CINXE.COM
M M Postnikov in nLab
<!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> M M Postnikov 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[/*><!--*/ .toc ul {margin: 0; padding: 0;} .toc ul ul {margin: 0; padding: 0 0 0 10px;} .toc li > p {margin: 0} .toc ul li {list-style-type: none; position: relative;} .toc div {border-top:1px dotted #ccc;} .rightHandSide h2 {font-size: 1.5em;color:#008B26} table.plaintable { border-collapse:collapse; margin-left:30px; border:0; } .plaintable td {border:1px solid #000; padding: 3px;} .plaintable th {padding: 3px;} .plaintable caption { font-weight: bold; font-size:1.1em; text-align:center; margin-left:30px; } /* Query boxes for questioning and answering mechanism */ div.query{ background: #f6fff3; border: solid #ce9; border-width: 2px 1px; padding: 0 1em; margin: 0 1em; max-height: 20em; overflow: auto; } /* Standout boxes for putting important text */ div.standout{ background: #fff1f1; border: solid black; border-width: 2px 1px; padding: 0 1em; margin: 0 1em; overflow: auto; } /* Icon for links to n-category arXiv documents (commented out for now i.e. disabled) a[href*="http://arxiv.org/"] { background-image: url(../files/arXiv_icon.gif); background-repeat: no-repeat; background-position: right bottom; padding-right: 22px; } */ /* Icon for links to n-category cafe posts (disabled) a[href*="http://golem.ph.utexas.edu/category"] { background-image: url(../files/n-cafe_5.gif); background-repeat: no-repeat; background-position: right bottom; padding-right: 25px; } */ /* Icon for links to pdf files (disabled) a[href$=".pdf"] { background-image: url(../files/pdficon_small.gif); background-repeat: no-repeat; background-position: right bottom; padding-right: 25px; } */ /* Icon for links to pages, etc. -inside- pdf files (disabled) a[href*=".pdf#"] { background-image: url(../files/pdf_entry.gif); background-repeat: no-repeat; background-position: right bottom; padding-right: 25px; } */ a.existingWikiWord { color: #226622; } a.existingWikiWord:visited { color: #226622; } a.existingWikiWord[title] { border: 0px; color: #aa0505; text-decoration: none; } a.existingWikiWord[title]:visited { border: 0px; color: #551111; text-decoration: none; } a[href^="http://"] { border: 0px; color: #003399; } a[href^="http://"]:visited { border: 0px; color: #330066; } a[href^="https://"] { border: 0px; color: #003399; } a[href^="https://"]:visited { border: 0px; color: #330066; } div.dropDown .hide { display: none; } div.dropDown:hover .hide { display:block; } div.clickDown .hide { display: none; } div.clickDown:focus { outline:none; } div.clickDown:focus .hide, div.clickDown:hover .hide { display: block; } div.clickDown .clickToReveal, div.clickDown:focus .clickToHide { display:block; } div.clickDown:focus .clickToReveal, div.clickDown .clickToHide { display:none; } div.clickDown .clickToReveal:after { content: "A(Hover to reveal, click to "hold")"; font-size: 60%; } div.clickDown .clickToHide:after { content: "A(Click to hide)"; font-size: 60%; } div.clickDown .clickToHide, div.clickDown .clickToReveal { white-space: pre-wrap; } .un_theorem, .num_theorem, .un_lemma, .num_lemma, .un_prop, .num_prop, .un_cor, .num_cor, .un_defn, .num_defn, .un_example, .num_example, .un_note, .num_note, .un_remark, .num_remark { margin-left: 1em; } span.theorem_label { margin-left: -1em; } .proof span.theorem_label { margin-left: 0em; } :target { background-color: #BBBBBB; border-radius: 5pt; } /*]]>*/--></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[//><!-- MathJax.Ajax.config.path["Contrib"] = "/MathJax"; MathJax.Hub.Config({ MathML: { useMathMLspacing: true }, "HTML-CSS": { scale: 90, extensions: ["handle-floats.js"] } }); MathJax.Hub.Queue( function () { var fos = document.getElementsByTagName('foreignObject'); for (var i = 0; i < fos.length; i++) { MathJax.Hub.Typeset(fos[i]); } }); //--><!]]> </script> <script type="text/javascript"> <!--//--><![CDATA[//><!-- window.addEventListener("DOMContentLoaded", function () { var div = document.createElement('div'); var math = document.createElementNS('http://www.w3.org/1998/Math/MathML', 'math'); document.body.appendChild(div); div.appendChild(math); // Test for MathML support comparable to WebKit version https://trac.webkit.org/changeset/203640 or higher. div.setAttribute('style', 'font-style: italic'); var mathml_unsupported = !(window.getComputedStyle(div.firstChild).getPropertyValue('font-style') === 'normal'); div.parentNode.removeChild(div); if (mathml_unsupported) { // MathML does not seem to be supported... var s = document.createElement('script'); s.src = "https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=MML_HTMLorMML-full"; document.querySelector('head').appendChild(s); } else { document.head.insertAdjacentHTML("beforeend", '<style>svg[viewBox] {max-width: 100%}</style>'); } }); //--><!]]> </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> M M Postnikov </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/discussions/?CategoryID=0" title="Discuss this page on the nForum. It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" 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></title></head> <body> <ul> <li><a href="https://mathshistory.st-andrews.ac.uk/Biographies/Postnikov/">MacTutor biography</a></li> </ul> <p><strong>Михаил Михайлович Постников</strong>, (that is <em>Mihail Mihajlovič Postnikov</em> (in a slavistic transliteration as in ISO9 or G. Shevelov, A prehistory of Slavic, Columbia 1965) or <em>Mikhail Mikhailovich Postnikov</em>) is a former Russian (Soviet) topologist (27.10.1927–27.5.2004) and a professor at MGU until his sudden death.</p> <p>His thesis under the mentorship of <a class="existingWikiWord" href="/nlab/show/L.+S.+Pontrjagin">L. S. Pontrjagin</a> (<a href="http://ru.wikipedia.org/wiki/&#1055;&#1086;&#1085;&#1090;&#1088;&#1103;&#1075;&#1080;&#1085;,_&#1051;&#1077;&#1074;_&#1057;&#1077;&#1084;&#1105;&#1085;&#1086;&#1074;&#1080;&#1095;">ru.wiki</a>), contained a construction of a homotopical decomposition of a topological space into a tower of <a class="existingWikiWord" href="/nlab/show/fibration">fibration</a>s (now called a <a class="existingWikiWord" href="/nlab/show/Postnikov+tower">Postnikov tower</a>) and cohomological invariants (<a class="existingWikiWord" href="/nlab/show/Postnikov+invariant">Postnikov invariant</a>s) which together with the tower form a “system” (<a class="existingWikiWord" href="/nlab/show/Postnikov+system">Postnikov system</a>). The use of these data enabled him to effectively encode the full <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of a space.</p> <p>After his early work, Postnikov dedicated himself to teaching, leading a famous seminar at MGU, and advising students (among most successful of his students is Rudyak) and largely neglected his own research. As a clear and superb lecturer, he also wrote several volumes of advanced textbooks in geometry and topology, and a booklet on Galois theory. Unfortunately, he also dedicated much of his efforts and time to an alternative history, considered pseudoscience by the mainstream. Some of his mathematical memoirs may be found in the book <em>Golden Years of Moscow Mathematics</em>, by Smilka Zdravkovska (AMS, 2nd ed. 2007).</p> <p>See also <a href="http://ru.wikipedia.org/wiki/&#1055;&#1086;&#1089;&#1090;&#1085;&#1080;&#1082;&#1086;&#1074;,_&#1052;&#1080;&#1093;&#1072;&#1080;&#1083;_&#1052;&#1080;&#1093;&#1072;&#1081;&#1083;&#1086;&#1074;&#1080;&#1095;">Postnikov at ru.wikipedia</a> and <a href="http://higeom.math.msu.su/people/postnikov">msu page</a></p> <h2 id="selected_writings">Selected writings</h2> <p>Introducing <a class="existingWikiWord" href="/nlab/show/Postnikov+systems">Postnikov systems</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/M.+M.+Postnikov">M. M. Postnikov</a>, <em>Determination of the homology groups of a space by means of the homotopy invariants</em>, Doklady Akad. Nauk SSSR (N.S.) 76: 359–362 (1951)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M.+M.+Postnikov">M. M. Postnikov</a>, <em>Issledovaniya po gomotopičeskoĭ teorii nepreryvnyh otobraženiĭ. I. Algebraičeskaya teoriya sistem. II. Naturalʹnaya sistema i gomotopičeskiĭ tip</em>. (Russian) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><em>Investigations in homotopy theory of continuous mappings. I. The algebraic theory of systems. II. The natural system and homotopy type.</em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> Trudy Mat. Inst. Steklov. no. 46. Izdat. Akad. Nauk SSSR, Moscow, 1955. (<a href="http://mi.mathnet.ru/eng/tm1182">mathnet:tm1182</a>)</p> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a> and <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/M.+M.+Postnikov">M. M. Postnikov</a>, <em>Lectures on geometry: Semester V, Lie groups and algebras</em> (1986) [<a href="https://archive.org/details/postnikov-lectures-in-geometry-semester-v-lie-group-and-lie-algebras">ark:/13960/t4cp9jn4p</a>]</li> </ul> <h2 id="related_entries">Related entries</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+space">compactly generated topological space</a></li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/people">people</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on September 3, 2024 at 10:14:05. See the <a href="/nlab/history/M+M+Postnikov" 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/M+M+Postnikov" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussions/?CategoryID=0">Discuss</a><span class="backintime"><a href="/nlab/revision/M+M+Postnikov/18" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/M+M+Postnikov" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/M+M+Postnikov" accesskey="S" class="navlink" id="history" rel="nofollow">History (18 revisions)</a> <a href="/nlab/show/M+M+Postnikov/cite" style="color: black">Cite</a> <a href="/nlab/print/M+M+Postnikov" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/M+M+Postnikov" id="view_source" rel="nofollow">Source</a> </div> </div> <!-- Content --> </div> <!-- Container --> </body> </html>