CINXE.COM
[math/0602210] Generalized string topology operations
<!DOCTYPE html> <html lang="en"> <head> <title>[math/0602210] Generalized string topology operations</title> <meta name="viewport" content="width=device-width, initial-scale=1"> <link rel="apple-touch-icon" sizes="180x180" href="/static/browse/0.3.4/images/icons/apple-touch-icon.png"> <link rel="icon" type="image/png" sizes="32x32" href="/static/browse/0.3.4/images/icons/favicon-32x32.png"> <link rel="icon" type="image/png" sizes="16x16" href="/static/browse/0.3.4/images/icons/favicon-16x16.png"> <link rel="manifest" href="/static/browse/0.3.4/images/icons/site.webmanifest"> <link rel="mask-icon" href="/static/browse/0.3.4/images/icons/safari-pinned-tab.svg" color="#5bbad5"> <meta name="msapplication-TileColor" content="#da532c"> <meta name="theme-color" content="#ffffff"> <link rel="stylesheet" type="text/css" media="screen" href="/static/browse/0.3.4/css/arXiv.css?v=20241206" /> <link rel="stylesheet" type="text/css" media="print" href="/static/browse/0.3.4/css/arXiv-print.css?v=20200611" /> <link rel="stylesheet" type="text/css" media="screen" href="/static/browse/0.3.4/css/browse_search.css" /> <script language="javascript" src="/static/browse/0.3.4/js/accordion.js" /></script> <link rel="stylesheet" type="text/css" media="screen" href="/static/browse/0.3.4/css/slider.css?v=20250312" /> <script src="//code.jquery.com/jquery-latest.min.js" type="text/javascript"></script> <script type="text/javascript" src="/static/browse/0.3.4/js/donate.js?v=121124"></script><link rel="canonical" href="https://arxiv.org/abs/math/0602210"/> <meta name="description" content="Abstract page for arXiv paper math/0602210: Generalized string topology operations"><meta property="og:type" content="website" /> <meta property="og:site_name" content="arXiv.org" /> <meta property="og:title" content="Generalized string topology operations" /> <meta property="og:url" content="https://arxiv.org/abs/math/0602210v1" /> <meta property="og:image" content="/static/browse/0.3.4/images/arxiv-logo-fb.png" /> <meta property="og:image:secure_url" content="/static/browse/0.3.4/images/arxiv-logo-fb.png" /> <meta property="og:image:width" content="1200" /> <meta property="og:image:height" content="700" /> <meta property="og:image:alt" content="arXiv logo"/> <meta property="og:description" content="We show that the Chas-Sullivan loop product, a combination of the Pontrjagin product on the fiber and intersection product on the base, makes sense on the total space homology of any fiberwise monoid E over a closed oriented manifold M. More generally the Thom spectrum E^{-TM} is a ring spectrum. Similarly a fiberwise module over E defines a module over E^{-TM}. Fiberwise monoids include adjoint bundles of principal bundles, and the construction is natural with respect to maps of principal bundles. This naturality implies homotopy invariance of the algebra structure on H_*(LM) arising from the loop product. If M=BG is the infinite dimensional classifying space of a compact Lie group, then we get a well-defined pro-ring spectrum, which we define to be the string topology of BG. If E has a fiberwise action of the little n-cubes operad then E^{-TM} is an E_n-ring spectrum. This gives homology operations combining Dyer-Lashof operations on the fiber and Steenrod operations on the base. We give several examples where the new operations give homological insight, borrowed from knot theory, complex geometry, gauge theory, and homotopy theory."/> <meta name="twitter:site" content="@arxiv"/> <meta name="twitter:card" content="summary"/> <meta name="twitter:title" content="Generalized string topology operations"/> <meta name="twitter:description" content="We show that the Chas-Sullivan loop product, a combination of the Pontrjagin product on the fiber and intersection product on the base, makes sense on the total space homology of any fiberwise..."/> <meta name="twitter:image" content="https://static.arxiv.org/icons/twitter/arxiv-logo-twitter-square.png"/> <meta name="twitter:image:alt" content="arXiv logo"/> <link rel="stylesheet" media="screen" type="text/css" href="/static/browse/0.3.4/css/tooltip.css"/><link rel="stylesheet" media="screen" type="text/css" href="https://static.arxiv.org/js/bibex-dev/bibex.css?20200709"/> <script src="/static/browse/0.3.4/js/mathjaxToggle.min.js" type="text/javascript"></script> <script src="//code.jquery.com/jquery-latest.min.js" type="text/javascript"></script> <script src="//cdn.jsdelivr.net/npm/js-cookie@2/src/js.cookie.min.js" type="text/javascript"></script> <script src="//cdn.jsdelivr.net/npm/dompurify@2.3.5/dist/purify.min.js"></script> <script src="/static/browse/0.3.4/js/toggle-labs.js?20241022" type="text/javascript"></script> <script src="/static/browse/0.3.4/js/cite.js" type="text/javascript"></script><meta name="citation_title" content="Generalized string topology operations" /><meta name="citation_author" content="Gruher, Kate" /><meta name="citation_author" content="Salvatore, Paolo" /><meta name="citation_doi" content="10.1112/plms/pdm030" /><meta name="citation_date" content="2006/02/10" /><meta name="citation_online_date" content="2006/02/10" /><meta name="citation_pdf_url" content="http://arxiv.org/pdf/math/0602210" /><meta name="citation_arxiv_id" content="math/0602210" /><meta name="citation_abstract" content=" We show that the Chas-Sullivan loop product, a combination of the Pontrjagin product on the fiber and intersection product on the base, makes sense on the total space homology of any fiberwise monoid E over a closed oriented manifold M. More generally the Thom spectrum E^{-TM} is a ring spectrum. Similarly a fiberwise module over E defines a module over E^{-TM}. Fiberwise monoids include adjoint bundles of principal bundles, and the construction is natural with respect to maps of principal bundles. This naturality implies homotopy invariance of the algebra structure on H_*(LM) arising from the loop product. If M=BG is the infinite dimensional classifying space of a compact Lie group, then we get a well-defined pro-ring spectrum, which we define to be the string topology of BG. If E has a fiberwise action of the little n-cubes operad then E^{-TM} is an E_n-ring spectrum. This gives homology operations combining Dyer-Lashof operations on the fiber and Steenrod operations on the base. We give several examples where the new operations give homological insight, borrowed from knot theory, complex geometry, gauge theory, and homotopy theory." /> </head> <body class="with-cu-identity"> <aside class="slider-wrapper bps-banner forum blue"> <div class="columns"> <img role="presentation" class="bps-banner-image" src="/static/browse/0.3.4/images/icons/HF-Smiley-Bones-Banner.png" alt="arXiv smile and hugging face"> <div class="copy-donation bps-banner"> <h2>Support arXiv on Giving Day with Hugging Face!</h2> <p>Today Hugging Face is TRIPLING your donation to arXiv with a 2:1 match! Give today to keep science open for all.</p> </div> <div class="amount-donation bps-banner"> <div class="donate-cta"><a class="banner_link banner-btn-grad" target="_blank" href="https://givingday.cornell.edu/campaigns/arxiv"><b>Donate!</b></a></div> </div> </div> </aside> <div class="flex-wrap-footer"> <header> <a href="#content" class="is-sr-only">Skip to main content</a> <!-- start desktop header --> <div class="columns is-vcentered is-hidden-mobile" id="cu-identity"> <div class="column" id="cu-logo"> <a href="https://www.cornell.edu/"><img src="/static/browse/0.3.4/images/icons/cu/cornell-reduced-white-SMALL.svg" alt="Cornell University" /></a> </div><div class="column" id="support-ack"> <span id="support-ack-url">We gratefully acknowledge support from the Simons Foundation, <a href="https://info.arxiv.org/about/ourmembers.html">member institutions</a>, and all contributors.</span> <a href="https://info.arxiv.org/about/donate.html" class="btn-header-donate">Donate</a> </div> </div> <div id="header" class="is-hidden-mobile"> <a aria-hidden="true" tabindex="-1" href="/IgnoreMe"></a> <div class="header-breadcrumbs is-hidden-mobile"> <a href="/"><img src="/static/browse/0.3.4/images/arxiv-logo-one-color-white.svg" alt="arxiv logo" style="height:40px;"/></a> <span>></span> <a href="/list/math/recent">math</a> <span>></span> arXiv:math/0602210 </div> <div class="search-block level-right"> <form class="level-item mini-search" method="GET" action="https://arxiv.org/search"> <div class="field has-addons"> <div class="control"> <input class="input is-small" type="text" name="query" placeholder="Search..." aria-label="Search term or terms" /> <p class="help"><a href="https://info.arxiv.org/help">Help</a> | <a href="https://arxiv.org/search/advanced">Advanced Search</a></p> </div> <div class="control"> <div class="select is-small"> <select name="searchtype" aria-label="Field to search"> <option value="all" selected="selected">All fields</option> <option value="title">Title</option> <option value="author">Author</option> <option value="abstract">Abstract</option> <option value="comments">Comments</option> <option value="journal_ref">Journal reference</option> <option value="acm_class">ACM classification</option> <option value="msc_class">MSC classification</option> <option value="report_num">Report number</option> <option value="paper_id">arXiv identifier</option> <option value="doi">DOI</option> <option value="orcid">ORCID</option> <option value="author_id">arXiv author ID</option> <option value="help">Help pages</option> <option value="full_text">Full text</option> </select> </div> </div> <input type="hidden" name="source" value="header"> <button class="button is-small is-cul-darker">Search</button> </div> </form> </div> </div><!-- /end desktop header --> <div class="mobile-header"> <div class="columns is-mobile"> <div class="column logo-arxiv"><a href="https://arxiv.org/"><img src="/static/browse/0.3.4/images/arxiv-logomark-small-white.svg" alt="arXiv logo" style="height:60px;" /></a></div> <div class="column logo-cornell"><a href="https://www.cornell.edu/"> <picture> <source media="(min-width: 501px)" srcset="/static/browse/0.3.4/images/icons/cu/cornell-reduced-white-SMALL.svg 400w" sizes="400w" /> <source srcset="/static/browse/0.3.4/images/icons/cu/cornell_seal_simple_black.svg 2x" /> <img src="/static/browse/0.3.4/images/icons/cu/cornell-reduced-white-SMALL.svg" alt="Cornell University Logo" /> </picture> </a></div> <div class="column nav" id="toggle-container" role="menubar"> <button class="toggle-control"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-white"><title>open search</title><path d="M505 442.7L405.3 343c-4.5-4.5-10.6-7-17-7H372c27.6-35.3 44-79.7 44-128C416 93.1 322.9 0 208 0S0 93.1 0 208s93.1 208 208 208c48.3 0 92.7-16.4 128-44v16.3c0 6.4 2.5 12.5 7 17l99.7 99.7c9.4 9.4 24.6 9.4 33.9 0l28.3-28.3c9.4-9.4 9.4-24.6.1-34zM208 336c-70.7 0-128-57.2-128-128 0-70.7 57.2-128 128-128 70.7 0 128 57.2 128 128 0 70.7-57.2 128-128 128z"/></svg></button> <div class="mobile-toggle-block toggle-target"> <form class="mobile-search-form" method="GET" action="https://arxiv.org/search"> <div class="field has-addons"> <input class="input" type="text" name="query" placeholder="Search..." aria-label="Search term or terms" /> <input type="hidden" name="source" value="header"> <input type="hidden" name="searchtype" value="all"> <button class="button">GO</button> </div> </form> </div> <button class="toggle-control"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512" class="icon filter-white" role="menu"><title>open navigation menu</title><path d="M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z"/ ></svg></button> <div class="mobile-toggle-block toggle-target"> <nav class="mobile-menu" aria-labelledby="mobilemenulabel"> <h2 id="mobilemenulabel">quick links</h2> <ul> <li><a href="https://arxiv.org/login">Login</a></li> <li><a href="https://info.arxiv.org/help">Help Pages</a></li> <li><a href="https://info.arxiv.org/about">About</a></li> </ul> </nav> </div> </div> </div> </div><!-- /end mobile-header --> </header> <main> <div id="content"> <!-- rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:trackback="http://madskills.com/public/xml/rss/module/trackback/"> <rdf:Description rdf:about="/abs/math/0602210" dc:identifier="/abs/math/0602210" dc:title="Generalized string topology operations" trackback:ping="/trackback/math/0602210" /> </rdf:RDF> --><div id="abs-outer"> <div class="leftcolumn"> <div class="subheader"> <h1>Mathematics > Algebraic Topology</h1> </div> <div class="header-breadcrumbs-mobile"> <strong>arXiv:math/0602210</strong> (math) </div> <link rel="stylesheet" type="text/css" href="/static/base/1.0.1/css/abs.css"> <div id="content-inner"> <div id="abs"> <div class="dateline"> [Submitted on 10 Feb 2006]</div> <h1 class="title mathjax"><span class="descriptor">Title:</span>Generalized string topology operations</h1> <div class="authors"><span class="descriptor">Authors:</span><a href="https://arxiv.org/search/math?searchtype=author&query=Gruher,+K" rel="nofollow">Kate Gruher</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Salvatore,+P" rel="nofollow">Paolo Salvatore</a></div> <div id="download-button-info" hidden>View a PDF of the paper titled Generalized string topology operations, by Kate Gruher and Paolo Salvatore</div> <a class="mobile-submission-download" href="/pdf/math/0602210">View PDF</a> <blockquote class="abstract mathjax"> <span class="descriptor">Abstract:</span> We show that the Chas-Sullivan loop product, a combination of the Pontrjagin product on the fiber and intersection product on the base, makes sense on the total space homology of any fiberwise monoid E over a closed oriented manifold M. More generally the Thom spectrum E^{-TM} is a ring spectrum. Similarly a fiberwise module over E defines a module over E^{-TM}. Fiberwise monoids include adjoint bundles of principal bundles, and the construction is natural with respect to maps of principal bundles. This naturality implies homotopy invariance of the algebra structure on H_*(LM) arising from the loop product. If M=BG is the infinite dimensional classifying space of a compact Lie group, then we get a well-defined pro-ring spectrum, which we define to be the string topology of BG. If E has a fiberwise action of the little n-cubes operad then E^{-TM} is an E_n-ring spectrum. This gives homology operations combining Dyer-Lashof operations on the fiber and Steenrod operations on the base. We give several examples where the new operations give homological insight, borrowed from knot theory, complex geometry, gauge theory, and homotopy theory. </blockquote> <!--CONTEXT--> <div class="metatable"> <table summary="Additional metadata"> <tr> <td class="tablecell label">Comments:</td> <td class="tablecell comments mathjax">27 pages</td> </tr> <tr> <td class="tablecell label">Subjects:</td> <td class="tablecell subjects"> <span class="primary-subject">Algebraic Topology (math.AT)</span>; Geometric Topology (math.GT)</td> </tr> <tr> <td class="tablecell label"><abbr title="Mathematical Subject Classification">MSC</abbr> classes:</td> <td class="tablecell msc-classes">55R35; 57R19; 55R70; 55P43; 55S12</td> </tr> <tr> <td class="tablecell label">Cite as:</td> <td class="tablecell arxivid"><span class="arxivid"><a href="https://arxiv.org/abs/math/0602210">arXiv:math/0602210</a> [math.AT]</span></td> </tr> <tr> <td class="tablecell label"> </td> <td class="tablecell arxividv">(or <span class="arxivid"> <a href="https://arxiv.org/abs/math/0602210v1">arXiv:math/0602210v1</a> [math.AT]</span> for this version) </td> </tr> <tr> <td class="tablecell label"> </td> <td class="tablecell arxivdoi"> <a href="https://doi.org/10.48550/arXiv.math/0602210" id="arxiv-doi-link">https://doi.org/10.48550/arXiv.math/0602210</a><div class="button-and-tooltip"> <button class="more-info" aria-describedby="more-info-desc-1"> <svg height="15" role="presentation" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M256 8C119.043 8 8 119.083 8 256c0 136.997 111.043 248 248 248s248-111.003 248-248C504 119.083 392.957 8 256 8zm0 110c23.196 0 42 18.804 42 42s-18.804 42-42 42-42-18.804-42-42 18.804-42 42-42zm56 254c0 6.627-5.373 12-12 12h-88c-6.627 0-12-5.373-12-12v-24c0-6.627 5.373-12 12-12h12v-64h-12c-6.627 0-12-5.373-12-12v-24c0-6.627 5.373-12 12-12h64c6.627 0 12 5.373 12 12v100h12c6.627 0 12 5.373 12 12v24z" class=""></path></svg> <span class="visually-hidden">Focus to learn more</span> </button> <!-- tooltip description --> <div role="tooltip" id="more-info-desc-1"> <span class="left-corner"></span> arXiv-issued DOI via DataCite</div> </div> </td> </tr> <tr> <td class="tablecell label"> <abbr title="Digital Object Identifier">Related DOI</abbr>: </td> <td class="tablecell doi"><a href="https://doi.org/10.1112/plms/pdm030" data-doi="10.1112/plms/pdm030" class="link-https link-external" rel="external noopener nofollow">https://doi.org/10.1112/plms/pdm030</a> <!-- accessible tooltip example --> <div class="button-and-tooltip"> <button class="more-info" aria-describedby="more-info-desc-1"> <svg height="15" role="presentation" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M256 8C119.043 8 8 119.083 8 256c0 136.997 111.043 248 248 248s248-111.003 248-248C504 119.083 392.957 8 256 8zm0 110c23.196 0 42 18.804 42 42s-18.804 42-42 42-42-18.804-42-42 18.804-42 42-42zm56 254c0 6.627-5.373 12-12 12h-88c-6.627 0-12-5.373-12-12v-24c0-6.627 5.373-12 12-12h12v-64h-12c-6.627 0-12-5.373-12-12v-24c0-6.627 5.373-12 12-12h64c6.627 0 12 5.373 12 12v100h12c6.627 0 12 5.373 12 12v24z" class=""></path></svg> <span class="visually-hidden">Focus to learn more</span> </button> <!-- tooltip description --> <div role="tooltip" id="more-info-desc-1"> <span class="left-corner"></span> DOI(s) linking to related resources </div> </div> </td> </tr> </table> </div> </div> </div> <div class="submission-history"> <h2>Submission history</h2> From: Paolo Salvatore [<a href="/show-email/956ccb1a/math/0602210" rel="nofollow">view email</a>] <br/> <strong>[v1]</strong> Fri, 10 Feb 2006 10:15:55 UTC (23 KB)<br/> </div> </div> <!--end leftcolumn--> <div class="extra-services"> <div class="full-text"> <a name="other"></a> <span class="descriptor">Full-text links:</span> <h2>Access Paper:</h2> <ul> <div id="download-button-info" hidden> View a PDF of the paper titled Generalized string topology operations, by Kate Gruher and Paolo Salvatore</div><li><a href="/pdf/math/0602210" aria-describedby="download-button-info" accesskey="f" class="abs-button download-pdf">View PDF</a></li><li><a href="/src/math/0602210" class="abs-button download-eprint">TeX Source</a></li><li><a href="/format/math/0602210" class="abs-button download-format">Other Formats</a></li></ul> <div class="abs-license"><a href="http://arxiv.org/licenses/assumed-1991-2003/" title="Rights to this article">view license</a></div> </div> <!--end full-text--> <div class="browse"> Current browse context: <div class="current">math.AT</div> <div class="prevnext"> <span class="arrow"> <a class="abs-button prev-url" href="/prevnext?id=math/0602210&function=prev&context=math.AT" accesskey="p" title="previous in math.AT (accesskey p)" rel="nofollow">< prev</a> </span> <span class="is-hidden-mobile"> | </span> <span class="arrow"> <a class="abs-button next-url" href="/prevnext?id=math/0602210&function=next&context=math.AT" accesskey="n" title="next in math.AT (accesskey n)" rel="nofollow">next ></a> </span><br/> </div><div class="list"> <a class="abs-button abs-button-grey abs-button-small context-new" href="/list/math.AT/new" rel="nofollow">new</a> <span class="is-hidden-mobile"> | </span> <a class="abs-button abs-button-grey abs-button-small context-recent" href="/list/math.AT/recent" rel="nofollow">recent</a> <span class="is-hidden-mobile"> | </span><a class="abs-button abs-button-grey abs-button-small context-id" href="/list/math.AT/2006-02" rel="nofollow">2006-02</a> </div> </div> <div class="extra-ref-cite"> <h3>References & Citations</h3> <ul> <li><a class="abs-button abs-button-small cite-ads" href="https://ui.adsabs.harvard.edu/abs/arXiv:math%2F0602210">NASA ADS</a></li><li><a class="abs-button abs-button-small cite-google-scholar" href="https://scholar.google.com/scholar_lookup?arxiv_id=math/0602210" target="_blank" rel="noopener">Google Scholar</a></li> <li><a class="abs-button abs-button-small cite-semantic-scholar" href="https://api.semanticscholar.org/arXiv:math/0602210" target="_blank" rel="noopener">Semantic Scholar</a></li> </ul> <div style="clear:both;"></div> </div> <div class='extra-ref-cite'> <a id='bib-cite-css' hidden='true' href='/static/browse/0.3.4/css/cite.css'>a</a> <span id='bib-cite-trigger' class="bib-cite-button abs-button">export BibTeX citation</span> <span id='bib-cite-loading' hidden='true'>Loading...</span> </div> <div id='bib-cite-modal' class='bib-modal' hidden='true'> <div class='bib-modal-content'> <div class='bib-modal-title'> <h2>BibTeX formatted citation</h2> <span class='bib-modal-close' >×</span> </div> <div> <textarea id='bib-cite-target' class="bib-citation-content" aria-label="loading the citation">loading...</textarea> </div> <div> <span>Data provided by: </span> <a id='bib-cite-source-api'></a> </div> </div> </div><div class="bookmarks"> <div><h3>Bookmark</h3></div><a class="abs-button abs-button-grey abs-button-small" href="http://www.bibsonomy.org/BibtexHandler?requTask=upload&url=https://arxiv.org/abs/math/0602210&description=Generalized string topology operations" title="Bookmark on BibSonomy"> <img src="/static/browse/0.3.4/images/icons/social/bibsonomy.png" alt="BibSonomy logo"/> </a> <a class="abs-button abs-button-grey abs-button-small" href="https://reddit.com/submit?url=https://arxiv.org/abs/math/0602210&title=Generalized string topology operations" title="Bookmark on Reddit"> <img src="/static/browse/0.3.4/images/icons/social/reddit.png" alt="Reddit logo"/> </a> </div> </div> <!--end extra-services--> <!-- LABS AREA --> <div id="labstabs"> <div class="labstabs"><input type="radio" name="tabs" id="tabone"checked="checked"> <label for="tabone">Bibliographic Tools</label> <div class="tab labs-display-bib"> <h1>Bibliographic and Citation Tools</h1> <div class="toggle"> <div class="columns is-mobile lab-row"> <div class="column lab-switch"> <label class="switch"> <input id="bibex-toggle" type="checkbox" class="lab-toggle" data-script-url="/static/browse/0.3.4/bibex/bibex.js?20241202"> <span class="slider"></span> <span class="is-sr-only">Bibliographic Explorer Toggle</span> </label> </div> <div class="column lab-name"> <span id="label-for-bibex">Bibliographic Explorer</span> <em>(<a href="https://info.arxiv.org/labs/showcase.html#arxiv-bibliographic-explorer">What is the Explorer?</a>)</em> </div> </div> <div class="columns is-mobile lab-row"> <div class="column lab-switch"> <label class="switch"> <input id="connectedpapers-toggle" type="checkbox" class="lab-toggle" data-script-url="/static/browse/0.3.4/js/connectedpapers.js" aria-labelledby="label-for-connected-papers"> <span class="slider"></span> <span class="is-sr-only">Connected Papers Toggle</span> </label> </div> <div class="column lab-name"> <span id="label-for-connected-papers">Connected Papers</span> <em>(<a href="https://www.connectedpapers.com/about" target="_blank">What is Connected Papers?</a>)</em> </div> </div><div class="columns is-mobile lab-row"> <div class="column lab-switch"> <label class="switch"> <input id="litmaps-toggle" type="checkbox" class="lab-toggle" data-script-url="/static/browse/0.3.4/js/litmaps.js?20210617" aria-labelledby="label-for-litmaps"> <span class="slider"></span> <span class="is-sr-only">Litmaps Toggle</span> </label> </div> <div class="column lab-name"> <span id="label-for-litmaps">Litmaps</span> <em>(<a href="https://www.litmaps.co/" target="_blank">What is Litmaps?</a>)</em> </div> </div> <div class="columns is-mobile lab-row"> <div class="column lab-switch"> <label class="switch"> <input id="scite-toggle" type="checkbox" class="lab-toggle" data-script-url="/static/browse/0.3.4/js/scite.js?20210617" aria-labelledby="label-for-scite"> <span class="slider"></span> <span class="is-sr-only">scite.ai Toggle</span> </label> </div> <div class="column lab-name"> <span id="label-for-scite">scite Smart Citations</span> <em>(<a href="https://www.scite.ai/" target="_blank">What are Smart Citations?</a>)</em> </div> </div> </div> <div class="labs-content-placeholder labs-display" style="display: none;"></div> <div style="min-height: 15px" id="connectedpapers-output"></div> <div style="min-height: 15px" id="litmaps-open-in"></div> <div style="min-height: 15px" id="scite-open-in"></div> </div> <input type="radio" name="tabs" id="tabtwo"> <label for="tabtwo">Code, Data, Media</label> <div class="tab"> <h1>Code, Data and Media Associated with this Article</h1> <div class="toggle"> <div class="columns is-mobile lab-row"> <div class="column lab-switch"> <label class="switch"> <input id="alphaxiv-toggle" data-script-url="/static/browse/0.3.4/js/alphaxiv.js" type="checkbox" class="lab-toggle" aria-labelledby="label-for-alphaxiv"> <span class="slider"></span> <span class="is-sr-only">alphaXiv Toggle</span> </label> </div> <div class="column lab-name"> <span id="label-for-alphaxiv">alphaXiv</span> <em>(<a href="https://alphaxiv.org/" target="_blank">What is alphaXiv?</a>)</em> </div> </div> <div class="columns is-mobile lab-row"> <div class="column lab-switch"> <label class="switch"> <input id="catalyzex-toggle" data-script-url="/static/browse/0.3.4/js/catalyzex.js" type="checkbox" class="lab-toggle" aria-labelledby="label-for-cx"> <span class="slider"></span> <span class="is-sr-only">Links to Code Toggle</span> </label> </div> <div class="column lab-name"> <span id="label-for-cx">CatalyzeX Code Finder for Papers</span> <em>(<a href="https://www.catalyzex.com" target="_blank">What is CatalyzeX?</a>)</em> </div> </div> <div class="columns is-mobile lab-row"> <div class="column lab-switch"> <label class="switch"> <input id="dagshub-toggle" data-script-url="/static/browse/0.3.4/js/dagshub.js" type="checkbox" class="lab-toggle" aria-labelledby="label-for-dagshub"> <span class="slider"></span> <span class="is-sr-only">DagsHub Toggle</span> </label> </div> <div class="column lab-name"> <span id="label-for-dagshub">DagsHub</span> <em>(<a href="https://dagshub.com/" target="_blank">What is DagsHub?</a>)</em> </div> </div> <div class="columns is-mobile lab-row"> <div class="column lab-switch"> <label class="switch"> <input id="gotitpub-toggle" data-script-url="/static/browse/0.3.4/js/gotitpub.js" type="checkbox" class="lab-toggle" aria-labelledby="label-for-gotitpub"> <span class="slider"></span> <span class="is-sr-only">GotitPub Toggle</span> </label> </div> <div class="column lab-name"> <span id="label-for-gotitpub">Gotit.pub</span> <em>(<a href="http://gotit.pub/faq" target="_blank">What is GotitPub?</a>)</em> </div> </div> <div class="columns is-mobile lab-row"> <div class="column lab-switch"> <label class="switch"> <input id="huggingface-toggle" data-script-url="/static/browse/0.3.4/js/huggingface.js" type="checkbox" class="lab-toggle" aria-labelledby="label-for-huggingface"> <span class="slider"></span> <span class="is-sr-only">Huggingface Toggle</span> </label> </div> <div class="column lab-name"> <span id="label-for-huggingface">Hugging Face</span> <em>(<a href="https://huggingface.co/huggingface" target="_blank">What is Huggingface?</a>)</em> </div> </div> <div class="columns is-mobile lab-row"> <div class="column lab-switch"> <label class="switch"> <input id="paperwithcode-toggle" data-script-url="/static/browse/0.3.4/js/paperswithcode.js" type="checkbox" class="lab-toggle" aria-labelledby="label-for-pwc"> <span class="slider"></span> <span class="is-sr-only">Links to Code Toggle</span> </label> </div> <div class="column lab-name"> <span id="label-for-pwc">Papers with Code</span> <em>(<a href="https://paperswithcode.com/" target="_blank">What is Papers with Code?</a>)</em> </div> </div> <div class="columns is-mobile lab-row"> <div class="column lab-switch"> <label class="switch"> <input id="sciencecast-toggle" data-script-url="/static/browse/0.3.4/js/sciencecast.js" type="checkbox" class="lab-toggle" aria-labelledby="label-for-sciencecast"> <span class="slider"></span> <span class="is-sr-only">ScienceCast Toggle</span> </label> </div> <div class="column lab-name"> <span id="label-for-sciencecast">ScienceCast</span> <em>(<a href="https://sciencecast.org/welcome" target="_blank">What is ScienceCast?</a>)</em> </div> </div> </div> <div id="alphaxiv-output" style="display:none"></div> <div id="catalyzex-output" style="display:none"></div> <div id="dagshub-output" style="display:none"></div> <div id="gotitpub-output" style="display:none"></div> <div id="pwc-output" style="display:none"></div> <div id="pwc-data-output" style="display:none"></div> <div id="sciencecast-output" style="display:none"></div> <div id="huggingface-output" style="display:none"></div> </div> <input type="radio" name="tabs" id="labstabs-demos-input"> <label for="labstabs-demos-input" id="labstabs-demos-label">Demos</label> <div class="tab"> <h1>Demos</h1> <div class="toggle"> <div class="columns is-mobile lab-row"> <div class="column lab-switch"> <label class="switch"> <input id="replicate-toggle" data-script-url="/static/browse/0.3.4/js/replicate.js" type="checkbox" class="lab-toggle" aria-labelledby="label-for-replicate"> <span class="slider"></span> <span class="is-sr-only">Replicate Toggle</span> </label> </div> <div class="column lab-name"> <span id="label-for-replicate">Replicate</span> <em>(<a href="https://replicate.com/docs/arxiv/about" target="_blank">What is Replicate?</a>)</em> </div> </div> <div class="columns is-mobile lab-row"> <div class="column lab-switch"> <label class="switch"> <input id="spaces-toggle" data-script-url="/static/browse/0.3.4/js/spaces.js" type="checkbox" class="lab-toggle" aria-labelledby="label-for-spaces"> <span class="slider"></span> <span class="is-sr-only">Spaces Toggle</span> </label> </div> <div class="column lab-name"> <span id="label-for-spaces">Hugging Face Spaces</span> <em>(<a href="https://huggingface.co/docs/hub/spaces" target="_blank">What is Spaces?</a>)</em> </div> </div> <div class="columns is-mobile lab-row"> <div class="column lab-switch"> <label class="switch"> <input id="txyz-toggle" data-script-url="/static/browse/0.3.4/js/txyz.js" type="checkbox" class="lab-toggle" aria-labelledby="label-for-txyz"> <span class="slider"></span> <span class="is-sr-only">Spaces Toggle</span> </label> </div> <div class="column lab-name"> <span id="label-for-txyz">TXYZ.AI</span> <em>(<a href="https://txyz.ai" target="_blank">What is TXYZ.AI?</a>)</em> </div> </div> </div> <div id="replicate-output"></div> <div id="spaces-output"></div> <div id="txyz-output"></div> </div> <input type="radio" name="tabs" id="tabfour"> <label for="tabfour">Related Papers</label> <div class="tab"> <h1>Recommenders and Search Tools</h1> <div class="toggle"> <div class="columns is-mobile lab-row"> <div class="column lab-switch"> <label class="switch"> <input id="influenceflower-toggle" data-script-url="/static/browse/0.3.4/js/influenceflower.js" type="checkbox" class="lab-toggle" aria-labelledby="label-for-influenceflower"> <span class="slider"></span> <span class="is-sr-only">Link to Influence Flower</span> </label> </div> <div class="column lab-name"> <span id="label-for-influenceflower">Influence Flower</span> <em>(<a href="https://influencemap.cmlab.dev/" target="_blank">What are Influence Flowers?</a>)</em> </div> </div> <div class="columns is-mobile lab-row"> <div class="column lab-switch"> <label class="switch"> <input id="core-recommender-toggle" type="checkbox" class="lab-toggle" aria-labelledby="label-for-core"> <span class="slider"></span> <span class="is-sr-only">Core recommender toggle</span> </label> </div> <div class="column lab-name"> <span id="label-for-core">CORE Recommender</span> <em>(<a href="https://core.ac.uk/services/recommender">What is CORE?</a>)</em> </div> </div></div> <div id="influenceflower-output"></div> <div id="influenceflower-output-graph" style="display:none"> <ul class="flower-tabs"> <li class="active"><a class="btn tab-btn" onclick="openTab(event, 'tab-author')">Author</a></li> <li><a class="btn tab-btn" onclick="openTab(event, 'tab-venue')">Venue</a></li> <li><a class="btn tab-btn" onclick="openTab(event, 'tab-inst')">Institution</a></li> <li><a class="btn tab-btn" onclick="openTab(event, 'tab-topic')">Topic</a></li> </ul> <div class="flower-tab-content"> <div class="tab-flower active" id="tab-author"><svg id="flower-graph-author"></svg></div> <div class="tab-flower" id="tab-venue"><svg id="flower-graph-venue"></svg></div> <div class="tab-flower" id="tab-inst"><svg id="flower-graph-inst"></svg></div> <div class="tab-flower" id="tab-topic"><svg id="flower-graph-topic"></svg></div> </div> </div> <div id="coreRecommenderOutput"></div> <div id="iarxivOutput"></div> </div> <input type="radio" name="tabs" id="tabfive"> <label for="tabfive"> About arXivLabs </label> <div class="tab"> <div class="columns"> <div class="column"> <h1>arXivLabs: experimental projects with community collaborators</h1> <p>arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.</p> <p>Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.</p> <p>Have an idea for a project that will add value for arXiv's community? <a href="https://info.arxiv.org/labs/index.html"><strong>Learn more about arXivLabs</strong></a>.</p> </div> <div class="column is-narrow is-full-mobile"> <p class="icon-labs"><svg xmlns="http://www.w3.org/2000/svg" role="presentation" viewBox="0 0 635.572 811"><path d="M175.6 676v27h-27v-27zm-54 27v27h27v-27zm-27 27v27h27v-27zm396-54v27h-27v-27zm0 27v27h27v-27zm27 27v27h27v-27zm-27-414h27v27h-27zm27 0h27v-27h-27zm27-27h27v-27h-27zm-396 45h-27v-27h27zm-27-54h-27v27h27zm-27-27h-27v27h27z"/><path d="M94.6 730v27h-27v-27zm477 0v27h-27v-27zm-27-495h27v27h-27zm-450 18h-27v-27h27zm477 9h27v27h-27zm-54 495h27v27h-27zm-423 0h27v27h-27zm-54-504h27v27h-27z" fill="#666"/><path d="M67.6 730v27h-27v-27zm54 54v27h-27v-27zm0-108v27h27v-27zm-27 27v27h27v-27zm-81 0v27h27v-27zm585 27v27h-27v-27zm-108-54v27h27v-27zm27 27v27h27v-27zm81 0v27h27v-27zm-54-495h27v27h-27zm-54 108h27v-27h-27zm27-27h27v-27h-27zm0-81h27v-27h-27zm-423 18h-27v-27h27zm54 54h-27v27h27zm-27-27h-27v27h27zm0-81h-27v27h27zm423 612v27h-27v-27zm81-522v27h-27v-27zm-585-9v27h-27v-27z" fill="#999"/><path d="M94.6 784v27h-27v-27zm-27-27v27h27v-27zm-27-54v27h27v-27zm27 0v27h27v-27zm0-27v27h27v-27zm27 0v27h27v-27zm0-27v27h27v-27zm27 0v27h27v-27zm-108 81v27h27v-27zm558 54v27h-27v-27zm-27-27v27h27v-27zm27-54v27h27v-27zm-27 0v27h27v-27zm0-27v27h27v-27zm-27 0v27h27v-27zm0-27v27h27v-27zm-27 0v27h27v-27zm108 81v27h27v-27zm0-495h27v27h-27zm-27 27h27v-27h-27zm-54-27h27v-27h-27zm0 27h27v-27h-27zm-27 0h27v-27h-27zm0 27h27v-27h-27zm-27 0h27v-27h-27zm0 27h27v-27h-27zm81-108h27v-27h-27zm-504 45h-27v-27h27zm27-27h-27v27h27zm54-27h-27v27h27zm0 27h-27v27h27zm27 0h-27v27h27zm0 27h-27v27h27zm27 0h-27v27h27zm0 27h-27v27h27zm-81-108h-27v27h27z" fill="#ccc"/><path d="M598.6 665.1H41.5C-76.5 667 176 280.2 176 280.2h53a46.5 46.5 0 0162.8-56.3 29.2 29.2 0 1128.5 35.9h-1a46.5 46.5 0 01-1.5 20.3l142.5-.1s255.3 387 138.3 385.1zM291 181a29.3 29.3 0 10-29.2-29.3A29.3 29.3 0 00291 181zm65.4-66.8a22.4 22.4 0 10-22.5-22.4 22.4 22.4 0 0022.5 22.4z" fill="#fc0"/><path d="M245.5 172V10h153v162s324 495 198 495h-558c-126 0 207-495 207-495zm126 54h56m-13 72h56m-9 72h56m-20 72h56m-22 72h56m-29 72h56m-457-45c20.8 41.7 87.3 81 160.7 81 72.1 0 142.1-38.2 163.4-81" fill="none" stroke="#000" stroke-miterlimit="10" stroke-width="20"/><path d="M273.3 421.7c0 31-9.8 56.3-21.9 56.3s-21.8-25.2-21.8-56.3 9.8-56.3 21.8-56.3 21.9 25.2 21.9 56.3zm114.4-56.3c-12 0-21.8 25.2-21.8 56.3s9.7 56.3 21.8 56.3 21.9-25.2 21.9-56.3-9.8-56.3-21.9-56.3zM150.1 526.6c-18.2 6.7-27.5 22.9-23.2 30.2s14.8-5.5 33-12.2 37.4-4.9 33-12.2-24.5-12.6-42.8-5.8zm296 5.8c-4.2 7.3 14.9 5.5 33.1 12.2s28.7 19.5 33 12.2-5-23.5-23.2-30.2-38.5-1.5-42.8 5.8z"/></svg></p> </div> </div> </div> </div> </div> <!-- END LABS AREA --> <div class="endorsers"> <a href="/auth/show-endorsers/math/0602210" class="endorser-who" rel="nofollow">Which authors of this paper are endorsers?</a> | <a id="mathjax_toggle" href="javascript:setMathjaxCookie()">Disable MathJax</a> (<a href="https://info.arxiv.org/help/mathjax.html">What is MathJax?</a>) <span class="help" style="font-style: normal; float: right; margin-top: 0; margin-right: 1em;"></span> </div> <script type="text/javascript" language="javascript">mathjaxToggle();</script> </div> </div> </main> <footer style="clear: both;"> <div class="columns is-desktop" role="navigation" aria-label="Secondary" style="margin: -0.75em -0.75em 0.75em -0.75em"> <!-- Macro-Column 1 --> <div class="column" style="padding: 0;"> <div class="columns"> <div class="column"> <ul style="list-style: none; line-height: 2;"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul style="list-style: none; line-height: 2;"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click here to contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg> <a href="https://info.arxiv.org/help/contact.html"> Contact</a> </li> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>subscribe to arXiv mailings</title><desc>Click here to subscribe</desc><path d="M476 3.2L12.5 270.6c-18.1 10.4-15.8 35.6 2.2 43.2L121 358.4l287.3-253.2c5.5-4.9 13.3 2.6 8.6 8.3L176 407v80.5c0 23.6 28.5 32.9 42.5 15.8L282 426l124.6 52.2c14.2 6 30.4-2.9 33-18.2l72-432C515 7.8 493.3-6.8 476 3.2z"/></svg> <a href="https://info.arxiv.org/help/subscribe"> Subscribe</a> </li> </ul> </div> </div> </div> <!-- End Macro-Column 1 --> <!-- Macro-Column 2 --> <div class="column" style="padding: 0;"> <div class="columns"> <div class="column"> <ul style="list-style: none; line-height: 2;"> <li><a href="https://info.arxiv.org/help/license/index.html">Copyright</a></li> <li><a href="https://info.arxiv.org/help/policies/privacy_policy.html">Privacy Policy</a></li> </ul> </div> <div class="column sorry-app-links"> <ul style="list-style: none; line-height: 2;"> <li><a href="https://info.arxiv.org/help/web_accessibility.html">Web Accessibility Assistance</a></li> <li> <p class="help"> <a class="a11y-main-link" href="https://status.arxiv.org" target="_blank">arXiv Operational Status <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 256 512" class="icon filter-dark_grey" role="presentation"><path d="M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z"/></svg></a><br> Get status notifications via <a class="is-link" href="https://subscribe.sorryapp.com/24846f03/email/new" target="_blank"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg>email</a> or <a class="is-link" href="https://subscribe.sorryapp.com/24846f03/slack/new" target="_blank"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512" class="icon filter-black" role="presentation"><path d="M94.12 315.1c0 25.9-21.16 47.06-47.06 47.06S0 341 0 315.1c0-25.9 21.16-47.06 47.06-47.06h47.06v47.06zm23.72 0c0-25.9 21.16-47.06 47.06-47.06s47.06 21.16 47.06 47.06v117.84c0 25.9-21.16 47.06-47.06 47.06s-47.06-21.16-47.06-47.06V315.1zm47.06-188.98c-25.9 0-47.06-21.16-47.06-47.06S139 32 164.9 32s47.06 21.16 47.06 47.06v47.06H164.9zm0 23.72c25.9 0 47.06 21.16 47.06 47.06s-21.16 47.06-47.06 47.06H47.06C21.16 243.96 0 222.8 0 196.9s21.16-47.06 47.06-47.06H164.9zm188.98 47.06c0-25.9 21.16-47.06 47.06-47.06 25.9 0 47.06 21.16 47.06 47.06s-21.16 47.06-47.06 47.06h-47.06V196.9zm-23.72 0c0 25.9-21.16 47.06-47.06 47.06-25.9 0-47.06-21.16-47.06-47.06V79.06c0-25.9 21.16-47.06 47.06-47.06 25.9 0 47.06 21.16 47.06 47.06V196.9zM283.1 385.88c25.9 0 47.06 21.16 47.06 47.06 0 25.9-21.16 47.06-47.06 47.06-25.9 0-47.06-21.16-47.06-47.06v-47.06h47.06zm0-23.72c-25.9 0-47.06-21.16-47.06-47.06 0-25.9 21.16-47.06 47.06-47.06h117.84c25.9 0 47.06 21.16 47.06 47.06 0 25.9-21.16 47.06-47.06 47.06H283.1z"/></svg>slack</a> </p> </li> </ul> </div> </div> </div> <!-- end MetaColumn 2 --> <!-- End Macro-Column 2 --> </div> </footer> </div> <script src="/static/base/1.0.1/js/member_acknowledgement.js"></script> </body> </html>