TOPOLOGICAL CRYSTALS 1. Introduction the 'Maximal Abelian Cover' of a Graph Plays a Key Role in Sunada's Work on Topologic
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
COVERING SPACES, GRAPHS, and GROUPS Contents 1. Covering
COVERING SPACES, GRAPHS, AND GROUPS CARSON COLLINS Abstract. We introduce the theory of covering spaces, with emphasis on explaining the Galois correspondence of covering spaces and the deck trans- formation group. We focus especially on the topological properties of Cayley graphs and the information these can give us about their corresponding groups. At the end of the paper, we apply our results in topology to prove a difficult theorem on free groups. Contents 1. Covering Spaces 1 2. The Fundamental Group 5 3. Lifts 6 4. The Universal Covering Space 9 5. The Deck Transformation Group 12 6. The Fundamental Group of a Cayley Graph 16 7. The Nielsen-Schreier Theorem 19 Acknowledgments 20 References 20 1. Covering Spaces The aim of this paper is to introduce the theory of covering spaces in algebraic topology and demonstrate a few of its applications to group theory using graphs. The exposition assumes that the reader is already familiar with basic topological terms, and roughly follows Chapter 1 of Allen Hatcher's Algebraic Topology. We begin by introducing the covering space, which will be the main focus of this paper. Definition 1.1. A covering space of X is a space Xe (also called the covering space) equipped with a continuous, surjective map p : Xe ! X (called the covering map) which is a local homeomorphism. Specifically, for every point x 2 X there is some open neighborhood U of x such that p−1(U) is the union of disjoint open subsets Vλ of Xe, such that the restriction pjVλ for each Vλ is a homeomorphism onto U. -
Convex Polytopes and Tilings with Few Flag Orbits
Convex Polytopes and Tilings with Few Flag Orbits by Nicholas Matteo B.A. in Mathematics, Miami University M.A. in Mathematics, Miami University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Abstract of Dissertation The amount of symmetry possessed by a convex polytope, or a tiling by convex polytopes, is reflected by the number of orbits of its flags under the action of the Euclidean isometries preserving the polytope. The convex polytopes with only one flag orbit have been classified since the work of Schläfli in the 19th century. In this dissertation, convex polytopes with up to three flag orbits are classified. Two-orbit convex polytopes exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron, and the rhombic triacontahedron. Two-orbit face-to-face tilings by convex polytopes exist on E1, E2, and E3; the only ones which are also combinatorially two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the tetrahedral-octahedral honeycomb, and the rhombic dodecahedral honeycomb. Moreover, any combinatorially two-orbit convex polytope or tiling is isomorphic to one on the above list. Three-orbit convex polytopes exist in two through eight dimensions. There are infinitely many in three dimensions, including prisms over regular polygons, truncated Platonic solids, and their dual bipyramids and Kleetopes. There are infinitely many in four dimensions, comprising the rectified regular 4-polytopes, the p; p-duoprisms, the bitruncated 4-simplex, the bitruncated 24-cell, and their duals. -
Configuration Spaces of Graphs
Configuration Spaces of Graphs Daniel Lütgehetmann Master’s Thesis Berlin, October 2014 Supervisor: Prof. Dr. Holger Reich Selbstständigkeitserklärung Hiermit versichere ich, dass ich die vorliegende Masterarbeit selbstständig und nur unter Zuhilfenahme der angegebenen Quellen erstellt habe. Daniel Lütgehetmann Erstkorrektor: Prof. Dr. Holger Reich Zweitkorrektor: Prof. Dr. Elmar Vogt Contents Introduction iii 1 Preliminaries1 1.1 Configuration Spaces . .1 1.2 Reduction to Connected Spaces . .3 1.3 The Rational Cohomology as Σn-Representation . .3 1.4 Some Representation Theory of the Symmetric Group . .4 2 A Deformation Retraction7 2.1 The Poset PnΓ ................................8 2.2 Cube Complexes and Cube Posets . 11 2.3 The Geometric Realization of a Cube Poset . 14 2.4 The Cube Complex KnΓ .......................... 16 2.5 The Embedding into the Configuration Space . 18 2.6 The Deformation Retraction . 21 2.7 The Path Metric on KnΓ .......................... 25 2.8 The Curvature of KnΓ ........................... 26 3 The Homology of Confn(Γ) 29 3.1 Representation Stability . 32 3.2 The Generalized Euler Characteristic of Confn(Γ) ........... 34 3.3 Graphs With At Most One Branched Vertex . 35 3.4 The General Case of Finite Graphs . 38 3.5 Applications . 39 3.6 Next Steps . 41 4 A Sheaf-Theoretic Approach 43 4.1 Sheaf Theory . 43 4.2 The Spectral Sequence . 46 4.3 The Stalks . 50 i Contents 4.4 Decomposition of Rq(inc )(Q) into Easier Sheaves . 52 ∗ 4.5 Sketch of the Proof for Manifolds . 53 BibliographyII Appendix III ii Introduction Let Γ be a graph and n a natural number. We want to understand the ordered configuration space Confn(Γ) consisting of all n-tuples x = (x1; : : : ; xn) of elements in Γ such that x = x for i = j, endowed with the subspace topology induced by the i 6 j 6 inclusion Conf (Γ) Γ n. -
Arxiv:2003.07426V2 [Math.CT] 22 Jul 2021 Modifying Some Constructions of Adams [1] for Connected CW Complexes with Base Point
BROWN REPRESENTABILITY FOR DIRECTED GRAPHS ZACHARY MCGUIRK † AND BYUNGDO PARK ‡ ABSTRACT. We prove that any contravariant functor from the homotopy category of finite directed graphs to abelian groups satisfying the additivity axiom and the Mayer-Vietoris axiom is repre- sentable. 1. INTRODUCTION The homotopy theory of directed graphs is a discrete analogue of homotopy theory in algebraic topology. In topology, a homotopy between two continuous maps is defined by an interpolating family of continuous maps parametrized by a closed interval [0; 1]. Its discrete analogue we study uses a directed line graph keeping track of a discrete change of directed graph maps. See for example Grigor’yan, Lin, Muranov, and Yau [8]. Efforts towards a homotopy theory for graphs extend back to the 1970’s and 1980’s with some results by Gianella [6] and Malle [16]. However, the most recent variant of graph homotopy theory appears to have taken off with a paper by Chen, Yau, and Yeh from 2001 [2], before culminating in the 2014 work of Grigor’yan, Lin, Muranov, and Yau [8]. Recently, there is increased interest in this new notion of homotopy for directed graphs because it was shown by Grigor’yan, Jimenez, Muranov, and Yau in [7] that the path-space homology theory of directed graphs is invariant under this version of graph homotopy. The path-space homology and cohomology theories for directed graphs were studied by Grigor’yan, Lin, Muranov, and Yau [9] and by Grigor’yan, Muranov, and Yau in [10] and [11]. The homology theory developed in the references above is quite natural, computable, and can be non-trivial for degrees greater than one, depending on the lengths of admissible, @-invariant paths in a directed graph. -
Hyperrogue: Playing with Hyperbolic Geometry
Bridges 2017 Conference Proceedings HyperRogue: Playing with Hyperbolic Geometry Eryk Kopczynski´ Dorota Celinska´ Marek Ctrnˇ act´ University of Warsaw, Poland University of Warsaw, Poland [email protected] [email protected] [email protected] Abstract HyperRogue is a computer game whose action takes place in the hyperbolic plane. We discuss how HyperRogue is relevant for mathematicians, artists, teachers, and game designers interested in hyperbolic geometry. Introduction a) b) c) d) Figure 1 : Example lands in HyperRogue: (a) Crossroads II, (b) Galapagos,´ (c) Windy Plains, (d) Reptiles. Hyperbolic geometry and tesselations of the hyperbolic plane have been of great interest to mathematical artists [16], most notably M.C. Escher and his Circle Limit series [4]. Yet, there were almost no attempts to bring more complexity and life to the hyperbolic plane. This is what our game, HyperRogue, sets out to do. We believe that it is relevant to mathematical artists for several reasons. First, fans of Escher’s art, or the famous Hofstadter’s book Godel,¨ Escher, Bach: an Eternal Golden Braid [10], will find many things appealing to them. The graphical style of HyperRogue is directly inspired by Escher’s tesselations, some of the music is based on Shepherd tones and crab canons if the gameplay in the given area is based on similar principles. As for Godel,¨ there is also one game mechanic reminding of a logical paradox: magical Orbs normally lose their power each turn, but Orb of Time prevents this, as long as the given Orb had no effect. Does Orb of Time lose its power if it is the only Orb you have? Second, we believe that exploring the world of HyperRogue is one of the best ways to understand hyperbolic geometry. -
New York Journal of Mathematics on Configuration Spaces and Simplicial
New York Journal of Mathematics New York J. Math. 25 (2019) 723{744. On configuration spaces and simplicial complexes Andrew A. Cooper, Vin de Silva and Radmila Sazdanovic Abstract. The n-point configuration space of a space M is a well- known object in topology, geometry, and combinatorics. We introduce a generalization, the simplicial configuration space MS , which takes as its data a simplicial complex S on n vertices, and explore the properties of its homology, considered as an invariant of S. As in Eastwood-Huggett's geometric categorification of the chromatic polynomial, our construction gives rise to a polynomial invariant of the simplicial complex S, which generalizes and shares several formal prop- erties with the chromatic polynomial. Contents 1. Introduction 723 2. The simplicial configuration space 725 3. Homology of MS 728 4. Properties of the simplicial configuration space 730 5. M matters 733 6. The simplicial chromatic polynomial 737 7. Comparison to other invariants 740 8. Questions and future directions 742 References 743 1. Introduction The configuration space of n points in the space M, n [ i j Confn(M) = M n x = x i6=j was introduced by Fadell and Neuwirth [14]. Its topology has since been the subject of extensive study (e.g. [16, 18, 23]), with particular focus on Received August 15, 2018. 2010 Mathematics Subject Classification. 55U10, 05C15, 05C31. Key words and phrases. simplicial complex, homology, chromatic polynomial, categorification. ISSN 1076-9803/2019 723 724 ANDREW A. COOPER, VIN DE SILVA AND RADMILA SAZDANOVIC how to relate the homology and cohomology of this (noncompact) space to the homology or cohomology of the space M, which may be taken to be a closed manifold or algebraic variety. -
The Eightfold Way: a Mathematical Sculpture by Helaman Ferguson
The Eightfold Way MSRI Publications Volume 35, 1998 The Eightfold Way: A Mathematical Sculpture by Helaman Ferguson WILLIAM P. THURSTON This introduction to The Eightfold Way andtheKleinquarticwaswritten for the sculpture’s inauguration. On that occasion it was distributed, to- gether with the illustration on Plate 2, to a public that included not only mathematicians but many friends of MSRI and other people with an in- terest in mathematics. Thurston was the Director of MSRI from 1992 to 1997. Mathematics is full of amazing beauty, yet the beauty of mathematics is far removed from most people’s everyday experience. The Mathematical Sciences Research Institute is committed to the search for ways to convey the beauty and spirit of mathematics beyond the circles of professional mathematicians. As a step in this effort, MSRI (pronounced “Emissary”) has installed a first mathematical sculpture, The Eightfold Way, by Helaman Ferguson. The sculp- ture represents a beautiful mathematical construction that has been studied by mathematicians for more than a century, from many points of view: geometry, symmetry, group theory, algebraic geometry, topology, number theory, complex analysis. The surface depicted by the sculpture was discovered, along with many of its amazing properties, by the German mathematician Felix Klein in 1879, and is often referred to as the Klein quartic or the Klein curve in his honor. The abstract surface is impossible to render exactly in three-dimensional space, so the sculpture should be thought of as a kind of topological sketch. Ridges and valleys carved into the white marble surface divide it into 24 regions. Each region has 7 sides, and represents the ideal of a regular heptagon (7-gon). -
Configuration Spaces, Braids, and Robotics
CONFIGURATION SPACES, BRAIDS, AND ROBOTICS R. GHRIST ABSTRACT. Braids are intimately related to configuration spaces of points. These configuration spaces give a useful model of autonomous agents (or robots) in an environment. Problems of relevance to autonomous engineering systems (e.g., motion planning, coordination, cooperation, assembly) are di- rectly related to topological and geometric properties of configuration spaces, including their braid groups. These notes detail this correspondence, and ex- plore several novel examples of configuration spaces relevant to applications in robotics. No familiarity with robotics is assumed. These notes shadow a two-hour set of tutorial talks given at the National University of Singapore, June 2007, for the IMS Program on Braids. CONTENTS 1. Configuration spaces and braids 2 2. Planning problems in robotics 3 2.1. Motion planning 3 2.2. Motion planning on tracks 4 2.3. Shape planning for metamorphic robots 5 2.4. Digital microfluidics 5 3. Configuration spaces of graphs 6 3.1. Visualization 6 3.2. Simplification: deformation 8 3.3. Simplification: simplicial 9 3.4. Simplification: discretization 11 4. Discretization 12 4.1. Cubes and collisions 12 4.2. Manifold examples 13 4.3. Everything that rises must converge 15 4.4. Curvature 15 5. Reconfigurable systems 17 5.1. Generators and relations 17 5.2. State complexes 18 5.3. Examples 19 6. Curvature again 24 6.1. State complexes are NPC 24 6.2. Efficient state planning 25 6.3. Back to braids 26 7. Last strands 27 7.1. Configuration spaces 27 1 2 R. GHRIST 7.2. Graph braid groups 29 7.3. -
Covering Spaces and Subgroups of the Free Group
COVERING SPACES AND SUBGROUPS OF THE FREE GROUP SAMANTHA NIEVEEN AND ALLISON SMITH Adviser: Dennis Garity Oregon State University Abstract. In this paper we will use the known link between covering spaces of a topological space and subgroups of its fundamental group to investigate the subgroup structures of the free groups on two and more generators. Some of our conclusions, such as Marshall Hall’s formula for the number of subgroups of a free group of finite index, have been shown before. (Hopefully others, such as some calculations for the number of normal subgroups of finite index, have not been. That would be cool.) 1. Introduction The correspondence between the covering spaces of a space X and subgroups of π1(X, x0), the fundamental group of X at a base point x0 is well known from topology. In particular, for any subgroup H of π1(X, x0), there exists a unique (up to covering equivalence) covering −1 p :E→X and a point e0 in p (x0) so that π1(E, e0) is homomorphic to H under a homo- morphism induced by p and E is path connected. Also, each covering space E corresponds −1 in this way to a conjugacy class of subgroups of π1(X, x0) as e0 ranges through p (x0). For more information see Munkres[4] Chapter 8. Throughout this paper we say that a subgroup corresponds to a particular covering and choice of base point or that a covering and choice of base point corresponds to a particular subgroup, referring to the correspondence described here. -
Patterns on the Genus-3 Klein Quartic
Patterns on the Genus-3 Klein Quartic Carlo H. Séquin Computer Science Division, EECS Department University of California, Berkeley, CA 94720 E-mail: [email protected] Abstract Projections of Klein's quartic surface of genus 3 into 3D space are used as canvases on which we present regular tessellations, Escher tilings, knot- and graph-embedding problems, Hamiltonian cycles, Petrie polygons and equatorial weaves derived from them. Many of the solutions found have also been realized as small physical models made on rapid-prototyping machines. Figure 1: Quilt made by Eveline Séquin showing regular tiling with 24 heptagons (a); virtual tetrus shape with cracked ceramic glazing programmed by Hayley Iben (b); and dual tiling with 56 triangles (c). 1. Introduction The Klein quartic, discovered in 1878 [5] has been called one of the most important mathematical struc- tures [6]. It emerges from the equation x3y + y3 + x = 0, if the variables are given complex values and the result is interpreted in 4-dimensional space. This structure has 168 automorphisms, where, with suitable variable substitution, the structure maps back onto itself. To make this more visible, we can cover the 4D surface with 24 heptagons. Every automorphism then maps a particular heptagon onto one of the 24 instances, in any one of 7 rotational positions. This means that all 24 heptagons, all 56 vertices, and all 84 edges are equivalent to each other. In 4D this is a completely regular structure in the same sense that the Platonic solids are completely regular polyhedral meshes. If we try to embed this construct in 3D so that we can make a physical model of it, we lose most of its metric symmetries, the regular heptagons get distorted, and only the symmetries of a regular tetrahedron are maintained. -
Homotopy Theory of Graphs
J Algebr Comb (2006) 24:31–44 DOI 10.1007/s10801-006-9100-0 Homotopy theory of graphs Eric Babson · H´el`ene Barcelo∗ · Mark de Longueville · Reinhard Laubenbacher Received: 11 February 2004 / Accepted: 30 November 2005 C Springer Science + Business Media, LLC 2006 1. Introduction Recently a new homotopy theory for graphs and simplicial complexes was defined (cf. [3, 4]). The motivation for the definition came initially from a desire to find invariants for dynamic processes that could be encoded via (combinatorial) simplicial complexes. The invariants were supposed to be topological in nature, but should at the same time be sensitive to the combinatorics encoded in the complex, in particular to the level of connectivity of simplices (see [7]). Namely, let be a simplicial complex of dimension d, let 0 ≤ q ≤ d be an integer, and let σ0 ∈ be a simplex of dimension greater than or equal to q. One obtains a family of groups q ,σ , ≥ , An ( 0) n 1 E. Babson () Department of Mathematics, University of Washington, Seattle, WA e-mail: [email protected] H. Barcelo Department of Mathematics and Statistics, Arizona State University, Tempe, Arizona 85287–1804 e-mail: [email protected] M. Longueville Fachbereich Mathematik, Freie Universit¨at Berlin, Arnimallee 3–5, D-14195 Berlin, Germany e-mail: [email protected] R. Laubenbacher Virginia Bioinformatics Institute, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061 e-mail: [email protected] ∗ Supported by NSA grant, #MSPF-04G-110 Springer 32 J Algebr Comb (2006) 24:31–44 Fig. 1 A 2-dimensional 1 complex with nontrivial A1. -
Covering Graphs
Chapter 5 Covering Graphs Another central concept in topological crystallography is the “covering map”. Roughly speaking, a covering map in general is a surjective continuous map between topological spaces which preserves the local topological structure (see Notes in this chapter for the exact definition). Figure 5.1 exhibits a covering map from the line R onto the circle S1, which somehow suggests an idea why we use the term “covering”.1 The canonical projection π : Rd −→ Rd/L, introduced in Example 2.5, is another example of a covering map. The theory of covering maps is regarded as a geometric analogue of Galois theory in algebra that describes a beautiful relationship between field extensions2 and Galois groups. Indeed, in a similar manner as Galois theory, covering maps are completely governed by what we call covering transformation groups.Inthis chapter, sacrificing full generality, we shall deal with covering maps for graphs from a combinatorial viewpoint. See [107] for the general theory. Our concern in crystallography is only special covering graphs (more specifically abelian covering graphs introduced in Chap. 6). However the notion of general covering graphs turns out to be of particular importance when we consider a “non- commutative” version of crystals, a mathematical object related to discrete groups [see Notes (IV) in Chap. 9]. Some problems in statistical mechanics are also well set up in terms of covering graph. 1More explicitly, the covering map is given by √ x ∈ R → e2π −1x ∈ S1 = {z ∈ C||z| = 1}. 2A field is an algebraic system with addition, subtraction, multiplication and division such as rational numbers, real numbers and complex numbers.