[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! 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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>  <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 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[math/0602210] Generalized string topology operations