Algebraic Topology

<!DOCTYPE html> <html lang="en"> <head> <title>Algebraic Topology </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> <script src="/static/browse/0.3.4/js/mathjaxToggle.min.js" type="text/javascript"></script> <script type="text/javascript" language="javascript">mathjaxToggle();</script> </head> <body class="with-cu-identity"> <div class="flex-wrap-footer"> <header> <a href="#content" class="is-sr-only">Skip to main content</a>  <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"> <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.AT/recent">math.AT</a> </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> <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> </header> <main> <div id="content"> <div id='content-inner'> <div id='dlpage'> <h1>Algebraic Topology</h1> <ul> <li><a href="#item0">New submissions</a></li> <li><a href="#item3">Cross-lists</a></li> <li><a href="#item6">Replacements</a></li> </ul> <p>See <a id="recent-math.AT" aria-labelledby="recent-math.AT" href="/list/math.AT/recent">recent</a> articles</p> <h3>Showing new listings for Friday, 4 April 2025</h3> <div class='paging'>Total of 10 entries </div> <div class='morefewer'>Showing up to 2000 entries per page: <a href=/list/math.AT/new?skip=0&show=1000 rel="nofollow"> fewer</a> | <span style="color: #454545">more</span> | <span style="color: #454545">all</span> </div> <dl id='articles'> <h3>New submissions (showing 2 of 2 entries)</h3> <dt> <a name='item1'>[1]</a> <a href ="/abs/2504.02143" title="Abstract" id="2504.02143"> arXiv:2504.02143 </a> [<a href="/pdf/2504.02143" title="Download PDF" id="pdf-2504.02143" aria-labelledby="pdf-2504.02143">pdf</a>, <a href="/format/2504.02143" title="Other formats" id="oth-2504.02143" aria-labelledby="oth-2504.02143">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> On tensor products with equivariant commutative operads </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Stewart,+N">Natalie Stewart</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 59 pages, comments welcome </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Topology (math.AT)</span>; Category Theory (math.CT) </div> <p class='mathjax'> We affirm and generalize a conjecture of Blumberg and Hill: unital weak $\mathcal{N}_\infty$-operads are closed under $\infty$-categorical Boardman-Vogt tensor products and the resulting tensor products correspond with joins of weak indexing systems; in particular, we acquire a natural $G$-symmetric monoidal equivalence <br>\[ <br>\underline{\mathrm{CAlg}}^{\otimes}_{I} \underline{\mathrm{CAlg}}^{\otimes}_{J} \mathcal{C} \simeq \underline{\mathrm{CAlg}}^{\otimes}_{I \vee J} \mathcal{C}. <br>\] <br>We accomplish this by showing that $\mathcal{N}_{I\infty}^{\otimes}$ is $\otimes$-idempotent and $\mathcal{O}^{\otimes}$ is local for the corresponding smashing localization if and only if $\mathcal{O}$-monoid $G$-spaces satisfy $I$-indexed Wirthm眉ller isomorphisms. <br>Ultimately, we accomplish this by advancing the equivariant higher algebra of cartesian and cocartesian $I$-symmetric monoidal $\infty$-categories. Additionally, we acquire a number of structural results concerning $G$-operads, including a canonical lift of $\otimes$ to a presentably symmetric monoidal structure and a general disintegration and assembly procedure for computing tensor products of non-reduced unital $G$-operads. All such results are proved in the generality of atomic orbital $\infty$-categories. <br>We also achieve the expected corollaries for (iterated) Real topological Hochschild and cyclic homology and construct a natural $I$-symmetric monoidal structure on right modules over an $\mathcal{N}_{I\infty}$-algebra. </p> </div> </dd> <dt> <a name='item2'>[2]</a> <a href ="/abs/2504.02366" title="Abstract" id="2504.02366"> arXiv:2504.02366 </a> [<a href="/pdf/2504.02366" title="Download PDF" id="pdf-2504.02366" aria-labelledby="pdf-2504.02366">pdf</a>, <a href="/format/2504.02366" title="Other formats" id="oth-2504.02366" aria-labelledby="oth-2504.02366">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Non-Koszulness in a family of properads </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=N%C3%A9d%C3%A9lec,+S">Silv猫re N茅d茅lec</a> (Nantes Univ, LMJL)</div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Topology (math.AT)</span> </div> <p class='mathjax'> Proving Koszulness of a properad can be very hard, but sometimes one can look at its Koszul complex to look for obstructions for Koszulness. In this paper, we present a method and tools to prove non-Koszulness of many properads in a family of quadratic properads. We illustrate this method on a family of associative and coassociative properads with one quadratic compatibility relation. </p> </div> </dd> </dl> <dl id='articles'> <h3>Cross submissions (showing 3 of 3 entries)</h3> <dt> <a name='item3'>[3]</a> <a href ="/abs/2504.02049" title="Abstract" id="2504.02049"> arXiv:2504.02049 </a> (cross-list from math.OC) [<a href="/pdf/2504.02049" title="Download PDF" id="pdf-2504.02049" aria-labelledby="pdf-2504.02049">pdf</a>, <a href="https://arxiv.org/html/2504.02049v1" title="View HTML" id="html-2504.02049" aria-labelledby="html-2504.02049" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2504.02049" title="Other formats" id="oth-2504.02049" aria-labelledby="oth-2504.02049">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Distributed Multi-agent Coordination over Cellular Sheaves </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Hanks,+T">Tyler Hanks</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Riess,+H">Hans Riess</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Cohen,+S">Samuel Cohen</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Gross,+T">Trevor Gross</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Hale,+M">Matthew Hale</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Fairbanks,+J">James Fairbanks</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Optimization and Control (math.OC)</span>; Multiagent Systems (cs.MA); Algebraic Topology (math.AT) </div> <p class='mathjax'> Techniques for coordination of multi-agent systems are vast and varied, often utilizing purpose-built solvers or controllers with tight coupling to the types of systems involved or the coordination goal. In this paper, we introduce a general unified framework for heterogeneous multi-agent coordination using the language of cellular sheaves and nonlinear sheaf Laplacians, which are generalizations of graphs and graph Laplacians. Specifically, we introduce the concept of a nonlinear homological program encompassing a choice of cellular sheaf on an undirected graph, nonlinear edge potential functions, and constrained convex node objectives. We use the alternating direction method of multipliers to derive a distributed optimization algorithm for solving these nonlinear homological programs. To demonstrate the wide applicability of this framework, we show how hybrid coordination goals including combinations of consensus, formation, and flocking can be formulated as nonlinear homological programs and provide numerical simulations showing the efficacy of our distributed solution algorithm. </p> </div> </dd> <dt> <a name='item4'>[4]</a> <a href ="/abs/2504.02720" title="Abstract" id="2504.02720"> arXiv:2504.02720 </a> (cross-list from math.AG) [<a href="/pdf/2504.02720" title="Download PDF" id="pdf-2504.02720" aria-labelledby="pdf-2504.02720">pdf</a>, <a href="https://arxiv.org/html/2504.02720v1" title="View HTML" id="html-2504.02720" aria-labelledby="html-2504.02720" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2504.02720" title="Other formats" id="oth-2504.02720" aria-labelledby="oth-2504.02720">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> On the topology of real algebraic stacks </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Ambrosi,+E">Emiliano Ambrosi</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=de+Gaay+Fortman,+O">Olivier de Gaay Fortman</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 41 pages, 6 figures, comments welcome </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span>; Algebraic Topology (math.AT); General Topology (math.GN) </div> <p class='mathjax'> Motivated by questions arising in the theory of moduli spaces in real algebraic geometry, we develop a range of methods to study the topology of the real locus of a Deligne-Mumford stack over the real numbers. As an application, we verify in several cases the Smith-Thom type inequality for stacks that we conjectured in an earlier work. This requires combining techniques from group theory, algebraic geometry, and topology. </p> </div> </dd> <dt> <a name='item5'>[5]</a> <a href ="/abs/2504.02760" title="Abstract" id="2504.02760"> arXiv:2504.02760 </a> (cross-list from math.AG) [<a href="/pdf/2504.02760" title="Download PDF" id="pdf-2504.02760" aria-labelledby="pdf-2504.02760">pdf</a>, <a href="https://arxiv.org/html/2504.02760v1" title="View HTML" id="html-2504.02760" aria-labelledby="html-2504.02760" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2504.02760" title="Other formats" id="oth-2504.02760" aria-labelledby="oth-2504.02760">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Topological groupoids with involution and real algebraic stacks </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Ambrosi,+E">Emiliano Ambrosi</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=de+Gaay+Fortman,+O">Olivier de Gaay Fortman</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 29 pages, comments welcome </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span>; Algebraic Topology (math.AT); General Topology (math.GN) </div> <p class='mathjax'> To a topological groupoid endowed with an involution, we associate a topological groupoid of fixed points, generalizing the fixed-point subspace of a topological space with involution. We prove that when the topological groupoid with involution arises from a Deligne-Mumford stack over $\mathbb{R}$, this fixed locus coincides with the real locus of the stack. This provides a topological framework to study real algebraic stacks, and in particular real moduli spaces. Finally, we propose a Smith-Thom type conjecture in this setting, generalizing the Smith-Thom inequality for topological spaces endowed with an involution. </p> </div> </dd> </dl> <dl id='articles'> <h3>Replacement submissions (showing 5 of 5 entries)</h3> <dt> <a name='item6'>[6]</a> <a href ="/abs/2301.02636" title="Abstract" id="2301.02636"> arXiv:2301.02636 </a> (replaced) [<a href="/pdf/2301.02636" title="Download PDF" id="pdf-2301.02636" aria-labelledby="pdf-2301.02636">pdf</a>, <a href="https://arxiv.org/html/2301.02636v3" title="View HTML" id="html-2301.02636" aria-labelledby="html-2301.02636" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2301.02636" title="Other formats" id="oth-2301.02636" aria-labelledby="oth-2301.02636">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Central H-spaces and banded types </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Buchholtz,+U">Ulrik Buchholtz</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Christensen,+J+D">J. Daniel Christensen</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Flaten,+J+G+T">Jarl G. Taxer氓s Flaten</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Rijke,+E">Egbert Rijke</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> v1: 22 pages; v2: 25 pages, with many improvements and additions; v3: 27 pages, accepted version to appear in JPAA </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Topology (math.AT)</span>; Logic in Computer Science (cs.LO); Logic (math.LO) </div> <p class='mathjax'> We introduce and study central types, which are generalizations of Eilenberg-Mac Lane spaces. A type is central when it is equivalent to the component of the identity among its own self-equivalences. From centrality alone we construct an infinite delooping in terms of a tensor product of banded types, which are the appropriate notion of torsor for a central type. Our constructions are carried out in homotopy type theory, and therefore hold in any $\infty$-topos. Even when interpreted into the $\infty$-topos of spaces, our approach to constructing these deloopings is new. <br>Along the way, we further develop the theory of H-spaces in homotopy type theory, including their relation to evaluation fibrations and Whitehead products. These considerations let us, for example, rule out the existence of H-space structures on the $2n$-sphere for $n > 0$. We also give a novel description of the moduli space of H-space structures on an H-space. Using this description, we generalize a formula of Arkowitz-Curjel and Copeland for counting the number of path components of this moduli space. As an application, we deduce that the moduli space of H-space structures on the $3$-sphere is $\Omega^6 \mathbb{S}^3$. </p> </div> </dd> <dt> <a name='item7'>[7]</a> <a href ="/abs/2309.05039" title="Abstract" id="2309.05039"> arXiv:2309.05039 </a> (replaced) [<a href="/pdf/2309.05039" title="Download PDF" id="pdf-2309.05039" aria-labelledby="pdf-2309.05039">pdf</a>, <a href="/format/2309.05039" title="Other formats" id="oth-2309.05039" aria-labelledby="oth-2309.05039">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> The inverse limit topology and profinite descent on Picard groups in $K(n)$-local homotopy theory </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Li,+G">Guchuan Li</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Zhang,+N">Ningchuan Zhang</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 46 pages. Improved expositions and fixed typos following the referee report. Comments welcome! </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Topology (math.AT)</span> </div> <p class='mathjax'> In this paper, we study profinite descent theory for Picard groups in $K(n)$-local homotopy theory through their inverse limit topology. Building upon Burklund's result on the multiplicative structures of generalized Moore spectra, we prove that the module category over a $K(n)$-local commutative ring spectrum is equivalent to the limit of its base changes by a tower of generalized Moore spectra of type $n$. As a result, the $K(n)$-local Picard groups are endowed with a natural inverse limit topology. This topology allows us to identify the entire $E_1$ and $E_2$-pages of a descent spectral sequence for Picard spaces of $K(n)$-local profinite Galois extensions. <br>Our main examples are $K(n)$-local Picard groups of homotopy fixed points $E_n^{hG}$ of the Morava $E$-theory $E_n$ for all closed subgroups $G$ of the Morava stabilizer group $\mathbb{G}_n$. The $G=\mathbb{G}_n$ case has been studied by Heard and Mor. At height $1$, we compute Picard groups of $E_1^{hG}$ for all closed subgroups $G$ of $\mathbb{G}_1=\mathbb{Z}_p^\times$ at all primes as a Mackey functor. </p> </div> </dd> <dt> <a name='item8'>[8]</a> <a href ="/abs/2503.03173" title="Abstract" id="2503.03173"> arXiv:2503.03173 </a> (replaced) [<a href="/pdf/2503.03173" title="Download PDF" id="pdf-2503.03173" aria-labelledby="pdf-2503.03173">pdf</a>, <a href="/format/2503.03173" title="Other formats" id="oth-2503.03173" aria-labelledby="oth-2503.03173">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> The $RO(\mathcal{K})$-graded Coefficients of $H\underline{A}$ </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Keyes,+J">Jesse Keyes</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> Corrected an argument for showing an extension is split. Charts were also updated </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Topology (math.AT)</span> </div> <p class='mathjax'> In $G$-equivariant stable homotopy theory, it is known that the equivariant Eilenberg-Mac Lane spectra representing ordinary equivariant cohomology have nontrivial $RO(G)$-graded homotopy corresponding to the equivariant (co)homology of representation spheres. We will compute the universal case of this ordinary $RO(G)$-graded homotopy in the case of $G=\mathcal{K}$, where $\mathcal{K}$ is the Klein-four group. In particular, we will compute a subring of the $RO(\mathcal{K})$-graded homotopy of $H\underline{A}$ for $\underline{A}$ the Burnside Mackey functor. </p> </div> </dd> <dt> <a name='item9'>[9]</a> <a href ="/abs/2503.20052" title="Abstract" id="2503.20052"> arXiv:2503.20052 </a> (replaced) [<a href="/pdf/2503.20052" title="Download PDF" id="pdf-2503.20052" aria-labelledby="pdf-2503.20052">pdf</a>, <a href="https://arxiv.org/html/2503.20052v2" title="View HTML" id="html-2503.20052" aria-labelledby="html-2503.20052" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.20052" title="Other formats" id="oth-2503.20052" aria-labelledby="oth-2503.20052">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> On Poincar茅 Surgery </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Klein,+J+R">John R. Klein</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> Fixed typos and provided additional references </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Topology (math.AT)</span>; Geometric Topology (math.GT) </div> <p class='mathjax'> We exhibit a homotopy theoretic proof of the Fundamental Theorem of Poincar茅 surgery in the simply connected case. We also deduce the Poincar茅 transversality exact sequence. </p> </div> </dd> <dt> <a name='item10'>[10]</a> <a href ="/abs/2401.14254" title="Abstract" id="2401.14254"> arXiv:2401.14254 </a> (replaced) [<a href="/pdf/2401.14254" title="Download PDF" id="pdf-2401.14254" aria-labelledby="pdf-2401.14254">pdf</a>, <a href="/format/2401.14254" title="Other formats" id="oth-2401.14254" aria-labelledby="oth-2401.14254">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Diagrammatic representations of 3-periodic entanglements </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Andriamanalina,+T">Toky Andriamanalina</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Evans,+M+E">Myfanwy E. Evans</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Mahmoudi,+S">Sonia Mahmoudi</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> Published </div> <div class='list-journal-ref'><span class='descriptor'>Journal-ref:</span> Topology and its Applications 368 (2025) 109346 </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Geometric Topology (math.GT)</span>; Algebraic Topology (math.AT) </div> <p class='mathjax'> Diagrams enable the use of various algebraic and geometric tools for analysing and classifying knots. In this paper we introduce a new diagrammatic representation of triply periodic entangled structures (TP tangles), which are embeddings of simple curves in $\mathbb{R}^3$ that are invariant under translations along three non-coplanar axes. As such, these entanglements can be seen as preimages of links embedded in the 3-torus $\mathbb{T}^3 = \mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{S}^1$ in its universal cover $\mathbb{R}^3$, where two non-isotopic links in $\mathbb{T}^3$ may possess the same TP tangle preimage. We consider the equivalence of TP tangles in $\mathbb{R}^3$ through the use of diagrams representing links in $\mathbb{T}^3$. These diagrams require additional moves beyond the classical Reidemeister moves, which we define and show that they preserve ambient isotopies of links in $\mathbb{T}^3$. The final definition of a tridiagram of a link in $\mathbb{T}^3$ allows us to then consider additional notions of equivalence relating non-isotopic links in $\mathbb{T}^3$ that possess the same TP tangle preimage. </p> </div> </dd> </dl> <div class='paging'>Total of 10 entries </div> <div class='morefewer'>Showing up to 2000 entries per page: <a href=/list/math.AT/new?skip=0&show=1000 rel="nofollow"> fewer</a> | <span style="color: #454545">more</span> | <span style="color: #454545">all</span> </div> </div> </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">  <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>   <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>   </div> </footer> </div> <script src="/static/base/1.0.1/js/member_acknowledgement.js"></script> </body> </html>

CINXE.COM

Algebraic Topology