Systolic Geometry and Topology
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The Motivic Fundamental Group of the Punctured Projective Line
THE MOTIVIC FUNDAMENTAL GROUP OF THE PUNCTURED PROJECTIVE LINE BERTRAND J. GUILLOU Abstract. We describe a construction of an object associated to the fundamental group of P1 −{0, 1, ∞} in the Bloch-Kriz category of mixed Tate motives. This description involves Massey products of Steinberg symbols in the motivic cohomology of the ground field. Contents 1. Introduction 2 2. Motivic Cohomology 4 2.1. The conjectural picture 4 2.2. Bloch’s cycle complex 6 2.3. Friedlander-Suslin-Voevodsky variant 8 2.4. Cocycles for functions 11 3. Mixed Tate Motives 13 3.1. Mixed Tate categories 13 3.2. The approach of Bloch-Kriz 14 3.3. The homotopy category of cell modules 17 3.4. The approach of Kriz-May 19 3.5. Minimal modules 20 4. The motivic fundamental group 21 4.1. The n = 2 case 21 arXiv:0903.1705v1 [math.AG] 10 Mar 2009 4.2. Massey products 23 4.3. The n = 3 case 24 4.4. The general case 28 4.5. Trimming the fat 31 4.6. Vanishing of Massey Products 33 4.7. Bloch-Totaro cycles 35 4.8. 3-fold and 4-fold Massey products 35 References 36 Date: October 15, 2018. 1 2 BERTRAND J. GUILLOU 1. Introduction The importance of the algebraic fundamental group of P1 − {0, 1, ∞} has been known for some time: “[A. Grothendieck] m’a aussi dit, avec force, que le compl´et´e 1 profiniπ ˆ1 du groupe fondamental de X := P (C)−{0, 1, ∞}, avec son action de Gal(Q/Q) est un objet remarquable, et qu’il faudrait l’´etudier.” -[Del] Indeed, Belyi’s theorem implies that the canonical action of Gal(Q/Q) is faithful. -
HE WANG Abstract. a Mini-Course on Rational Homotopy Theory
RATIONAL HOMOTOPY THEORY HE WANG Abstract. A mini-course on rational homotopy theory. Contents 1. Introduction 2 2. Elementary homotopy theory 3 3. Spectral sequences 8 4. Postnikov towers and rational homotopy theory 16 5. Commutative differential graded algebras 21 6. Minimal models 25 7. Fundamental groups 34 References 36 2010 Mathematics Subject Classification. Primary 55P62 . 1 2 HE WANG 1. Introduction One of the goals of topology is to classify the topological spaces up to some equiva- lence relations, e.g., homeomorphic equivalence and homotopy equivalence (for algebraic topology). In algebraic topology, most of the time we will restrict to spaces which are homotopy equivalent to CW complexes. We have learned several algebraic invariants such as fundamental groups, homology groups, cohomology groups and cohomology rings. Using these algebraic invariants, we can seperate two non-homotopy equivalent spaces. Another powerful algebraic invariants are the higher homotopy groups. Whitehead the- orem shows that the functor of homotopy theory are power enough to determine when two CW complex are homotopy equivalent. A rational coefficient version of the homotopy theory has its own techniques and advan- tages: 1. fruitful algebraic structures. 2. easy to calculate. RATIONAL HOMOTOPY THEORY 3 2. Elementary homotopy theory 2.1. Higher homotopy groups. Let X be a connected CW-complex with a base point x0. Recall that the fundamental group π1(X; x0) = [(I;@I); (X; x0)] is the set of homotopy classes of maps from pair (I;@I) to (X; x0) with the product defined by composition of paths. Similarly, for each n ≥ 2, the higher homotopy group n n πn(X; x0) = [(I ;@I ); (X; x0)] n n is the set of homotopy classes of maps from pair (I ;@I ) to (X; x0) with the product defined by composition. -
Arxiv:Math/0401075V1 [Math.AT] 8 Jan 2004 Ojcue1.1
CONFIGURATION SPACES ARE NOT HOMOTOPY INVARIANT RICCARDO LONGONI AND PAOLO SALVATORE Abstract. We present a counterexample to the conjecture on the homotopy invariance of configuration spaces. More precisely, we consider the lens spaces L7,1 and L7,2, and prove that their configuration spaces are not homotopy equivalent by showing that their universal coverings have different Massey products. 1. Introduction The configuration space Fn(M) of pairwise distinct n-tuples of points in a man- ifold M has been much studied in the literature. Levitt reported in [4] as “long- standing” the following Conjecture 1.1. The homotopy type of Fn(M), for M a closed compact smooth manifold, depends only on the homotopy type of M. There was some evidence in favor: Levitt proved that the loop space ΩFn(M) is a homotopy invariant of M. Recently Aouina and Klein [1] have proved that a suitable iterated suspension of Fn(M) is a homotopy invariant. For example the double suspension of F2(M) is a homotopy invariant. Moreover F2(M) is a homotopy invariant when M is 2-connected (see [4]). A rational homotopy theoretic version of this fact appears in [3]. On the other hand there is a similar situation n suggesting that the conjecture might fail: the Euclidean configuration space F3(R ) has the homotopy type of a bundle over Sn−1 with fiber Sn−1 ∨ Sn−1 but it does n not split as a product in general [6]. However the loop spaces of F3(R ) and of the product Sn−1 × (Sn−1 ∨ Sn−1) are homotopy equivalent and also the suspensions of the two spaces are homotopic. -
February 19, 2016
February 19, 2016 Professor HELGE HOLDEN SECRETARY OF THE INTERNATIONAL MATHEMATICAL UNION Dear Professor Holden, The Turkish Mathematical Society (TMD) as the Adhering Organization, applies to promote Turkey from Group I to Group II as a member of IMU. We attach an overview of the developments of Mathematics in Turkey during the last 10 years (2005- 2015) preceded by a historical account. With best regards, Betül Tanbay President of the Turkish Mathematical Society Report on the state of mathematics in Turkey (2005-2015) This is an overview of the status of Mathematics in Turkey, prepared for the IMU for promotion from Group I to Group II by the adhering organization, the Turkish Mathematical Society (TMD). 1-HISTORICAL BACKGROUND 2-SOCIETIES AND CENTERS RELATED TO MATHEMATICAL SCIENCES 3-NUMBER OF PUBLICATIONS BY SUBJECT CATEGORIES 4-NATIONAL CONFERENCES AND WORKSHOPS HELD IN TURKEY BETWEEN 2013-2015 5-INTERNATIONAL CONFERENCES AND WORKSHOPS HELD IN TURKEY BETWEEN 2013-2015 6-NUMBERS OF STUDENTS AND TEACHING STAFF IN MATHEMATICAL SCIENCES IN TURKEY FOR THE 2013-2014 ACADEMIC YEAR AND THE 2014-2015 ACADEMIC YEAR 7- RANKING AND DOCUMENTS OF TURKEY IN MATHEMATICAL SCIENCES 8- PERIODICALS AND PUBLICATIONS 1-HISTORICAL BACKGROUND Two universities, the Istanbul University and the Istanbul Technical University have been influential in creating a mathematical community in Turkey. The Royal School of Naval Engineering, "Muhendishane-i Bahr-i Humayun", was established in 1773 with the responsibility to educate chart masters and ship builders. Gaining university status in 1928, the Engineering Academy continued to provide education in the fields of engineering and architecture and, in 1946, Istanbul Technical University became an autonomous university which included the Faculties of Architecture, Civil Engineering, Mechanical Engineering, and Electrical and Electronic Engineering. -
Systolic Inequalities, Discrete Or Not
Systolic inequalities, discrete or not Arnaud de Mesmay Gipsa-lab, CNRS, Université Grenoble Alpes Based on joint work with Éric Colin de Verdière and Alfredo Hubard. 1 / 53 A primer on surfaces We deal with connected, compact and orientable surfaces of genus g without boundary. Discrete metric Riemannian metric Triangulation G. Scalar product m on the Length of a curve jγjG : tangent space. Number of edges. Riemannian length jγjm. 2 / 53 Systoles and pants decompositions We study the length of topologically interesting curves for discrete and continuous metrics. Non-contractible curves Pants decompositions 3 / 53 Motivations Why should we care ? Topological graph theory: If the shortest non-contractible cycle is long, the surface is planar-like. ) Uniqueness of embeddings, colourability, spanning trees. Riemannian geometry: René Thom: “Mais c’est fondamental !”. Links with isoperimetry, topological dimension theory, number theory. Algorithms for surface-embedded graphs: Cookie-cutter algorithm for surface-embedded graphs: Decompose the surface, solve the planar case, recover the solution. More practical sides: texture mapping, parameterization, meshing ... 4 / 53 Part 1: Length of shortest curves 5 / 53 Intuition p p It should have length O( A) or O( n), but what is the dependency on g ? On shortest noncontractible curves Discrete setting Continuous setting What is the length of the red curve? 6 / 53 On shortest noncontractible curves Discrete setting Continuous setting What is the length of the red curve? Intuition p p It should have length O( A) or O( n), but what is the dependency on g ? 7 / 53 Discrete Setting: Topological graph theory The edgewidth of a triangulated surface is the length of the shortest noncontractible cycle. -
Combination of Cubic and Quartic Plane Curve
IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 2 (Mar. - Apr. 2013), PP 43-53 www.iosrjournals.org Combination of Cubic and Quartic Plane Curve C.Dayanithi Research Scholar, Cmj University, Megalaya Abstract The set of complex eigenvalues of unistochastic matrices of order three forms a deltoid. A cross-section of the set of unistochastic matrices of order three forms a deltoid. The set of possible traces of unitary matrices belonging to the group SU(3) forms a deltoid. The intersection of two deltoids parametrizes a family of Complex Hadamard matrices of order six. The set of all Simson lines of given triangle, form an envelope in the shape of a deltoid. This is known as the Steiner deltoid or Steiner's hypocycloid after Jakob Steiner who described the shape and symmetry of the curve in 1856. The envelope of the area bisectors of a triangle is a deltoid (in the broader sense defined above) with vertices at the midpoints of the medians. The sides of the deltoid are arcs of hyperbolas that are asymptotic to the triangle's sides. I. Introduction Various combinations of coefficients in the above equation give rise to various important families of curves as listed below. 1. Bicorn curve 2. Klein quartic 3. Bullet-nose curve 4. Lemniscate of Bernoulli 5. Cartesian oval 6. Lemniscate of Gerono 7. Cassini oval 8. Lüroth quartic 9. Deltoid curve 10. Spiric section 11. Hippopede 12. Toric section 13. Kampyle of Eudoxus 14. Trott curve II. Bicorn curve In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a rational quartic curve defined by the equation It has two cusps and is symmetric about the y-axis. -
Discrete Systolic Inequalities and Decompositions of Triangulated Surfaces
Discrete Systolic Inequalities and Decompositions of Triangulated Surfaces Éric Colin de Verdière 1 Alfredo Hubard 2 Arnaud de Mesmay 3 1École normale supérieure, CNRS 2INRIA, Laboratoire d’Informatique Gaspard Monge, Université Paris-Est Marne-la-Vallée 3IST Austria, Autriche 1 / 62 A primer on surfaces We deal with connected, compact and orientable surfaces of genus g without boundary. Discrete metric Riemannian metric Triangulation G. Scalar product m on the Length of a curve jγjG : tangent space. Number of edges. Riemannian length jγjm. 2 / 62 Systoles and surface decompositions We study the length of topologically interesting curves and graphs, for discrete and continuous metrics. 1.Non-contractible curves 2.Pants decompositions 3.Cut-graphs 3 / 62 Part 0: Why should we care.. 4 / 62 .. about graphs embedded on surfaces ? The easy answer: because they are “natural”. They occur in multiple settings: Graphics, computer-aided design, network design. The algorithmic answer: because they are “general”. Every graph is embeddable on some surface, therefore the genus of this surface is a natural parameter of a graph (similarly as tree-width, etc.). The hard answer: because of Robertson-Seymour theory. Theorem (Graph structure theorem, roughly) Every minor-closed family of graphs can be obtained from graphs k-nearly embedded on a surface S, for some constant k. 5 / 62 ) We need algorithms to do this cutting efficiently. ) We need good bounds on the lengths of the cuttings. ... about cutting surfaces along cycles/graphs ? Algorithms for surface-embedded graphs: Cookie-cutter algorithm for surface-embedded graphs: Cut the surface into the plane. Solve the planar case. -
Lusternik-Schnirelmann Category and Systolic Category of Low Dimensional Manifolds
LUSTERNIK-SCHNIRELMANN CATEGORY AND SYSTOLIC CATEGORY OF LOW DIMENSIONAL MANIFOLDS1 MIKHAIL G. KATZ∗ AND YULI B. RUDYAK† Abstract. We show that the geometry of a Riemannian mani- fold (M, G) is sensitive to the apparently purely homotopy-theoretic invariant of M known as the Lusternik-Schnirelmann category, denoted catLS(M). Here we introduce a Riemannian analogue of catLS(M), called the systolic category of M. It is denoted catsys(M), and defined in terms of the existence of systolic in- equalities satisfied by every metric G, as initiated by C. Loewner and later developed by M. Gromov. We compare the two cate- gories. In all our examples, the inequality catsys M ≤ catLS M is satisfied, which typically turns out to be an equality, e.g. in dimen- sion 3. We show that a number of existing systolic inequalities can be reinterpreted as special cases of such equality, and that both categories are sensitive to Massey products. The comparison with the value of catLS(M) leads us to prove or conjecture new systolic inequalities on M. Contents Introduction 2 1. Systoles 3 2. Systolic categories 4 3. Categories agree in dimension 2 6 4. Essential manifolds and detecting elements 7 arXiv:math/0410456v2 [math.DG] 20 Dec 2004 5. Inessential manifolds and pullback metrics 8 6. Manifolds of dimension 3 9 1Communications on Pure and Applied Mathematics, to appear. Available at arXiv:math.DG/0410456 Date: August 31, 2018. 1991 Mathematics Subject Classification. Primary 53C23; Secondary 55M30, 57N65 . Key words and phrases. detecting element, essential manifolds, isoperimetric quotient, Lusternik-Schnirelmann category, Massey product, systole. -
Arxiv:1012.2020V1 [Math.CV]
TRANSITIVITY ON WEIERSTRASS POINTS ZOË LAING AND DAVID SINGERMAN 1. Introduction An automorphism of a Riemann surface will preserve its set of Weier- strass points. In this paper, we search for Riemann surfaces whose automorphism groups act transitively on the Weierstrass points. One well-known example is Klein’s quartic, which is known to have 24 Weierstrass points permuted transitively by it’s automorphism group, PSL(2, 7) of order 168. An investigation of when Hurwitz groups act transitively has been made by Magaard and Völklein [19]. After a section on the preliminaries, we examine the transitivity property on several classes of surfaces. The easiest case is when the surface is hy- perelliptic, and we find all hyperelliptic surfaces with the transitivity property (there are infinitely many of them). We then consider surfaces with automorphism group PSL(2, q), Weierstrass points of weight 1, and other classes of Riemann surfaces, ending with Fermat curves. Basically, we find that the transitivity property property seems quite rare and that the surfaces we have found with this property are inter- esting for other reasons too. 2. Preliminaries Weierstrass Gap Theorem ([6]). Let X be a compact Riemann sur- face of genus g. Then for each point p ∈ X there are precisely g integers 1 = γ1 < γ2 <...<γg < 2g such that there is no meromor- arXiv:1012.2020v1 [math.CV] 9 Dec 2010 phic function on X whose only pole is one of order γj at p and which is analytic elsewhere. The integers γ1,...,γg are called the gaps at p. The complement of the gaps at p in the natural numbers are called the non-gaps at p. -
Small Values of Lusternik-Schnirelmann And
SMALL VALUES OF LUSTERNIK-SCHNIRELMANN AND SYSTOLIC CATEGORIES FOR MANIFOLDS ALEXANDER N. DRANISHNIKOV1, MIKHAIL G. KATZ2, AND YULI B. RUDYAK3 Abstract. We prove that manifolds of Lusternik-Schnirelmann category 2 necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larra˜naga and Gonzalez-Acu˜na, by generalizing their result in dimension 3, to all higher dimensions. We examine its ramifications in systolic topology, and provide a sufficient condition for ensuring a lower bound of 3 for systolic category. Contents 1. Introduction 1 2. Cohomology with local coefficients 4 3. Category weight and lower bounds for catLS 7 4.ManifoldsofLScategory2 8 5. Manifolds of higher LS category 9 6. Upperboundforsystoliccategory 14 7. Self-linking of fibers and a lower bound for catsys 16 References 20 1. Introduction arXiv:0706.1625v3 [math.AT] 23 Jul 2007 We follow the normalization of the Lusternik-Schnirelmann category (LS category) used in the recent monograph [CLOT03] (see Section 3 for a definition). Spaces of LS category 0 are contractible, while a Date: November 3, 2018. 2000 Mathematics Subject Classification. Primary 53C23; Secondary 55M30, 57N65 . Key words and phrases. Category weight, cohomological dimension, detecting element, essential manifolds, free fundamental group, Lusternik-Schnirelmann cat- egory, Massey product, self-linking class, systolic category. 1Supported by NSF, grant DMS-0604494. 2Supported by the Israel Science Foundation (grants no. 84/03 and 1294/06). 3Supported by NSF, grant 0406311. 1 2 A.DRANISHNIKOV,M.KATZ,ANDYU.RUDYAK closed manifold of LS category 1 is homotopy equivalent (and hence homeomorphic) to a sphere. The characterization of closed manifolds of LS category 2 was initi- ated in 1992 by J. -
Cohomological Dimension, Self-Linking, and Systolic Geometry
COHOMOLOGICAL DIMENSION, SELF-LINKING, AND SYSTOLIC GEOMETRY ALEXANDER N. DRANISHNIKOV1, MIKHAIL G. KATZ2, AND YULI B. RUDYAK3 Abstract. Given a closed manifold M, we prove the upper bound of 1 2 (dim M + cd(π1M)) for the number of systolic factors in a curvature-free lower bound for the total volume of M, in the spirit of M. Gromov’s systolic inequalities. Here “cd” is the cohomological dimension. We ap- ply this upper bound to show that, in the case of a 4-manifold, the Lusternik–Schnirelmann category is an upper bound for the systolic category. Furthermore we prove a systolic inequality on a manifold M with b1(M) = 2 in the presence of a nontrivial self- linking class of a typical fiber of its Abel–Jacobi map to the 2-torus. Contents 1. Systolic inequalities and LS category 2 2. A systolic introduction 4 3. Gromov’s inequalities 6 4. Fundamental group and systolic category 7 5. Systolic category, LS category, and cd(π) 8 6. Self-linking of fibers and a lower bound for catsys 9 arXiv:0807.5040v3 [math.DG] 14 Dec 2009 References 12 Date: June 9, 2018. 2000 Mathematics Subject Classification. Primary 53C23; Secondary 55M30, 57N65. Key words and phrases. Category weight, cohomological dimension, detect- ing element, Eilenberg–Ganea conjecture, essential manifolds, free fundamental group, Lusternik–Schnirelmann category, Massey product, self-linking class, sys- tolic category. 1Supported by the NSF, grant DMS-0604494. 2Supported by the Israel Science Foundation (grants 84/03 and 1294/06) and the BSF (grant 2006393). 3Supported by the NSF, grant 0406311. -
Arxiv:1406.2663V2 [Hep-Th]
Multiboundary Wormholes and Holographic Entanglement Vijay Balasubramaniana;b, Patrick Haydenc, Alexander Maloneyd;e, Donald Marolff , Simon F. Rossg aDavid Rittenhouse Laboratories, University of Pennsylvania 209 S 33rd Street, Philadelphia, PA 19104, USA bCUNY Graduate Center, Initiative for the Theoretical Sciences 365 Fifth Avenue, New York, NY 10016, USA cDepartment of Physics, Stanford University Palo Alto, CA 94305, USA dDepartment of Physics, McGill University 3600 rue Universit´e,Montreal H3A2T8, Canada eCenter for the Fundamental Laws of Nature, Harvard University Cambridge, MA 02138, USA f Department of Physics, University of California, Santa Barbara, CA 93106, USA gCentre for Particle Theory, Department of Mathematical Sciences Durham University, South Road, Durham DH1 3LE, UK Abstract The AdS/CFT correspondence relates quantum entanglement between boundary Conformal Field Theories and geometric connections in the dual asymptotically Anti- de Sitter space-time. We consider entangled states in the n−fold tensor product of a 1+1 dimensional CFT Hilbert space defined by the Euclidean path integral over a Riemann surface with n holes. In one region of moduli space, the dual bulk state is arXiv:1406.2663v2 [hep-th] 23 Jun 2014 a black hole with n asymptotically AdS3 regions connected by a common wormhole, while in other regions the bulk fragments into disconnected components. We study the entanglement structure and compute the wave function explicitly in the puncture limit of the Riemann surface in terms of CFT n-point functions. We also use AdS minimal surfaces to measure entanglement more generally. In some regions of the moduli space the entanglement is entirely multipartite, though not of the GHZ type.