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> EHP spectral sequence 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> EHP spectral sequence </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/6857/#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="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <p><strong><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></strong></p> <p>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>…</p> <p>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>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <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> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><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></p> </li> <li> <p><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></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><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></p> </li> <li> <p><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></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <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> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> <h4 id="homological_algebra">Homological algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></strong></p> <p>(also <a class="existingWikiWord" href="/nlab/show/nonabelian+homological+algebra">nonabelian homological algebra</a>)</p> <p><em><a class="existingWikiWord" href="/schreiber/show/Introduction+to+Homological+Algebra">Introduction</a></em></p> <p><strong>Context</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-additive+category">pre-additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+category">Grothendieck category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-abelian+category">semi-abelian category</a></p> </li> </ul> <p><strong>Basic definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>, <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex">complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential">differential</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homology+and+cohomology">chain homology and cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">simplicial homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>, <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a>, <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+resolution">flat resolution</a></p> </li> </ul> </li> </ul> <p><strong>Stable homotopy theory notions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>, <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated dg-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E-category">A-∞-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+chain+complexes">(∞,1)-category of chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>, <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">derived functor in homological algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>, <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lim%5E1+and+Milnor+sequences">lim^1 and Milnor sequences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fr-code">fr-code</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+complex">BRST-BV complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Serre+spectral+sequence">Serre spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Serre+spectral+sequence">Hochschild-Serre spectral sequence</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Lemmas</strong></p> <p><a class="existingWikiWord" href="/nlab/show/diagram+chasing">diagram chasing</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/3x3+lemma">3x3 lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/four+lemma">four lemma</a>, <a class="existingWikiWord" href="/nlab/show/five+lemma">five lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/snake+lemma">snake lemma</a>, <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/horseshoe+lemma">horseshoe lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baer%27s+criterion">Baer's criterion</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Schanuel%27s+lemma">Schanuel's lemma</a></p> <p><strong>Homology theories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> / <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal</a>, <a class="existingWikiWord" href="/nlab/show/operadic+Dold-Kan+correspondence">operadic</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a>, <a class="existingWikiWord" href="/nlab/show/Alexander-Whitney+map">Alexander-Whitney map</a>, <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+theorem">Eilenberg-Zilber theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">cochain on a simplicial set</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#constructions_of_'>Constructions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></a></li> <ul> <li><a href='#via_the_james_model'>Via the James model</a></li> <li><a href='#via_pushoutpullback_comparisons'>Via pushout/pullback comparisons</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>The <em>EHP spectral sequence</em> (we follow <a href="#Mahowald85">Mahowald 85</a>) is the <a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a> for computation of <a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+spheres">homotopy groups of spheres</a> induced from the <a class="existingWikiWord" href="/nlab/show/filtration">filtration</a> of the underlying homotopy type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mn>∞</mn></msup><msup><mi>Σ</mi> <mn>∞</mn></msup><msup><mi>S</mi> <mn>0</mn></msup><mo>=</mo><msup><mi>Ω</mi> <mn>∞</mn></msup><mi>𝕊</mi></mrow><annotation encoding="application/x-tex">\Omega^\infty \Sigma^\infty S^0 = \Omega^\infty \mathbb{S}</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a> by <a class="existingWikiWord" href="/nlab/show/suspensions">suspensions</a> (German: <em>E</em>inhängung):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>n</mi></msup><msup><mi>S</mi> <mi>n</mi></msup><mover><mo>⟶</mo><mi>E</mi></mover><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mi>S</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega^n S^n \stackrel{E}{\longrightarrow} \Omega^{n+1} S^{n+1} \,. </annotation></semantics></math></div> <p>More concretely, (<a href="#James57">James 57</a>) constructed maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><msup><mi>S</mi> <mi>n</mi></msup><mover><mo>⟶</mo><mi>H</mi></mover><mi>Ω</mi><msup><mi>S</mi> <mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> \Omega S^n \stackrel{H}{\longrightarrow} \Omega S^{2n-1} </annotation></semantics></math></div> <p>(for <em>H</em>opf as in <a class="existingWikiWord" href="/nlab/show/Hopf+invariant">Hopf invariant</a>) and showed that <a class="existingWikiWord" href="/nlab/show/p-localization">2-locally</a> these fit with <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> into <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequences">homotopy fiber sequences</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msup><msup><mi>S</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mover><mo>⟶</mo><mi>P</mi></mover><msup><mi>Ω</mi> <mi>n</mi></msup><msup><mi>S</mi> <mi>n</mi></msup><mover><mo>⟶</mo><mi>E</mi></mover><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mi>S</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mover><mo>⟶</mo><mi>H</mi></mover><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mi>S</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega^{n+2} S^{2n+1} \stackrel{P}{\longrightarrow} \Omega^n S^n \stackrel{E}{\longrightarrow} \Omega^{n+1} S^{n+1} \stackrel{H}{\longrightarrow} \Omega^{n+1}S^{2n+1} \,. </annotation></semantics></math></div> <p>(Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> is by definition the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of <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>, the notation refers to <em><a class="existingWikiWord" href="/nlab/show/Whitehead+product">Whitehead product</a></em>.)</p> <p>This “EHP-<a class="existingWikiWord" href="/nlab/show/long+homotopy+fiber+sequence">long homotopy fiber sequence</a>” gives rise to the corresponding <a class="existingWikiWord" href="/nlab/show/long+exact+sequence+of+homotopy+groups">long exact sequence of homotopy groups</a> and so to an <a class="existingWikiWord" href="/nlab/show/exact+couple">exact couple</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo>⊕</mo><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></munder><msub><mi>π</mi> <mrow><mi>s</mi><mo>+</mo><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><msup><mi>Ω</mi> <mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mi>S</mi> <mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mi>i</mi></mover></mtd> <mtd></mtd> <mtd><msub><mi>π</mi> <mrow><mi>s</mi><mo>+</mo><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><msup><mi>Ω</mi> <mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mi>S</mi> <mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↖</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><munder><mo>⊕</mo><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></munder><msub><mi>π</mi> <mrow><mi>t</mi><mo>+</mo><mi>s</mi></mrow></msub><mo stretchy="false">(</mo><msup><mi>Ω</mi> <mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mi>S</mi> <mrow><mn>2</mn><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \underset{s,t}{\oplus} \pi_{s+t}(\Omega^{s+1}S^{s+1}) && \stackrel{i}{\longrightarrow} && \pi_{s+t}(\Omega^{s+1}S^{s+1}) \\ & \nwarrow && \swarrow \\ && \underset{s,t}{\oplus} \pi_{t+s}(\Omega^{s+1}S^{2s+1}) } \,. </annotation></semantics></math></div> <p>The corresponding <a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a> is the EHP spectral sequence proper. It converges, 2-locally, to the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups+of+spheres">stable homotopy groups of spheres</a>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">E_1</annotation></semantics></math>-page given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mn>1</mn> <mrow><mi>k</mi><mo>,</mo><mi>n</mi></mrow></msubsup><mo>=</mo><msub><mi>π</mi> <mrow><mi>k</mi><mo>+</mo><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><msubsup><mi>π</mi> <mo>•</mo> <mi>𝕊</mi></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E^{k,n}_1 = \pi_{k+n}(S^{2n-1}) \;\Rightarrow\; \pi_\bullet^{\mathbb{S}} \,. </annotation></semantics></math></div> <p>For more general <a class="existingWikiWord" href="/nlab/show/prime+numbers">prime numbers</a> than just 2, (<a href="#Toda62">Toda 62</a>) found analogous fibrations, which hence give EHP spectral sequences for general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>.</p> <p>The EHP spectral sequence is often used used in the context of the <a class="existingWikiWord" href="/nlab/show/Adams-Novikov+spectral+sequence">Adams-Novikov spectral sequence</a> for <a class="existingWikiWord" href="/nlab/show/p-localization">p-localization</a> at some prime <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>.</p> <h2 id="constructions_of_">Constructions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></h2> <p>For James’ fiber sequence, the essential property required of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> is to realize the isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>the</mi><mspace width="mediummathspace"></mspace><mi>map</mi><mspace width="mediummathspace"></mspace><mi>H</mi><msup><mo stretchy="false">)</mo> <mrow><mi>cohomology</mi><mspace width="mediummathspace"></mspace><mi>pullback</mi></mrow></msup><mo>:</mo><msup><mi>H</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mo stretchy="false">(</mo><mi>Ω</mi><msup><mi>𝕊</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mover><mo>→</mo><mo>∼</mo></mover><msup><mi>H</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mo stretchy="false">(</mo><mi>Ω</mi><msup><mi>𝕊</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> (the\:map\:H)^{cohomology\:pullback} : H^{2n}(\Omega\mathbb{S}^{2n+1})\overset{\sim}{\to} H^{2n}(\Omega \mathbb{S}^{n+1}) .</annotation></semantics></math></div> <p>The remaining corollaries then follow using the fact cohomology pullback is a ring homomorphism, and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>mod</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">mod 2</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Leray-Serre+spectral+sequence">Leray-Serre spectral sequence</a>.</p> <h3 id="via_the_james_model">Via the James model</h3> <p>Using the James model of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>Σ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Omega\Sigma X</annotation></semantics></math> as a quotient space of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>colim</mi> <mi>n</mi></msub><msup><mi>X</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">colim_n X^n </annotation></semantics></math>, a candidate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> is constructed by recursion:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>#</mo><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>∧</mo><msub><mi>x</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo>∧</mo><msub><mi>x</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H([x_0 , \dots, x_{n+1}]) = H([x_0,\dots,x_n]) \# ([x_0 \wedge x_{n+1} , \dots , x_n\wedge x_{n+1}]) </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>#</mo></mrow><annotation encoding="application/x-tex">\#</annotation></semantics></math> denotes concatenation and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∧</mo></mrow><annotation encoding="application/x-tex">\wedge</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a>. One checks that the ordering of product terms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>i</mi></msub><mo>∧</mo><msub><mi>x</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex"> x_i\wedge x_j </annotation></semantics></math> w.r.t. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>k</mi></msub><mo>∧</mo><msub><mi>x</mi> <mi>l</mi></msub></mrow><annotation encoding="application/x-tex">x_k\wedge x_l</annotation></semantics></math> depends only on the relative orders of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>l</mi></mrow><annotation encoding="application/x-tex">i,j,k,l</annotation></semantics></math>, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> is well-defined on the quotient space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>Σ</mi><mi>X</mi><mo>→</mo><mi>Ω</mi><mi>Σ</mi><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega\Sigma X \to \Omega \Sigma(X\wedge X)</annotation></semantics></math>.</p> <p>In particular, the restriction to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mn>2</mn></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">J_2 X</annotation></semantics></math> factors through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∧</mo><mi>X</mi><mo>→</mo><mi>Ω</mi><mi>Σ</mi><mi>X</mi><mo>∧</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> X\wedge X \to \Omega\Sigma X\wedge X</annotation></semantics></math> as the cofiber of the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msub><mi>J</mi> <mn>2</mn></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">X \to J_2 X</annotation></semantics></math>. In the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>≃</mo><msup><mi>𝕊</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">X\simeq \mathbb{S}^n</annotation></semantics></math>, the desired cohomology isomorphism is immediate.</p> <h3 id="via_pushoutpullback_comparisons">Via pushout/pullback comparisons</h3> <p>Starting with the three-legged cospan <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mover><mo>⟶</mo><mo>⟶</mo></mover><mo>⟶</mo></mover><mo>*</mo></mrow><annotation encoding="application/x-tex"> X \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} * </annotation></semantics></math>, construct the cube of all pushouts</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Σ</mi><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Σ</mi><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Σ</mi><mi>X</mi><mo>∨</mo><mi>Σ</mi><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↖</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Σ</mi><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X & & & \to & & & * \\ & \searrow & & & & \swarrow \\ & & * & \to & \Sigma X \\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ & & \Sigma X & \to & \Sigma X \vee \Sigma X \\ & \nearrow & & & & \nwarrow \\ * & & & \to & & & \Sigma X } </annotation></semantics></math></div> <p>Construct pullbacks in some pair of parallel squares, and compare them by naturality</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Ω</mi><mi>Σ</mi><mi>X</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Ω</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Ω</mi><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo><mo>⋆</mo><mo stretchy="false">(</mo><mi>Ω</mi><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Σ</mi><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Σ</mi><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Σ</mi><mi>X</mi><mo>∨</mo><mi>Σ</mi><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↖</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Σ</mi><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Omega\Sigma X & & & \to & & & * \\ & \searrow & & & & \swarrow \\ & & \Omega((\Omega\Sigma X)\star(\Omega\Sigma X)) & \to & \Sigma X \\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ & & \Sigma X & \to & \Sigma X \vee \Sigma X \\ & \nearrow & & & & \nwarrow \\ * & & & \to & & & \Sigma X } </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋆</mo></mrow><annotation encoding="application/x-tex">\star</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/join">reduced join</a>. On the other hand, the natural transformations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mi>Ω</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex"> \Sigma\Omega \to 1</annotation></semantics></math> give natural maps, e.g.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo stretchy="false">(</mo><mi>ev</mi><mo>∘</mo><mi>τ</mi><mo>∘</mo><mi>ev</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Ω</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Ω</mi><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo><mo>⋆</mo><mo stretchy="false">(</mo><mi>Ω</mi><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>Ω</mi><mo stretchy="false">(</mo><mi>X</mi><mo>⋆</mo><mi>X</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">\Omega( ev \circ \tau \circ ev ) : \Omega((\Omega\Sigma X) \star (\Omega\Sigma X)) \to \Omega( X \star X) .</annotation></semantics></math></div> <p>The composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>Σ</mi><mi>X</mi><mo>→</mo><mi>Ω</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Ω</mi><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo><mo>⋆</mo><mo stretchy="false">(</mo><mi>Ω</mi><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>Ω</mi><mo stretchy="false">(</mo><mi>X</mi><mo>⋆</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Omega \Sigma X \to \Omega((\Omega\Sigma X)\star(\Omega\Sigma X)) \to \Omega(X\star X) </annotation></semantics></math></div> <p>is a candidate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a> may be obtained from the EHP spectral sequence;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Goodwillie+spectral+sequence">Goodwillie spectral sequence</a> of the identity functor at the point also computes homotopy groups of spheres, the interplay of the two is discussed in (<a href="#Behrens10">Behrens 10</a>)</p> </li> </ul> <h2 id="references">References</h2> <p>Original articles include</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">Ioan Mackenzie James</a>, <em>Reduced product spaces</em>, Ann. of Math. (2) 62 (1955), 170-197.</p> </li> <li id="James57"> <p><a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">Ioan Mackenzie James</a>, <em>On the Suspension Sequence</em>, Annals of Mathematics Second Series, Vol. 65, No. 1 (Jan., 1957), pp. 74-107 (<a href="http://www.jstor.org/stable/1969666">jstor</a>)</p> </li> <li id="Toda62"> <p><a class="existingWikiWord" href="/nlab/show/Hiroshi+Toda">Hiroshi Toda</a>, <em>Composition methods in homotopy groups of spheres</em>, Princeton University Press (1962)</p> </li> <li id="Mahowald85"> <p><a class="existingWikiWord" href="/nlab/show/Mark+Mahowald">Mark Mahowald</a>, <em>Lin’s theorem and the EHP sequence</em>. Conference on algebraic topology in honor of <a class="existingWikiWord" href="/nlab/show/Peter+Hilton">Peter Hilton</a>, Contemp. Math. 37 (1985), 115–119. Amer. Math. Soc., Providence, RI.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Marcel+B%C3%B6kstedt">Marcel Bökstedt</a>, <a class="existingWikiWord" href="/nlab/show/Anne+Marie+Svane">Anne Marie Svane</a>, <em>A generalization of the stable EHP spectral sequence</em> (<a href="http://arxiv.org/abs/1208.3938">arXiv:1208.3938</a>)</p> </li> </ul> <p>Relation to the <a class="existingWikiWord" href="/nlab/show/Goodwillie+spectral+sequence">Goodwillie spectral sequence</a> is discussed in</p> <ul> <li id="Behrens10"><a class="existingWikiWord" href="/nlab/show/Mark+Behrens">Mark Behrens</a>, <em>The Goodwillie tower and the EHP sequence</em> (<a href="http://arxiv.org/abs/1009.1125">arXiv:1009.1125</a>)</li> </ul> <p>An algebraic version of the EHP spectral sequence for the <a class="existingWikiWord" href="/nlab/show/Lambda-algebra">Lambda-algebra</a> and used for computation of the second page of the <a class="existingWikiWord" href="/nlab/show/classical+Adams+spectral+sequence">classical Adams spectral sequence</a> (the <em><a class="existingWikiWord" href="/nlab/show/Curtis+algorithm">Curtis algorithm</a></em>), is discussed in</p> <ul> <li id="Kochmann96"><a class="existingWikiWord" href="/nlab/show/Stanley+Kochman">Stanley Kochman</a>, around prop. 5.2.6 of <em><a class="existingWikiWord" href="/nlab/show/Bordism%2C+Stable+Homotopy+and+Adams+Spectral+Sequences">Bordism, Stable Homotopy and Adams Spectral Sequences</a></em>, AMS 1996</li> </ul> <p>Review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a> (notes by <a class="existingWikiWord" href="/nlab/show/Akhil+Mathew">Akhil Mathew</a>), Lectures 6,7 in: <em>Spectra and stable homotopy theory</em>, 2012 (<a href="http://math.uchicago.edu/~amathew/256y.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/HopkinsMathewStableHomotopyTheory.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mark+Mahowald">Mark Mahowald</a>, <a class="existingWikiWord" href="/nlab/show/Doug+Ravenel">Doug Ravenel</a>, section 7 of <em>Towards a Global Understanding of the Homotopy Groups of Spheres</em> (<a href="http://www.math.rochester.edu/people/faculty/doug/mypapers/global.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Doug+Ravenel">Doug Ravenel</a>, chapter 1, section 5 of <em><a class="existingWikiWord" href="/nlab/show/Complex+cobordism+and+stable+homotopy+groups+of+spheres">Complex cobordism and stable homotopy groups of spheres</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/eom">eom</a>, <em><a href="https://www.encyclopediaofmath.org/index.php/EHP_spectral_sequence">EHP spectral sequence</a></em></p> </li> </ul> <p>See also:</p> <ul> <li>Wikipedia, <em><a href="http://en.wikipedia.org/wiki/EHP_spectral_sequence">EHP spectral sequence</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 24, 2021 at 10:41:05. See the <a href="/nlab/history/EHP+spectral+sequence" 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/EHP+spectral+sequence" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/6857/#Item_3">Discuss</a><span class="backintime"><a href="/nlab/revision/EHP+spectral+sequence/15" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/EHP+spectral+sequence" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/EHP+spectral+sequence" accesskey="S" class="navlink" id="history" rel="nofollow">History (15 revisions)</a> <a href="/nlab/show/EHP+spectral+sequence/cite" style="color: black">Cite</a> <a href="/nlab/print/EHP+spectral+sequence" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/EHP+spectral+sequence" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>