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<!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> Editing cylinder spectrum in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="noindex,nofollow" /> <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> Editing cylinder spectrum </h1> <div id="MarkupHelp"> <h3 style="font-size: 1.1em; font-style: normal; text-align: center">Syntax tips</h3> <ol style="margin-left: 0em; padding-left: 0em"> <li style="font-size: 0.8em">The basic syntax is <a href="https://www.markdownguide.org/cheat-sheet/">extended Markdown</a>. </li> <li style="font-size: 0.8em">Links to other nLab pages should be made by surrounding the name of the page in double square brackets: [[ name of page ]]. To link to an nLab page but show a different link text, do the following: [[ name of page | link text to show ]].</li> <li style="font-size: 0.8em">LaTeX can be used inside single dollar signs (inline) or double dollar signs or \[ and \], as usual. </li> <li style="font-size: 0.8em">To create a table of contents, add \tableofcontents on its own line.</li> <li style="font-size: 0.8em">For a theorem or proof, use \begin{theorem} \end{theorem} as you would in LaTeX. Labelling and referencing is exactly as in LaTeX, with use of \label and \ref. The full list of supported environments can be found in the <a href="/nlab/show/HowTo#DefinitionTheoremProofEnvironments">HowTo</a>. </li> <li style="font-size: 0.8em">Tikz can be used for figures almost exactly as in LaTeX. Similarly, tikz-cd and xymatrix can be used for commutative diagrams. See the <a href="/nlab/show/HowTo#diagrams">HowTo</a>.</li> <li style="font-size: 0.8em">As an alternative to the Markdown syntax for sections (headings), one can use the usual LaTeX syntax \section, \subsection, etc.</li> <li style="font-size: 0.8em">For further help, see the <a href="/nlab/show/HowTo">HowTo</a>, or you are very welcome to ask at the <a href="https://nforum.ncatlab.org/">nForum</a>.</li> </ol> </div> <form accept-charset="utf-8" action="/nlab/save/cylinder+spectrum" id="editForm" method="post"> <div style="display: none;"> <input name="see_if_human" id="see_if_human" style="tabindex: -1; autocomplete: off"/> </div> <div> <textarea name="content" id="content" style="height: 45em; width: 70%;"> #Contents# * table of contents {:toc} ## Idea For $S_\bullet$ a [[spectrum]], its _cylinder spectrum_ $Cyl(E)_\bullet$ has as component spaces the [[reduced cylinder]] spaces over the original component spaces: $$ Cyl(E)_n \coloneqq I_+ \wedge E_n \,, $$ i.e. the [[smash product]] (of [[pointed topological spaces]]) of $E_n$ with the unit inveral with a dijoint basepoint adjoined. Cylinder spectra serve to define [[left homotopy]] of spectra as for other [[cylinder objects]]. ## References Early lecture notes include * [[Frank Adams]], part III, section 2 _[[Stable homotopy and generalised homology]]_, 1974 [[!redirects cylinder spectra]]</textarea> <p> <input id="alter_title" name="alter_title" onchange="toggleVisibility();" type="checkbox" value="1" /> <label for="alter_title">Change page name.</label><br/> <span id="title_change" style="display:none"><label for="new_name">New name:</label> <input id="new_name" name="new_name" onblur="addRedirect();" type="text" value="cylinder spectrum" /></span> </p> <div> <p style="font-size: 0.8em; width: 70%;"> For non-trivial edits, please briefly describe your changes below. A discussion thread for this page does not yet exist at the <a href="https://nforum.ncatlab.org/discussions/?CategoryID=0">nForum</a>, but will be created upon submission of your changes, and your comments will be added to it. This discussion thread can also be used for further discussion related to this page. For trivial edits, such as correcting typos, please leave the box below empty; feel free to ask for advice at the nForum if you are unsure. </p> </div> <div> <textarea name="announcement" id="announcement" style="height: 10em; width: 70%"></textarea> </div> <div id="editFormButtons"> <input type="submit" value="Submit" accesskey="s"/> as <input id="author" name="author" onblur="this.value == '' ? this.value = 'Anonymous' : true" onfocus="this.value == 'Anonymous' ? this.value = '' : true;" type="text" value="Anonymous" /> | <span> <a href="/nlab/cancel_edit/cylinder+spectrum" accesskey="c">Cancel</a> <span class="unlock">(unlocks page)</span> </span> </div> </div> </form> <script type="text/javascript"> <![CDATA[//><!]]> </script> </div>  </div>  </body> </html>