Schwarz anomaly cancellation (<a href="https://arxiv.org/abs/2008.08544">arXiv:2008.08544</a>) </li> <li id="SS19a"> H. Sati, U. Schreiber, Equivariant Cohomotopy implies orientifold tadpole cancellation J. Geom. Phys. Vol 156, 2020, 103775 (<a href="https://arxiv.org/abs/1909.12277">arXiv:1909.12277</a>, <a href="https://doi.org/10.1016/j.geomphys.2020.103775">doi:10.1016/j.geomphys.2020.103775</a>) </li> <li id="SS19b"> H. Sati, U. Schreiber, Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams (<a href="https://arxiv.org/abs/1912.10425">arXiv:1912.10425</a>) </li> <li id="SS20a"> H. Sati, U. Schreiber, Twisted Cohomotopy implies M5-brane anomaly cancellation (<a href="https://arxiv.org/abs/2002.07737">arXiv:2002.07737</a>) </li> <li id="SS20b"> H. Sati, U. Schreiber, The character map in equivariant twistorial Cohomotopy implies the Green-Schwarz mechanism with heterotic M5-branes (<a href="https://arxiv.org/abs/2011.06533">arXiv:2011.06533</a>) </li> <li id="Sa20"> H. Sati, M-theory and cohomotopy, talk at M-Theory and Mathematics, NYUAD 2020 (<a href="https://ncatlab.org/nlab/files/SatiMTheoryCohomotopy2020.pdf">pdf</a>) </li> <li id="Sc20"> U. Schreiber, Microscopic brane physics from Cohomotopy theory, talk at M-Theory and Mathematics, NYUAD 2020 (<a href="https://ncatlab.org/schreiber/files/Schreiber-MTheoryMathematics2020-v200126a.pdf">pdf</a>) </li> </ul> <h2 id="References">References</h2> Introduction and survey: <ul> <li id="IntroductionToHypothsisH"><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/schreiber/show/Introduction+to+Hypothesis+H">Introduction to Hypothesis H</a>.</li> </ul> More references: <ul> <li> <a href="#ReferencesInRational">Motivation by results in rational Cohomotopy</a> </li> <li> <a href="#ReferencesFormulationInFullCohomotopyTheory">Formulation in full Cohomotopy</a> </li> <li> <a href="#ExpositionAndTalkNotes">Exposition and talk notes</a> </li> </ul> <h3 id="ReferencesInRational">Motivation by results in rational Cohomotopy</h3> <a href="#HypothesisH">Hypothesis H</a> is motivated by analysis of <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+fundamental+super+p-branes">super p-brane WZ terms</a> in <a class="existingWikiWord" href="/nlab/show/super+homotopy+theory">super homotopy theory</a>, which proves that – in the approximation of <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a> – <a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a> charge is in <a class="existingWikiWord" href="/nlab/show/rational+Cohomotopy">rational Cohomotopy</a> in exactly the same way that <a class="existingWikiWord" href="/nlab/show/D-brane+charge">D-brane charge</a> is in <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>. This is due to: <ul> <li id="Sati13"> <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>: Section 2.5 of: Framed M-branes, corners, and topological invariants J. Math. Phys. 59 (2018) 062304 <a href="https://doi.org/10.1063/1.5007185">doi:10.1063/1.5007185</a> <a href="http://arxiv.org/abs/1310.1060">arXiv:1310.1060</a> </li> </ul> and was further developed in: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/The+WZW+term+of+the+M5-brane+and+differential+cohomotopy">The WZW term of the M5-brane and differential cohomotopy</a> J. Math. Phys. 56 (2015) 102301 <a href="http://scitation.aip.org/content/aip/journal/jmp/56/10/10.1063/1.4932618">doi:10.1063/1.4932618</a> <a href="http://arxiv.org/abs/1506.07557">arXiv:1506.07557</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/Rational+sphere+valued+supercocycles+in+M-theory">Rational sphere valued supercocycles in M-theory and type IIA string theory</a> J. Geom. Phys. 114 (2017) 91-108 <a href="http://dx.doi.org/10.1016/j.geomphys.2016.11.024">doi:10.1016/j.geomphys.2016.11.024</a> <a href="http://arxiv.org/abs/1606.03206">arXiv:1606.03206</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/Higher+T-duality+of+super+M-branes">Higher T-duality of super M-branes</a> Phys. Lett. B 781 (2018) 694-698 <a href="https://doi.org/10.1016/j.physletb.2018.04.058">doi:10.1016/j.physletb.2018.04.058</a> <a href="https://arxiv.org/abs/1805.00233">arXiv:1805.00233</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/John+Huerta">John Huerta</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/Equivariant+homotopy+and+super+M-branes">Real ADE-equivariant (co)homotopy and Super M-branes</a> Comm. Math. Phys. 371 (2019) 425–524 <a href="https://doi.org/10.1007/s00220-019-03442-3">doi:10.1007/s00220-019-03442-3</a> <a href="https://arxiv.org/abs/1805.05987">arXiv:1805.05987</a> </li> <li id="BMSS19"> <a class="existingWikiWord" href="/nlab/show/Vincent+Braunack-Mayer">Vincent Braunack-Mayer</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/Gauge+enhancement+of+Super+M-Branes">Gauge enhancement of super M-branes via parametrized stable homotopy theory</a> Comm. Math. Phys. 371 (2019) 197–265 <a href="https://doi.org/10.1007/s00220-019-03441-4">doi:10.1007/s00220-019-03441-4</a> <a href="https://arxiv.org/abs/1806.01115">arXiv:1806.01115</a> </li> </ul> Exposition and review of the rational situation is in: <ul> <li id="FSS19a"> <a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/The+rational+higher+structure+of+M-theory">The rational higher structure of M-theory</a> Fort. Phys. (2019) Proceedings of the <a href="http://www.maths.dur.ac.uk/lms/">LMS-EPSRC Durham Symposium</a>: <a class="existingWikiWord" href="/nlab/show/Higher+Structures+in+M-Theory+2018">Higher Structures in M-Theory 2018</a> <a href="https://doi.org/10.1002/prop.201910017">doi:10.1002/prop.201910017</a> <a href="https://arxiv.org/abs/1903.02834">arXiv:1903.02834</a> </li> </ul> Partial derivation of the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a>-model at this rational level: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/Super-exceptional+embedding+construction+of+the+M5-brane">Super-exceptional embedding construction of the M5-brane</a> J. High Energy Phys. 2020 107 (2020). <a href="https://doi.org/10.1007/JHEP02(2020)107">doi:10.1007/JHEP02(2020)107</a> <a href="https://arxiv.org/abs/1908.00042">arxiv:1908.00042</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/Super-exceptional+M5-brane+model+--+Emergence+of+SU%282%29-flavor+sector">Super-exceptional M5-brane model – Emergence of SU(2)-flavor sector</a> J. Geom. Phys. 170(2021) 104349 <a href="https://doi.org/10.1016/j.geomphys.2021.104349">doi:10.1016/j.geomphys.2021.104349</a> <a href="https://arxiv.org/abs/2006.00012">arXiv:2006.00012</a> </li> </ul> On <a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a> (and possibly <a class="existingWikiWord" href="/nlab/show/mysterious+duality">mysterious duality</a>) as <a class="existingWikiWord" href="/nlab/show/automorphisms">automorphisms</a> of <a class="existingWikiWord" href="/nlab/show/iterated+loop+space">iterated</a> (<a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+loop+space">rational</a>) <a class="existingWikiWord" href="/nlab/show/cyclic+loop+spaces">cyclic loop spaces</a> of the (<a class="existingWikiWord" href="/nlab/show/rational+n-sphere">rational</a>) <a class="existingWikiWord" href="/nlab/show/4-sphere">4-sphere</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Voronov">Alexander Voronov</a>, Mysterious triality (<a href="https://arxiv.org/abs/2111.14810">arXiv:2111.14810</a>)</li> </ul> <h3 id="ReferencesFormulationInFullCohomotopyTheory">Formulation in full Cohomotopy theory</h3> The detailed form of the <a href="#HypothesisH">Hypothesis H</a> (that the above rational situation should lift to full <a class="existingWikiWord" href="/nlab/show/twisted+Cohomotopy">twisted</a> <a class="existingWikiWord" href="/nlab/show/equivariant+Cohomotopy">equivariant</a> <a class="existingWikiWord" href="/nlab/show/differential+Cohomotopy">differential</a> <a class="existingWikiWord" href="/nlab/show/Cohomotopy+theory">Cohomotopy theory</a>) was formulated, and first consistency checks were made, in: <ul> <li id="FSS19b"> <a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/Twisted+Cohomotopy+implies+M-theory+anomaly+cancellation+on+8-manifolds">Twisted Cohomotopy implies M-theory anomaly cancellation on 8-manifolds</a> Comm Math. Phys 377 3 (2020) 1961-2025 <a href="https://doi.org/10.1007/s00220-020-03707-2">doi:10.1007/s00220-020-03707-2</a> <a href="https://arxiv.org/abs/1904.10207">arXiv:1904.10207</a> </li> <li id="FSS19c"> <a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/Twisted+Cohomotopy+implies+M5+WZ+term+level+quantization">Twisted Cohomotopy implies M5 WZ term level quantization</a> Comm. Math. Phys. 384 403–432 (2021) <a href="https://doi.org/10.1007/s00220-021-03951-0">doi:10.1007/s00220-021-03951-0</a> <a href="https://arxiv.org/abs/1906.07417">arXiv:1906.07417</a> </li> <li id="GS20"> <a class="existingWikiWord" href="/nlab/show/Daniel+Grady">Daniel Grady</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>: Differential cohomotopy versus differential cohomology for M-theory J. Geom. Phys. 165 (2021) 104203 <a href="doi:10.1016/j.geomphys.2021.104203">https://doi.org/10.1016/j.geomphys.2021.104203</a> <a href="https://arxiv.org/abs/2001.07640">arXiv:2001.07640</a> </li> <li id="SS19a"> <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/Equivariant+Cohomotopy+implies+orientifold+tadpole+cancellation">Equivariant Cohomotopy implies orientifold tadpole cancellation</a> J. Geom. Phy. 156 (2020) 103775 <a href="https://doi.org/10.1016/j.geomphys.2020.103775">doi:10.1016/j.geomphys.2020.103775</a> <a href="https://arxiv.org/abs/1909.12277">arXiv:1909.12277</a> </li> <li id="SS19b"> <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/Lift+of+fractional+D-brane+charge+to+equivariant+Cohomotopy+theory">Lift of fractional D-brane charge to equivariant Cohomotopy theory</a> J. Geom. Phys. 161 (2021) 104034 <a href="https://doi.org/10.1016/j.geomphys.2020.104034">doi:10.1016/j.geomphys.2020.104034</a> <a href="https://arxiv.org/abs/1812.09679">arXiv:1812.09679</a> </li> <li id="SS20a"> <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/Twisted+Cohomotopy+implies+M5-brane+anomaly+cancellation">Twisted Cohomotopy implies M5-brane anomaly cancellation</a> Lett. Math. Phys. 111 120 (2021) <a href="https://doi.org/10.1007/s11005-021-01452-8">doi:10.1007/s11005-021-01452-8</a> <a href="https://arxiv.org/abs/2002.07737">arXiv:2002.07737</a> </li> <li id="FSS20a"> <a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/Twisted+Cohomotopy+implies+twisted+String+structure+on+M5-branes">Twisted Cohomotopy implies twisted String structure on M5-branes</a> J. Math. Phys. 62 042301 (2021) <a href="https://doi.org/10.1063/5.0037786">doi:10.1063/5.0037786</a> <a href="https://arxiv.org/abs/2002.11093">arXiv:2002.11093</a> </li> <li id="SS19c"> <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/Differential+Cohomotopy+implies+intersecting+brane+observables">Differential Cohomotopy implies intersecting brane observables</a> Adv. Theor. Math. Physics 26 4 (2022) <a href="https://arxiv.org/abs/1912.10425">arXiv:1912.10425</a> </li> <li id="CSS21a"> <a class="existingWikiWord" href="/nlab/show/David+Corfield">David Corfield</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/Fundamental+weight+systems+are+quantum+states">Fundamental weight systems are quantum states</a> <a href="https://arxiv.org/abs/2105.02871">arXiv:2105.02871</a> </li> <li id="SS20b"> <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/Proper+Orbifold+Cohomology">Proper Orbifold Cohomology</a> <a href="https://arxiv.org/abs/2008.01101">arXiv:2008.01101</a> </li> <li id="FSS20b"> <a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/Twistorial+Cohomotopy+implies+Green-Schwarz+anomaly+cancellation">Twistorial Cohomotopy implies Green-Schwarz anomaly cancellation</a> Rev. Math. Phys. 34 05 (2022) 2250013 <a href="https://arxiv.org/abs/2008.08544">arXiv:2008.08544</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/The+Character+Map+in+Twisted+Non-Abelian+Cohomology">The Character Map in Twisted Non-Abelian Cohomology</a> <a href="https://arxiv.org/abs/2009.11909">arXiv:2009.11909</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/The+Character+Map+in+Equivariant+Twistorial+Cohomotopy">The Character Map in Equivariant Twistorial Cohomotopy</a> <a href="https://arxiv.org/abs/2011.06533">arXiv:2011.06533</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/M-Theory+as+Mf-Theory">M/F-Theory as Mf-Theory</a> <a href="https://arxiv.org/abs/2103.01877">arXiv:2103.01877</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/Anyonic+defect+branes+in+TED-K-theory">Anyonic defect branes in TED-K-theory</a> </li> </ul> See also: <ul> <li> <a class="existingWikiWord" href="/nlab/show/David+Michael+Roberts">David Michael Roberts</a>: Topological sectors for heterotic M5-brane charges under Hypothesis H* J. High Energ. Phys. 2020 52 (2020) <a href=" https://doi.org/10.1007/JHEP06(2020)052">doi:10.1007/JHEP06(2020)052</a> <a href="https://arxiv.org/abs/2003.09832">arXiv:2003.09832</a> </li> </ul> <h3 id="ExpositionAndTalkNotes">Exposition and talk notes</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Luigi+Alfonsi">Luigi Alfonsi</a>, §4 in: Higher geometry in physics, in: <a class="existingWikiWord" href="/nlab/show/Encyclopedia+of+Mathematical+Physics+2nd+ed">Encyclopedia of Mathematical Physics 2nd ed</a>, Elsevier (2024) [<a href="https://arxiv.org/abs/2312.07308">arXiv:2312.07308</a>] </li> <li> <a class="existingWikiWord" href="/schreiber/show/Some+Quantum+States+of+M-Branes+under+Hypothesis+H">Some Quantum States of M-Branes under Hypothesis H</a> talk at <a href="https://strings.ph.qmul.ac.uk/welcome-crst">Centre for Research in String Theory</a> Queen Mary University London, 2021 </li> <li> <a class="existingWikiWord" href="/schreiber/show/Proper+Orbifold+Cohomotopy+for+M-Theory">Proper Orbifold Cohomotopy for M-Theory</a> talk at <a href="https://ims.nus.edu.sg/events/string-and-m-theory-the-new-geometry-of-the-21st-century-ii/">String and M-Theory: The New Geometry of the 21st Century – II</a> via NUS Singapore, 2021 </li> <li> <a class="existingWikiWord" href="/schreiber/show/Microscopic+Brane+Physics+from+Cohomotopy">Microscopic Brane Physics from Cohomotopy</a> talk at <a class="existingWikiWord" href="/nlab/show/M-Theory+and+Mathematics">M-Theory and Mathematics</a> NYU Abu Dhabi, 2020 </li> <li> <a class="existingWikiWord" href="/schreiber/show/Strings2019">Twisted Cohomotopy implies M-theory anomaly cancellation</a> presentation at <a href="https://sis-pc15.ulb.ac.be/event/2/">Strings2019</a> Brussels, 2019 </li> <li> <a class="existingWikiWord" href="/schreiber/show/StringMath2017">Super p-Brane Theory emerging from Super Homotopy Theory</a> talk at <a class="existingWikiWord" href="/nlab/show/String+Math+2017">String Math 2017</a>, Hamburg, 2017 </li> <li> <a class="existingWikiWord" href="/schreiber/show/Equivariant+Cohomotopy+and+Branes">Equivariant Cohomotopy and Branes</a> talk at <a href="http://ims.nus.edu.sg/events/2018/wstring/index.php">String and M-Theory: The New Geometry of the 21st Century</a> NUS Singapore, 2018 </li> <li> <a class="existingWikiWord" href="/schreiber/show/The+Higher+Structure+of+11d+Supergravity">The Higher Structure of 11d Supergravity</a> talk at <a href="http://souriau2019.fr/">Souriau 2019</a> IHP Paris, 2019 </li> <li> <a class="existingWikiWord" href="/schreiber/show/Equivariant+Cohomotopy+of+toroidal+orbifolds">Equivariant Cohomotopy of toroidal orbifolds</a> talk at <a class="existingWikiWord" href="/nlab/show/Sadok+Kallel">Prof. Sadok Kallel</a>‘s group seminar AUS Sharjah, 2019 </li> <li> <a class="existingWikiWord" href="/schreiber/show/Cohomotopy+Theory+and+Branes">Cohomotopy Theory and Branes</a> talk at Geometry, topology and Physics NYU Abu Dhabi, 2020 </li> <li> <a class="existingWikiWord" href="/schreiber/show/Equivariant+Stable+Cohomotopy+and+Branes">Equivariant Stable Cohomotopy and Branes</a> talk at Geometry, Topology and Physics NYU Abu Dhabi, 2018 </li> <li> <a class="existingWikiWord" href="/schreiber/show/Equivariant+cohomology+of+M2%2FM5-branes">Equivariant cohomology of M2/M5-branes</a> talk at Seminar on Higher Structures MPI Bonn, 2016 </li> <li> <a class="existingWikiWord" href="/schreiber/files/ToposesCohomologyPhysics201002.pdf" title="pdf">pdf</a> </li> <li> <a class="existingWikiWord" href="/schreiber/files/CenturyOfChargeQuantization201012.pdf" title="pdf">pdf</a> </li> </ul> See also: <ul> <li><a class="existingWikiWord" href="/schreiber/show/New+Foundations+for+TDA+--+Cohomotopy">New Foundations for TDA – Cohomotopy</a></li> </ul> </body></html> </div> <div class="revisedby"> Last revised on January 6, 2024 at 18:40:35. See the <a href="/nlab/history/Hypothesis+H" style="color: #005c19">history</a> of this page for a list of all contributions to it. </div> <div class="navigation navfoot"> <a href="/nlab/edit/Hypothesis+H" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/14819/#Item_20">Discuss</a><a href="/nlab/revision/Hypothesis+H/6" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a><a href="/nlab/show/diff/Hypothesis+H" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/Hypothesis+H" accesskey="S" class="navlink" id="history" rel="nofollow">History (6 revisions)</a> <a href="/nlab/show/Hypothesis+H/cite" style="color: black">Cite</a> <a href="/nlab/print/Hypothesis+H" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/Hypothesis+H" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>