1<!DOCTYPE 5> 2<html lang="en"> 3 <head> 4 <meta charset="utf-8" /> 5 <title>Research - k = 1 cat - Jim Fowler</title> 6 <meta name="viewport" content="width=device-width, initial-scale=1.0" /> 7 <meta name="description" content="Jim Fowler" /> 8 <meta name="author" content="Jim Fowler" /> 9 <meta name="keywords" content="mathematics, math, maths, courses, course, research, education, teaching, topology, geometry" /> 10 <link rel="alternate" type="application/atom+xml" title="Feed for kisonecat.com" href="../feed.xml" /> 11 <link rel="stylesheet" href="https://stackpath.bootstrapcdn.com/bootstrap/4.5.2/css/bootstrap.min.css" integrity="sha384-JcKb8q3iqJ61gNV9KGb8thSsNjpSL0n8PARn9HuZOnIxN0hoP+VmmDGMN5t9UJ0Z" crossorigin="anonymous"> 12 <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.2.0/css/all.min.css" /> 13 <link href="../css/base.css" rel="stylesheet" media="screen" /> 14 <link href="../css/cv.css" rel="stylesheet" media="screen" /> 15 <link href="../css/tango.css" rel="stylesheet" media="screen" /> 16 <link rel="apple-touch-icon" sizes="114x114" href="../images/icons/favicon-114x114.png" /> 17 <link rel="apple-touch-icon" sizes="72x72" href="../images/icons/favicon-72x72.png" /> 18 <link rel="apple-touch-icon" sizes="57x57" href="../images/icons/favicon-57x57.png" /> 19 <link rel="shortcut icon" type="image/x-icon" href="../favicon.ico" /> 20 </head> 21 <body> 22 <header> 23 <nav class="navbar navbar-expand-md navbar-dark fixed-top bg-dark"> 24 <a class="navbar-brand" href="../">Jim <strong>Fowler</strong></a> 25 <button class="navbar-toggler" type="button" data-toggle="collapse" data-target="#navbarsExampleDefault" aria-controls="navbarsExampleDefault" aria-expanded="false" aria-label="Toggle navigation"> 26 <span class="navbar-toggler-icon"></span> 27 </button> 28 29 <div class="collapse navbar-collapse" id="navbarsExampleDefault"> 30 <ul class="navbar-nav mr-auto"> 31 <li class="nav-item "> 32 <a class="nav-link" href="../cv">CV</a> 33 </li> 34 <li class="nav-item active "> 35 <a class="nav-link" href="../research">Research<span class="sr-only">(current)</span></a> 36 </li> 37 <li class="nav-item "> 38 <a class="nav-link" href="../teaching">Teaching</a> 39 </li> 40 <li class="nav-item "> 41 <a class="nav-link" href="../blog">Blog</a> 42 </li> 43 </ul> 44 </div> 45 </nav> 46 </header> 47 48 49 50<main class="container" role="main"> 51 <nav aria-label="breadcrumb"> 52 <ol class="breadcrumb"> 53 <li class="breadcrumb-item"><a href="../">Home</a></li> 54 <li class="breadcrumb-item active">Research</li> 55 </ol> 56</nav> 57 58<p>My research is in topology and geometryâspecifically, surgery theory 59and geometric group theory. A few of my favorite things include: 60aspherical manifolds, rational homotopy types of manifolds, group 61actions on manifolds, quantified versions of classical invariants.</p> 62 63<p>You may also be interested in seeing 64a <a href="../research/papers">list of my publications</a> or 65reading <a href="../research/statement">my research statement</a>.</p> 66 67<hr /> 68 69 70<a href="../research/projective-planes.html"><h1>Projective planes 71 72<small class="text-muted">Joint work with Zhixu Su</small> 73 74</h1></a> 75<p>The Hirzebruch <span class="math inline"><em>L</em></span>-polynomial is one place where number theory very strongly interacts with high-dimensional topology <span class="citation" data-cites="MR0339202"></span>. Recall that the Hirzebruch signature theorem relates the signature of a smooth closed manifold <span class="math inline"><em>M</em><sup>4<em>k</em></sup></span> to <span class="math inline">â<sub><em>I</em></sub><em>L</em><sub><em>I</em></sub><em>p</em><sub><em>I</em></sub>(<em>M</em>)</span>. Unfortunately, the naïve method to compute coefficients <span class="math inline"><em>L</em><sub><em>I</em></sub></span> of the Hirzebruch <span class="math inline"><em>L</em></span>-polynomial is much too slow for applications; Zhixu Su and I have discovered a recursive method which is fast enough to compute many coefficients. Solutions to some Diophantine equations related to these <span class="math inline"><em>L</em></span>-polynomials give rise to manifolds having a truncated polynomial algebra as their rational cohomology ring; such manifolds may exist even when the corresponding truncated polynomial algebra over <span class="math inline">â¤</span> is not the cohomology ring of any space. For instance, there is a manifold having the rational cohomology that <span class="math inline">ð<em>P</em><sup>4</sup></span> would be expected to have, if <span class="math inline">ð<em>P</em><sup>4</sup></span> existed.</p> 76<hr /> 77 78<a href="../research/nontriangulable-aspherical.html"><h1>Aspherical manifolds that cannot be triangulated 79 80<small class="text-muted">Joint work with Michael W. Davis, Jean-François Lafont</small> 81 82</h1></a> 83<p>Kirby and Siebenmann showed that there are manifolds that do not admit PL structures, and yet the possibility remained that all manifolds could be triangulated, meaning that for every manifold <span class="math inline"><em>M</em></span>, there is a simplicial complex <span class="math inline"><em>K</em></span>
83 so that the geometric realization of <span class="math inline"><em>K</em></span> is homeomorphic to <span class="math inline"><em>M</em></span>, but of course the simplicial complex <span class="math inline"><em>K</em></span> is not a PL triangulation, meaning the links are not spheres. Freedman showed that there are 4-manifolds that cannot be triangulated. Davis and Januszkiewicz applied a hyperbolization procedure to Freedmanâs 4-manifolds to get closed aspherical 4-manifolds that cannot be triangulated. What about higher dimensions?</p> 84<hr /> 85 86<a href="../research/no-three-in-line.html"><h1>The no-three-in-line problem on a torus 87 88<small class="text-muted">Joint work with Andrew Groot, Deven Pandya, Bart Snapp</small> 89 90</h1></a> 91<p>For a group <span class="math inline"><em>G</em></span>, let <span class="math inline"><em>T</em>(<em>G</em>)</span> denote the cardinality of the largest subset <span class="math inline"><em>S</em>âââ<em>G</em></span> so that no three elements of <span class="math inline"><em>S</em></span> are in the same coset of a cyclic subgroup. Undergraduates Andrew Groot and Deven Pandya, advised by myself and my colleague Bart Snapp, considered the case <span class="math inline"><em>G</em>â=ââ¤/<em>m</em>â¤â Ãâ â¤/<em>n</em>â¤</span>, and showed that <span class="math inline"><em>T</em>(â¤<sub><em>p</em></sub>Ãâ¤<sub><em>p</em><sup>2</sup></sub>)â=â2<em>p</em></span> and <span class="math inline"><em>T</em>(â¤<sub><em>p</em></sub>Ãâ¤<sub><em>p</em><em>q</em></sub>)â=â<em>p</em>â +â 1</span>.</p> 92<hr /> 93 94<a href="../research/finiteness.html"><h1>Finiteness properties 95 96</h1></a> 97<p>A combination of BestvinaâBrady Morse theory and an acyclic reflection group trick produces a torsion-free finitely presented <span class="math inline">â</span>-Poincaré duality group which is not the fundamental group of an aspherical closed ANR <span class="math inline">â</span>-homology manifold.</p> 98<hr /> 99 100<a href="../research/compute-ell-class.html"><h1>L-class computations 101 102</h1></a> 103<p>Computation of the total <span class="math inline"><em>L</em></span>-class would also solve some manifold recognition problems, such as recognizing whether a particular combinatorial 15-vertex triangulations of an 8-manifold is the quaternionic projective plane <span class="math inline">â<em>P</em><sup>8</sup></span>. One of these examples <span class="math inline"><em>X</em><sup>8</sup></span> is especially symmetric, and likely PL homeomorphic to <span class="math inline">â<em>P</em><sup>8</sup></span>.</p> 104<hr /> 105 106<a href="../research/bounded-homotopy-theory.html"><h1>Bounded homotopy theory 107 108<small class="text-muted">Joint work with Crichton Ogle</small> 109 110</h1></a> 111<p>
111Given a bounding class <span class="math inline">â¬</span>, we construct a bounded refinement <span class="math inline">â¬<em>K</em>(â)</span> of Quillenâs <span class="math inline"><em>K</em></span>-theory functor from rings to spaces. As defined, <span class="math inline">â¬<em>K</em>(â)</span> is a functor from weighted rings to spaces, and is equipped with a comparison map <span class="math inline"><em>B</em><em>K</em>âââ<em>K</em></span> induced by &ldquo;forgetting control.&rdquo; In contrast to the situation with <span class="math inline">â¬</span>-bounded cohomology, there is a functorial splitting <span class="math inline">â¬<em>K</em>(â)âââ<em>K</em>(â)â Ãâ â¬<em>K</em><sup><em>r</em><em>e</em><em>l</em></sup>(â)</span> where <span class="math inline">â¬<em>K</em><sup><em>r</em><em>e</em><em>l</em></sup>(â)</span> is the homotopy fiber of the comparison map.</p> 112<hr /> 113 114<a href="../research/thesis.html"><h1>Ph.D. Thesis 115 116</h1></a> 117<p>We say that a group <span class="math inline"><em>G</em></span> is <span class="math inline">â</span>-<span class="math inline">PD</span> if it satisfies Poincare duality with rational coefficients (i.e., if its classifying space <span class="math inline"><em>B</em><em>G</em></span> does). Examples include the fundamental groups of aspherical manifolds. But there are other geometric examples: if a group <span class="math inline"><em>G</em></span> acts freely on a rationally-acyclic, rational homology manifold, then <span class="math inline"><em>G</em></span> is <span class="math inline">â</span>-<span class="math inline">PD</span>. Does every <span class="math inline">â</span>-<span class="math inline">PD</span> arise in this way&mdash;does every <span class="math inline">â</span>-<span class="math inline">PD</span> group act on such an object? The answer is no: lattices with torsion in semisimple Lie groups are counterexamples.</p> 118<hr /> 119 120<a href="../research/cusp-size-bounds.html"><h1>Cusp size bounds from singular surfaces in hyperbolic 3-manifolds 121 122<small class="text-muted">Joint work with Colin Adams, Adam Colestock, William Gillam, Eric Katerman</small> 123 124</h1></a> 125<p>Singular maps of surfaces into a hyperbolic 3-manifold are utilized to find upper bounds on meridian length, <span class="math inline">â</span>-curve length and maximal cusp volume for the manifold. This allows a proof of the fact that there exist hyperbolic knots with arbitrarily small cusp density and that every closed orientable 3-manifold contains a knot whose complement is hyperbolic with maximal cusp volume less than or equal to 9.</p> 126<hr /> 127 128<a href="../research/clean-geodesics.html"><h1>Cleanliness of geodesics in hyperbolic 3-manifolds 129 130<small class="text-muted">Joint work with Colin Adams, Adam Colestock, William Gillam, Eric Katerman</small> 131 132</h1></a> 133<p>We derive conditions guaranteeing the existence of geodesics avoiding the cusps and use these geodesics to show that in âalmost allâ finite volume hyperbolic 3-manifolds, infinitely many horoballs in the universal cover corresponding to a cusp are visible in a fundamental domain of the cusp when viewed from infinity.</p> 134 135 136</main> 137 138<footer><div class="container-fluid"><hr /><ul class="pull-right"><li><a class="socialmedia" href="http://www.youtube.com/kisonecat" rel="me"><i class="fab fa-youtube"></i> YouTube</a></li><li><a class="socialmedia" href="https://github.com/kisonecat"><i class="fab fa-github-alt"></i> GitHub</a></li><li><a class="socialmedia" href="http://www.facebook.com/kisonecat" rel="me"><i class="fab fa-facebook-square"></i> Facebook</a></li><li><a class="socialmedia" href="http://www.weibo.com/u/3696948727" rel="me"><i class="fab fa-weibo"></i> Weibo</a></li><li><a class="socialmedia" href="http://arxiv.org/find/math/1/au:+Fowler_J/0/1/0/all/0/1" rel="me"><i class="fa fa-file-alt"></i> arXiv</a></li><li><a class="socialmedia" href="../feed.xml"><i class="fa fa-rss"></i> RSS</a></li><li><a class="socialmedia" href="http://www.twitter.com/kisonecat" rel="me"><i class="fab fa-twitter"></i> Twitter</a></li><li><a class="socialmedia" href="https://mathstodon.xyz/@kisonecat" rel="me"><i class="fab fa-mastodon"></i> Mastodon</a></li></ul><p class="muted">© 2022, Jim Fowler.</p><dl><dt>Phone</dt><dd><span class="phone">(773) xxx-5659</span></p></dd><dt>Email</dt><dd><a href="#" class="email"><span class="email">my last name at math dot osu dot edu</span></a></p></dd><dt>Postal Address</dt><dd><div class="postal-address"><a href="https://www.osu.edu/">The Ohio State University</a><br /><a href="https://www.math.osu.edu/">Department of Mathematics</a><br />100 Math Tower<br />231 West 18th Avenue<br />Columbus, OH 43210-1174</p></div></dd></dl></div></footer> 139
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