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> MR cohomology theory 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> MR cohomology theory </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/5836/#Item_3" 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="cobordism_theory">Cobordism theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cobordism+theory">cobordism theory</a></strong> = <a class="existingWikiWord" href="/nlab/show/manifolds+and+cobordisms+-+contents">manifolds and cobordisms</a> + <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a>/<a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/equivariant+cobordism+theory">equivariant cobordism theory</a></li> </ul> <p><strong>Concepts of cobordism theory</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold">manifold</a>, <a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangential+structure">tangential structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>, <a class="existingWikiWord" href="/nlab/show/cobordism+class">cobordism class</a></p> <p><a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/submanifold">submanifold</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pontrjagin%27s+theorem">Pontrjagin's theorem</a> (<a class="existingWikiWord" href="/nlab/show/equivariant+Pontrjagin+theorem">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/twisted+Pontrjagin+theorem">twisted</a>):</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mo>↔</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{\leftrightarrow}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↔</mo></mrow><annotation encoding="application/x-tex">\leftrightarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/cobordism+classes">cobordism classes</a> of <a class="existingWikiWord" href="/nlab/show/normally+framed+submanifolds">normally framed submanifolds</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom%27s+theorem">Thom's theorem</a>:</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mo>↔</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{\leftrightarrow}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of maps to <a class="existingWikiWord" href="/nlab/show/Thom+space">Thom space</a> <a class="existingWikiWord" href="/nlab/show/MO">MO</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↔</mo></mrow><annotation encoding="application/x-tex">\leftrightarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/cobordism+classes">cobordism classes</a> of <a class="existingWikiWord" href="/nlab/show/normally+oriented+submanifolds">normally oriented submanifolds</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/Thom+space">Thom space</a></p> <p><a class="existingWikiWord" href="/nlab/show/Thom+isomorphism">Thom isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a></p> <p><a class="existingWikiWord" href="/nlab/show/Pontryagin-Thom+collapse+construction">Pontryagin-Thom collapse construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/complex+cobordism+cohomology+theory">complex cobordism cohomology theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientation in generalized cohomology</a></p> <p><a class="existingWikiWord" href="/nlab/show/genus">genus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a></p> </li> </ul> <div> <p><strong>flavors of <a class="existingWikiWord" href="/nlab/show/bordism+homology+theories">bordism homology theories</a>/<a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theories">cobordism cohomology theories</a>, their <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">representing</a> <a class="existingWikiWord" href="/nlab/show/Thom+spectra">Thom spectra</a> and <a class="existingWikiWord" href="/nlab/show/cobordism+rings">cobordism rings</a></strong>:</p> <p><a class="existingWikiWord" href="/nlab/show/bordism+homology+theory">bordism theory</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/M%28B%2Cf%29">M(B,f)</a> (<a class="existingWikiWord" href="/nlab/show/B-bordism">B-bordism</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/MFr">MFr</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MO">MO</a>, <a class="existingWikiWord" href="/nlab/show/MSO">MSO</a>, <a class="existingWikiWord" href="/nlab/show/MSpin">MSpin</a>, <a class="existingWikiWord" href="/nlab/show/MString">MString</a>, …</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MU">MU</a>, <a class="existingWikiWord" href="/nlab/show/MSU">MSU</a>, …</p> <p><a class="existingWikiWord" href="/nlab/show/Ravenel%27s+spectrum">MΩΩSU(n)</a></p> <p><a class="existingWikiWord" href="/nlab/show/MP-theory">MP</a>, <a class="existingWikiWord" href="/nlab/show/MR-theory">MR</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MSpin%5Ec">MSpin<sup><i>c</i></sup></a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MSp">MSp</a></p> </li> </ul> <p>relative bordism theories:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/MOFr">MOFr</a>, <a class="existingWikiWord" href="/nlab/show/MUFr">MUFr</a>, <a class="existingWikiWord" href="/nlab/show/MSUFr">MSUFr</a></li> </ul> <p><a class="existingWikiWord" href="/nlab/show/equivariant+bordism+homology+theory">equivariant bordism theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MFr">equivariant MFr</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MO">equivariant MO</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MU">equivariant MU</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+bordism+homology+theory">global equivariant bordism theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+mO">global equivariant mO</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+mU">global equivariant mU</a></p> </li> </ul> <p>algebraic:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+cobordism">algebraic cobordism</a></li> </ul> </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='#properties'>Properties</a></li> <ul> <li><a href='#relation_to_real_manifolds'>Relation to real manifolds</a></li> <li><a href='#structure'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">E_\infty</annotation></semantics></math>-structure</a></li> <li><a href='#universal_real_orientation'>Universal real orientation</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>What is called <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MR</mi></mrow><annotation encoding="application/x-tex">MR</annotation></semantics></math> cohomology theory</em> or <em>Real cobordism</em> (<a href="#Landweber68">Landweber 68</a>, <a href="#Landweber69">Landweber 69</a>) is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/equivariant+cohomology+theory">equivariant cohomology theory</a> version of <a class="existingWikiWord" href="/nlab/show/complex+cobordism">complex cobordism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math>.</p> <p>There is an evident <a class="existingWikiWord" href="/nlab/show/action">action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/formal+group+laws">formal group laws</a> given by negation in the formal group (the <a class="existingWikiWord" href="/nlab/show/inversion+involution">inversion involution</a>), and this lifts to an <a class="existingWikiWord" href="/nlab/show/involution">involutive</a> <a class="existingWikiWord" href="/nlab/show/automorphism">automorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi><mover><mo>→</mo><mo>≃</mo></mover><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU \stackrel{\simeq}{\to} MU</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a> <a class="existingWikiWord" href="/nlab/show/MU">MU</a>. This induces an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/equivariant+spectrum">equivariant spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">M\mathbb{R}</annotation></semantics></math>, and real cobordism is the <a class="existingWikiWord" href="/nlab/show/cohomology+theory">cohomology theory</a> that it <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">represents</a>. This is directly analogous to how <a class="existingWikiWord" href="/nlab/show/complex+K-theory">complex K-theory</a> <a class="existingWikiWord" href="/nlab/show/KU">KU</a> gives <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-equivariant <a class="existingWikiWord" href="/nlab/show/KR-theory">KR-theory</a>, both are examples of <em><a class="existingWikiWord" href="/nlab/show/real-oriented+cohomology+theories">real-oriented cohomology theories</a></em>.</p> <p>A modern review in in (<a href="#Kriz01">Kriz 01, section 2</a>).</p> <h2 id="properties">Properties</h2> <h3 id="relation_to_real_manifolds">Relation to real manifolds</h3> <p>While <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>M</mi><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_\bullet(M \mathbb{R})</annotation></semantics></math> is <em>not</em> the <a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a> of <a class="existingWikiWord" href="/nlab/show/real+manifolds">real manifolds</a>, still every <a class="existingWikiWord" href="/nlab/show/real+manifold">real manifold</a> does give a class in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">M \mathbb{R}</annotation></semantics></math> (<a href="#Hu99">Hu 99</a>, <a href="#Kriz01">Kriz 01, p. 13</a>).</p> <h3 id="structure"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">E_\infty</annotation></semantics></math>-structure</h3> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">M \mathbb{R}</annotation></semantics></math> is naturally an <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a> spectrum. (reviewed as <a href="#Kriz01">Kriz 01, prop. 3.1</a>)</p> <h3 id="universal_real_orientation">Universal real orientation</h3> <p>In direct analogy with the situation for <a class="existingWikiWord" href="/nlab/show/complex+cobordism+theory">complex cobordism theory</a> in <a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theory">complex oriented cohomology theory</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">M \mathbb{R}</annotation></semantics></math> is the univeral <a class="existingWikiWord" href="/nlab/show/real+oriented+cohomology+theory">real oriented cohomology theory</a>:</p> <p>Equivalence classes of <a class="existingWikiWord" href="/nlab/show/real+oriented+cohomology+theory">real orientations</a> of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/equivariant+spectrum">equivariant</a> <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> are in <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> to equivalence classes of <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a> homomorphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>ℝ</mi><mo>⟶</mo><mi>E</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> M \mathbb{R}\longrightarrow E \,. </annotation></semantics></math></div> <p>(<a href="#Kriz01">Kriz 01, theorem 2.25</a>)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cobordism+cohomology+theory">equivariant cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/real-oriented+cohomology+theory">real-oriented cohomology theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/KR-theory">KR-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BPR-theory">BPR-theory</a></p> </li> </ul> </li> </ul> <p><br /></p> <div> <p><strong>flavors of <a class="existingWikiWord" href="/nlab/show/bordism+homology+theories">bordism homology theories</a>/<a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theories">cobordism cohomology theories</a>, their <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">representing</a> <a class="existingWikiWord" href="/nlab/show/Thom+spectra">Thom spectra</a> and <a class="existingWikiWord" href="/nlab/show/cobordism+rings">cobordism rings</a></strong>:</p> <p><a class="existingWikiWord" href="/nlab/show/bordism+homology+theory">bordism theory</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/M%28B%2Cf%29">M(B,f)</a> (<a class="existingWikiWord" href="/nlab/show/B-bordism">B-bordism</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/MFr">MFr</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MO">MO</a>, <a class="existingWikiWord" href="/nlab/show/MSO">MSO</a>, <a class="existingWikiWord" href="/nlab/show/MSpin">MSpin</a>, <a class="existingWikiWord" href="/nlab/show/MString">MString</a>, …</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MU">MU</a>, <a class="existingWikiWord" href="/nlab/show/MSU">MSU</a>, …</p> <p><a class="existingWikiWord" href="/nlab/show/Ravenel%27s+spectrum">MΩΩSU(n)</a></p> <p><a class="existingWikiWord" href="/nlab/show/MP-theory">MP</a>, <a class="existingWikiWord" href="/nlab/show/MR-theory">MR</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MSpin%5Ec">MSpin<sup><i>c</i></sup></a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MSp">MSp</a></p> </li> </ul> <p>relative bordism theories:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/MOFr">MOFr</a>, <a class="existingWikiWord" href="/nlab/show/MUFr">MUFr</a>, <a class="existingWikiWord" href="/nlab/show/MSUFr">MSUFr</a></li> </ul> <p><a class="existingWikiWord" href="/nlab/show/equivariant+bordism+homology+theory">equivariant bordism theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MFr">equivariant MFr</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MO">equivariant MO</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MU">equivariant MU</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+bordism+homology+theory">global equivariant bordism theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+mO">global equivariant mO</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+mU">global equivariant mU</a></p> </li> </ul> <p>algebraic:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+cobordism">algebraic cobordism</a></li> </ul> </div> <h2 id="References">References</h2> <p>The definition of Real cobordism cohomology goes back to:</p> <ul> <li id="Landweber67"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Landweber">Peter Landweber</a>, <em>Fixed point free conjugations on complex manifolds</em>, Annals of Mathematics <strong>86</strong> (2) (1967) 491-502 [<a href="https://www.jstor.org/stable/1970612">jstor:</a>]</p> </li> <li id="Landweber68"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Landweber">Peter Landweber</a>, <em>Conjugations on complex manifolds and equivariant homotopy of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math></em>, Bulletin of the American Mathematical Society <strong>74</strong> (1968) 271-274 [<a href="https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-74/issue-2/Conjugations-on-complex-manifolds-and-equivariant-homotopy-of-MU/bams/1183529527.full">Euclid</a>]</p> </li> </ul> <p>and in the broader context of <a class="existingWikiWord" href="/nlab/show/real-oriented+cohomology+theories">real-oriented cohomology theories</a>:</p> <ul> <li id="Araki78"> <p><a class="existingWikiWord" href="/nlab/show/Sh%C3%B4r%C3%B4+Araki">Shôrô Araki</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math>-Cohomology Theories</em>, Japanese Journal of Mathematics <strong>4</strong> 2 (1978) 363-416 [<a href="https://doi.org/10.4099/math1924.4.363">doi:10.4099/math1924.4.363</a>]</p> </li> <li id="Araki79"> <p><a class="existingWikiWord" href="/nlab/show/Sh%C3%B4r%C3%B4+Araki">Shôrô Araki</a>, <em>Forgetful spectral sequences</em>, Osaka Journal of Mathematics <strong>16</strong> 1 (1979) 173-199 [<a href="https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-16/issue-1/Forgetful-spectral-sequences/ojm/1200771837.full">Euclid</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sh%C3%B4r%C3%B4+Araki">Shôrô Araki</a>, <em>Orientations in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math>-cohomology theories</em>, Japan Journal of Mathematics (N.S.) <strong>5</strong> 2 (1979) 403-430 [<a href="https://doi.org/10.4099/math1924.5.403">doi:10.4099/math1924.5.403</a>]</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/Adams+spectral+sequence">Adams spectral sequence</a> for Real cobordism:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Po+Hu">Po Hu</a>, <em>The cobordism of Real manifolds and calculations with the Real Adams-Novikov spectral sequence</em>, (1998) [<a href="http://hdl.handle.net/2027.42/130996">hdl:2027.42/130996</a>, <a href="https://deepblue.lib.umich.edu/bitstream/handle/2027.42/130996/9825253.pdf?sequence=2&isAllowed=n">pdf</a>]</p> </li> <li id="Kriz01"> <p><a class="existingWikiWord" href="/nlab/show/Po+Hu">Po Hu</a>, <a class="existingWikiWord" href="/nlab/show/Igor+Kriz">Igor Kriz</a>, <em>Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence</em>, Topology 40 (2001) 317-399 [<a href="https://doi.org/10.1016/S0040-9383(99)00065-8">doi:10.1016/S0040-9383(99)00065-8</a>, <a href="http://www.math.rochester.edu/people/faculty/doug/otherpapers/hukriz.pdf">pdf</a>]</p> </li> </ul> <p>Further discussion:</p> <ul> <li id="Hu99"> <p><a class="existingWikiWord" href="/nlab/show/Po+Hu">Po Hu</a>, <em>The cobordism of Real manifolds</em>, Fundamenta Mathematicae (1999) Volume: 161, Issue: 1-2, page 119-136 [<a href="https://eudml.org/doc/212395">dml:212395</a>, <a href="http://matwbn.icm.edu.pl/ksiazki/fm/fm161/fm16115.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Po+Hu">Po Hu</a>, <a class="existingWikiWord" href="/nlab/show/Igor+Kriz">Igor Kriz</a>, <em>Some Remarks on Real and Algebraic Cobordism</em>, K-Theory <strong>22</strong> (2001) 335–366 [<a href="https://deepblue.lib.umich.edu/bitstream/handle/2027.42/43150/10977_2004_Article_321295.pdf?sequence=1">pdf</a>, <a href="http://dx.doi.org/10.1023/A:1011196901303">doi:10.1023/A:1011196901303</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Po+Hu">Po Hu</a>, <a class="existingWikiWord" href="/nlab/show/Igor+Kriz">Igor Kriz</a>, <em>Topological Hermitian Cobordism</em>, Journal of Homotopy and Related Structures, <strong>11</strong> (2016) 173–197 [<a href="https://arxiv.org/abs/1110.5608">arXiv:1110.5608</a>, <a href="https://doi.org/10.1007/s40062-014-0100-9">doi:10.1007/s40062-014-0100-9</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 22, 2023 at 17:34:11. See the <a href="/nlab/history/MR+cohomology+theory" 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/MR+cohomology+theory" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/5836/#Item_3">Discuss</a><span class="backintime"><a href="/nlab/revision/MR+cohomology+theory/11" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/MR+cohomology+theory" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/MR+cohomology+theory" accesskey="S" class="navlink" id="history" rel="nofollow">History (11 revisions)</a> <a href="/nlab/show/MR+cohomology+theory/cite" style="color: black">Cite</a> <a href="/nlab/print/MR+cohomology+theory" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/MR+cohomology+theory" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>