Moduli Spaces of Colored Graphs
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Computation of Cubical Homology, Cohomology, and (Co)Homological Operations Via Chain Contraction
Computation of cubical homology, cohomology, and (co)homological operations via chain contraction Paweł Pilarczyk · Pedro Real Abstract We introduce algorithms for the computation of homology, cohomology, and related operations on cubical cell complexes, using the technique based on a chain contraction from the original chain complex to a reduced one that represents its homology. This work is based on previous results for simplicial complexes, and uses Serre’s diagonalization for cubical cells. An implementation in C++ of the intro-duced algorithms is available at http://www.pawelpilarczyk.com/chaincon/ together with some examples. The paper is self-contained as much as possible, and is written at a very elementary level, so that basic knowledge of algebraic topology should be sufficient to follow it. Keywords Algorithm · Software · Homology · Cohomology · Computational homology · Cup product · Alexander-Whitney coproduct · Chain homotopy · Chain contraction · Cubical complex Mathematics Subject Classification (2010) 55N35 · 52B99 · 55U15 · 55U30 · 55-04 Communicated by: D. N. Arnold P. Pilarczyk () Universidade do Minho, Centro de Matema´tica, Campus de Gualtar, 4710-057 Braga, Portugal e-mail: [email protected] URL: http://www.pawelpilarczyk.com/ P. Real Departamento de Matema´ ticaAplicadaI,E.T. S.de Ingenier´ı a Informatica,´ Universidad de Sevilla, Avenida Reina Mercedes s/n, 41012 Sevilla, Spain e-mail: [email protected] URL: http://investigacion.us.es/sisius/sis showpub.php?idpers=1021 Present Address: P. Pilarczyk IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria 254 P. Pilarczyk, P. Real 1 Introduction A full cubical set in Rn is a finite union of n-dimensional boxes of fixed size (called cubes for short) aligned with a uniform rectangular grid in Rn. -
Metric Geometry, Non-Positive Curvature and Complexes
METRIC GEOMETRY, NON-POSITIVE CURVATURE AND COMPLEXES TOM M. W. NYE . These notes give mathematical background on CAT(0) spaces and related geometry. They have been prepared for use by students on the International PhD course in Non- linear Statistics, Copenhagen, June 2017. 1. METRIC GEOMETRY AND THE CAT(0) CONDITION Let X be a set. A metric on X is a map d : X × X ! R≥0 such that (1) d(x; y) = d(y; x) 8x; y 2 X, (2) d(x; y) = 0 , x = y, and (3) d(x; z) ≤ d(x; y) + d(y; z) 8x; y; z 2 X. On its own, a metric does not give enough structure to enable us to do useful statistics on X: for example, consider any set X equipped with the metric d(x; y) = 1 whenever x 6= y. A geodesic path between x; y 2 X is a map γ : [0; `] ⊂ R ! X such that (1) γ(0) = x, γ(`) = y, and (2) d(γ(t); γ(t0)) = jt − t0j 8t; t0 2 [0; `]. We will use the notation Γ(x; y) ⊂ X to be the image of the path γ and call this a geodesic segment (or just geodesic for short). A path γ : [0; `] ⊂ R ! X is locally geodesic if there exists > 0 such that property (2) holds whever jt − t0j < . This is a weaker condition than being a geodesic path. The set X is called a geodesic metric space if there is (at least one) geodesic path be- tween every pair of points in X. -
Normal Crossing Immersions, Cobordisms and Flips
NORMAL CROSSING IMMERSIONS, COBORDISMS AND FLIPS KARIM ADIPRASITO AND GAKU LIU Abstract. We study various analogues of theorems from PL topology for cubical complexes. In particular, we characterize when two PL home- omorphic cubulations are equivalent by Pachner moves by showing the question to be equivalent to the existence of cobordisms between generic immersions of hypersurfaces. This solves a question and conjecture of Habegger, Funar and Thurston. 1. Introduction Pachner’s influential theorem proves that every PL homeomorphim of tri- angulated manifolds can be written as a combination of local moves, the so-called Pachner moves (or bistellar moves). Geometrically, they correspond to exchanging, inside the given manifold, one part of the boundary of a simplex by the complementary part. For instance, in a two-dimensional manifold, a triangle can be exchanged by three triangles with a common interior vertex, and two triangles sharing an edge, can be exchanged by two different triangles, connecting the opposite two edges. In cubical complexes, popularized for their connection to low-dimensional topology and geometric group theory, the situation is not as simple. Indeed, a cubical analogue of the Pachner move can never change, for instance, the parity of the number of facets mod 2. But cubical polytopes, for instance in dimension 3, can have odd numbers of facets as well as even, see for instance [SZ04] for a few particularly notorious constructions. Hence, there are at least two classes of cubical 2-spheres that can never arXiv:2001.01108v1 [math.GT] 4 Jan 2020 be connected by cubical Pachner moves. Indeed, Funar [Fun99] provided a conjecture for a complete characterization for the cubical case, that revealed its immense depth. -
Homology of Hom Complexes
Rose-Hulman Undergraduate Mathematics Journal Volume 13 Issue 1 Article 10 Homology of Hom Complexes Mychael Sanchez New Mexico State University, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Sanchez, Mychael (2012) "Homology of Hom Complexes," Rose-Hulman Undergraduate Mathematics Journal: Vol. 13 : Iss. 1 , Article 10. Available at: https://scholar.rose-hulman.edu/rhumj/vol13/iss1/10 Rose- Hulman Undergraduate Mathematics Journal Homology of Hom Complexes Mychael Sancheza Volume 13, No. 1, Spring 2012 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 Email: [email protected] a http://www.rose-hulman.edu/mathjournal New Mexico State University Rose-Hulman Undergraduate Mathematics Journal Volume 13, No. 1, Spring 2012 Homology of Hom Complexes Mychael Sanchez Abstract. The hom complex Hom (G; K) is the order complex of the poset com- posed of the graph multihomomorphisms from G to K. We use homology to provide conditions under which the hom complex is not contractible and derive a lower bound on the rank of its homology groups. Acknowledgements: I acknowledge my advisor, Dr. Dan Ramras and the National Science Foundation for funding this research through Dr. Ramras' grants, DMS-1057557 and DMS-0968766. Page 212 RHIT Undergrad. Math. J., Vol. 13, No. 1 1 Introduction In 1978, László Lovász proved the Kneser Conjecture [8]. His proof involves associating to a graph G its neighborhood complex N (G) and using its topological properties to locate obstructions to graph colorings. This proof illustrates the fruitful interaction between graph theory, combinatorics and topology. -
Arxiv:Math/9804046V2 [Math.GT] 26 Jan 1999 CB R O Ubeeuvln.I at for Fact, Negativ in Is States, Equivalent
. CUBULATIONS, IMMERSIONS, MAPPABILITY AND A PROBLEM OF HABEGGER LOUIS FUNAR Abstract. The aim of this paper (inspired from a problem of Habegger) is to describe the set of cubical decompositions of compact manifolds mod out by a set of combinatorial moves analogous to the bistellar moves considered by Pachner, which we call bubble moves. One constructs a surjection from this set onto the the bordism group of codimension one immersions in the manifold. The connected sums of manifolds and immersions induce multiplicative structures which are respected by this surjection. We prove that those cubulations which map combinatorially into the standard decomposition of Rn for large enough n (called mappable), are equivalent. Finally we classify the cubulations of the 2-sphere. Keywords and phrases: Cubulation, bubble/np-bubble move, f-vector, immersion, bordism, mappable, embeddable, simple, standard, connected sum. 1. Introduction and statement of results 1.1. Outline. Stellar moves were first considered by Alexander ([1]) who proved that they can relate any two triangulations of a polyhedron. Alexander’s moves were refined to a finite set of local (bistellar) moves which still act transitively on the triangulations of manifolds, according to Pachner ([35]). Using Pachner’s result Turaev and Viro proved that certain state-sums associated to a triangulation yield topological invariants of 3-manifolds (see [39]). Recall that a bistellar move (in dimension n) excises B and replaces it with B′, where B and B′ are complementary balls, subcomplexes in the boundary of the standard (n + 1)-simplex. For a nice exposition of Pachner’s result and various extensions, see [29]. -
Constructing Covers of the Rose
Constructing Covers of the rose. Robert Kropholler April 25, 2017 This was discussed in class but does not appear in any of the lecture notes. I will outline the constructions and proof here. Definition 0.1. Let F (X) be the free group on the set X. Assume jXj = n. Let H be any subgroup of F (X), we define the Schrier graph G(H) as follows: • The vertices of G(H) are the left cosets of H in G. • Two cosets gH; g0H are joined by an edge if there exists an x 2 X such that xgH = g0H. It should be noted that the Schrier graph of a normal subgroup N is the Cayley graph of G=N with respect to the generating set X. Let Rn be a cell complex with 1 vertex and n edges. We identify the fundamental group of Rn with F (X). Pick a bijection between the edges of X and give each edge an orientation. Proposition 0.2. The graph G(H) is a cover of Rn. Proof. We can label each edge with an element of X and give it an orientation so that the edge (gH; xgH) points towards xgH. We can then define a map p : G(H) ! Rn. This sends each vertex of G(H) to the unique vertex of Rn and each edge to the edge with same label by an orientation preserving homeomorphism. This defines a covering map. We must check that every point has a neighbourhood such that p is a homeomorphism on this neighbourhood. -
Knot Theory and the Alexander Polynomial
Knot Theory and the Alexander Polynomial Reagin Taylor McNeill Submitted to the Department of Mathematics of Smith College in partial fulfillment of the requirements for the degree of Bachelor of Arts with Honors Elizabeth Denne, Faculty Advisor April 15, 2008 i Acknowledgments First and foremost I would like to thank Elizabeth Denne for her guidance through this project. Her endless help and high expectations brought this project to where it stands. I would Like to thank David Cohen for his support thoughout this project and through- out my mathematical career. His humor, skepticism and advice is surely worth the $.25 fee. I would also like to thank my professors, peers, housemates, and friends, particularly Kelsey Hattam and Katy Gerecht, for supporting me throughout the year, and especially for tolerating my temporary insanity during the final weeks of writing. Contents 1 Introduction 1 2 Defining Knots and Links 3 2.1 KnotDiagramsandKnotEquivalence . ... 3 2.2 Links, Orientation, and Connected Sum . ..... 8 3 Seifert Surfaces and Knot Genus 12 3.1 SeifertSurfaces ................................. 12 3.2 Surgery ...................................... 14 3.3 Knot Genus and Factorization . 16 3.4 Linkingnumber.................................. 17 3.5 Homology ..................................... 19 3.6 TheSeifertMatrix ................................ 21 3.7 TheAlexanderPolynomial. 27 4 Resolving Trees 31 4.1 Resolving Trees and the Conway Polynomial . ..... 31 4.2 TheAlexanderPolynomial. 34 5 Algebraic and Topological Tools 36 5.1 FreeGroupsandQuotients . 36 5.2 TheFundamentalGroup. .. .. .. .. .. .. .. .. 40 ii iii 6 Knot Groups 49 6.1 TwoPresentations ................................ 49 6.2 The Fundamental Group of the Knot Complement . 54 7 The Fox Calculus and Alexander Ideals 59 7.1 TheFreeCalculus ............................... -
NON-POSITIVELY CURVED CUBE COMPLEXES Henry Wilton Last Updated: June 29, 2012
Cube complexes Winter 2011 NON-POSITIVELY CURVED CUBE COMPLEXES Henry Wilton Last updated: June 29, 2012 1 Some basics of topological and geometric group theory 1.1 Presentations and complexes Let Γ be a discrete group, defined by a presentation P = hai j rji, say, or as the fundamental group of a connected CW-complex X. 0 1 Remark 1.1. Let XP be the CW-complex with a single 0-cell E , one 1-cell Ei 2 for each ai (oriented accordingly), and one 2-cell Ej for each rj, with attaching 2 (1) map @Ej ! XP that reads off the word rj in the generators faig. Then, by the Seifert{van Kampen Theorem, the fundamental group of XP is the group presented by P. Conversely, restricting attention to X(2) and contracting a maximal tree in (1) X , we obtain XP for some presentation P of Γ. So we see that these two situations are equivalent. We will call XP a pre- sentation complex for Γ. Here are two typical questions that we might want to be able to answer about Γ. ∗ Question 1.2. Can we compute the homology H∗(Γ) or cohomology H (Γ)? Question 1.3. Can we solve the word problem in Γ? That is, is there an algorithm to determine whether a word in the generators represents the trivial element in Γ? 1.2 Non-positively curved manifolds In general, the answer to both of these questions is `no'. Novikov (1955) and Boone (1959) constructed a finitely presentable group with an unsolvable word problem, and Cameron Gordon proved that the rank of H2(Γ) is not computable. -
1 SL2(R) Problems 1 2 1
1 SL2(R) problems 1 2 1. Show that SL2(R) is homeomorphic to S R . 2 × Hint 1: SL2(R) acts on R 0 transitively with stabilizer of (1, 0) equal − to the set of upper triangular matrices with 1’s on the diagonal, which is homeomorphic to R. This gives a “principal bundle”, i.e. a map 2 SL2(R) R 0 whose point inverses are lines. Construct a section 2 → − R 0 SL2(R) and then a homeomorphism to the product. − → Hint 2: Same thing, but for the action of SL2(R) on the upper half plane, with point inverses circles. Better yet (but uses more sophisticated math): PSL2(R)= SL2(R)/ I can be identified with the unit tangent bundle 2 ± 2 T1H of the hyperbolic plane H , which in turn can be identified with 2 1 H S∞, product with the circle at infinity. This shows that PSL2(R) × 1 2 is homeomorphic to S R , but SL2(R) is a double cover of PSL2(R). × 2. We have seen that trace Tr(A) determines the conjugacy class of A ∈ SL2(R) provided Tr(A) > 2. Show that this conjugacy class, as a subset | | 1 of SL2(R), is a closed subset homeomorphic to S R. × 3. The set of parabolics is not closed and consists of four conjugacy classes (two with trace 2 and two with trace 2), each homeomorphic to S1 R. − × 4. There are two conjugacy classes of elements of SL2(R) whose trace is a given number in ( 2, 2), each closed and homeomorphic to R2. -
Covering Spaces and the Fundamental Group
FUNDAMENTAL GROUPS AND COVERING SPACES NICK SALTER, SUBHADIP CHOWDHURY 1. Preamble The following exercises are intended to introduce you to some of the basic ideas in algebraic topology, namely the fundamental group and covering spaces. Exercises marked with a (B) are basic and fundamental. If you are new to the subject, your time will probably be best spent digesting the (B) exercises. If you have some familiarity with the material, you are under no obligation to attempt the (B) exercises, but you should at least convince yourself that you know how to do them. If you are looking to focus on specific ideas and techniques, I've made some attempt to label exercises with the sorts of ideas involved. 2. Two extremely important theorems If you get nothing else out of your quarter of algebraic topology, you should know and understand the following two theorems. The exercises on this sheet (mostly) exclusively rely on them, along with the ability to reason spatially and geometrically. Theorem 2.1 (Seifert-van Kampen). Let X be a topological space and assume that X = U[V with U; V ⊂ X open such that U \ V is path connected and let x0 2 U \ V . We consider the commutative diagram π1(U; x0) i1∗ j1∗ π1(U \ V; x0) π1(X; x0) i2∗ j2∗ π1(V; x0) where all maps are induced by the inclusions. Then for any group H and any group homomor- phisms φ1 : π1(U; x0) ! H and φ2 : π1(V; x0) ! H such that φ1i1∗ = φ2i2∗; Date: September 20, 2015. 1 2 NICK SALTER, SUBHADIP CHOWDHURY there exists a unique group homomorphism φ : π1(X; x0) ! H such that φ1 = φj1∗, φ2 = φj2∗. -
TDA: Lecture 2 Complexes and Filtrations
TDA: Lecture 2 Complexes and filtrations Wesley Hamilton University of North Carolina - Chapel Hill [email protected] 02/05/20 1 / 89 1 Complexes Simplicial complexes Cubical complexes 2 Filtrations 3 From data to filtrations Vietoris-Rips filtrations Sub/superlevel set filtrations Alpha shapes 2 / 89 Complexes Complexes Complexes are the combinatorial building blocks used in TDA. The two types of complexes we'll focus on are simplicial complexes and cubical complexes. Simplicial complexes are easier to \construct" with, and more common in algebraic topology. Cubical complexes are better adapted to image/pixelated/voxel data. 3 / 89 Complexes Simplicial complexes Simplices A geometric n-simplex σ is the set ( n ) X X σ = ai ei : ai = 1; ai ≥ 0 ; i=0 i n where the ei are standard basis vectors for R . 4 / 89 Complexes Simplicial complexes Examples of simplices, dim 0 and 1 5 / 89 Complexes Simplicial complexes Examples of simplices, dim 2 and 3 6 / 89 Complexes Simplicial complexes Simplices and barycentric coordinates Given an n-simplex σ, we can specify points within σ using barycentric coordinates: n X (a0; :::; a1) 7! ai ei 2 σ: i=0 7 / 89 The jth face of a simplex σ is the set @j σ. Complexes Simplicial complexes Faces and facets The face maps @j , defined on n-simplexes, are the restriction of a simplex σ's barycentric coordinates to aj = 0, i.e. n X X @j σ = f ai ei : ai = 1; ai ≥ 0; aj = 0g: i=0 i 8 / 89 Complexes Simplicial complexes Faces and facets The face maps @j , defined on n-simplexes, are the restriction of a simplex σ's barycentric coordinates to aj = 0, i.e. -
The Topology, Geometry, and Dynamics of Free Groups
The topology, geometry, and dynamics of free groups (containing most of): Part I: Outer space, fold paths, and the Nielsen/Whitehead problems Lee Mosher January 10, 2020 arXiv:2001.02776v1 [math.GR] 8 Jan 2020 Contents I Outer space, fold paths, and the Nielsen/Whitehead problems 9 1 Marked graphs and outer space: Topological and geometric structures for free groups 11 1.1 Free groups and free bases . 11 1.2 Graphs . 14 1.2.1 Graphs, trees, and their paths. 14 1.2.2 Trees are contractible . 22 1.2.3 Roses. 24 1.2.4 Notational conventions for the free group Fn............. 25 1.3 The Nielsen/Whitehead problems: free bases. 25 1.3.1 Statements of the central problems. 26 1.3.2 Negative tests, using abelianization. 27 1.3.3 Positive tests, using Nielsen transformations. 28 1.3.4 Topological versions of the central problems, using roses. 29 1.4 Marked graphs . 31 1.4.1 Core graphs and their ranks. 32 1.4.2 Based marked graphs. 34 1.4.3 Based marked graphs and Aut(Fn). 35 1.4.4 The Nielsen-Whitehead problems: Based marked graphs. 36 1.4.5 Marked graphs. 37 1.4.6 Marked graphs and Out(Fn). ..................... 38 1.4.7 Equivalence of marked graphs . 40 1.4.8 Applications: Conjugacy classes in free groups and circuits in marked graphs . 42 1.5 Finite subgroups of Out(Fn): Applying marked graphs . 44 1.5.1 Automorphism groups of finite graphs . 45 1.5.2 Automorphisms act faithfully on homology .