Introduction to Simplicial Homology (Work in Progress)
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K-THEORY of FUNCTION RINGS Theorem 7.3. If X Is A
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 69 (1990) 33-50 33 North-Holland K-THEORY OF FUNCTION RINGS Thomas FISCHER Johannes Gutenberg-Universitit Mainz, Saarstrage 21, D-6500 Mainz, FRG Communicated by C.A. Weibel Received 15 December 1987 Revised 9 November 1989 The ring R of continuous functions on a compact topological space X with values in IR or 0Z is considered. It is shown that the algebraic K-theory of such rings with coefficients in iZ/kH, k any positive integer, agrees with the topological K-theory of the underlying space X with the same coefficient rings. The proof is based on the result that the map from R6 (R with discrete topology) to R (R with compact-open topology) induces a natural isomorphism between the homologies with coefficients in Z/kh of the classifying spaces of the respective infinite general linear groups. Some remarks on the situation with X not compact are added. 0. Introduction For a topological space X, let C(X, C) be the ring of continuous complex valued functions on X, C(X, IR) the ring of continuous real valued functions on X. KU*(X) is the topological complex K-theory of X, KO*(X) the topological real K-theory of X. The main goal of this paper is the following result: Theorem 7.3. If X is a compact space, k any positive integer, then there are natural isomorphisms: K,(C(X, C), Z/kZ) = KU-‘(X, Z’/kZ), K;(C(X, lR), Z/kZ) = KO -‘(X, UkZ) for all i 2 0. -
Some Notes About Simplicial Complexes and Homology II
Some notes about simplicial complexes and homology II J´onathanHeras J. Heras Some notes about simplicial homology II 1/19 Table of Contents 1 Simplicial Complexes 2 Chain Complexes 3 Differential matrices 4 Computing homology groups from Smith Normal Form J. Heras Some notes about simplicial homology II 2/19 Simplicial Complexes Table of Contents 1 Simplicial Complexes 2 Chain Complexes 3 Differential matrices 4 Computing homology groups from Smith Normal Form J. Heras Some notes about simplicial homology II 3/19 Simplicial Complexes Simplicial Complexes Definition Let V be an ordered set, called the vertex set. A simplex over V is any finite subset of V . Definition Let α and β be simplices over V , we say α is a face of β if α is a subset of β. Definition An ordered (abstract) simplicial complex over V is a set of simplices K over V satisfying the property: 8α 2 K; if β ⊆ α ) β 2 K Let K be a simplicial complex. Then the set Sn(K) of n-simplices of K is the set made of the simplices of cardinality n + 1. J. Heras Some notes about simplicial homology II 4/19 Simplicial Complexes Simplicial Complexes 2 5 3 4 0 6 1 V = (0; 1; 2; 3; 4; 5; 6) K = f;; (0); (1); (2); (3); (4); (5); (6); (0; 1); (0; 2); (0; 3); (1; 2); (1; 3); (2; 3); (3; 4); (4; 5); (4; 6); (5; 6); (0; 1; 2); (4; 5; 6)g J. Heras Some notes about simplicial homology II 5/19 Chain Complexes Table of Contents 1 Simplicial Complexes 2 Chain Complexes 3 Differential matrices 4 Computing homology groups from Smith Normal Form J. -
Fixed Point Theory
Andrzej Granas James Dugundji Fixed Point Theory With 14 Illustrations %1 Springer Contents Preface vii §0. Introduction 1 1. Fixed Point Spaces 1 2. Forming New Fixed Point Spaces from Old 3 3. Topological Transversality 4 4. Factorization Technique 6 I. Elementary Fixed Point Theorems §1. Results Based on Completeness 9 1. Banach Contraction Principle 9 2. Elementary Domain Invariance 11 3. Continuation Method for Contractive Maps 12 4. Nonlinear Alternative for Contractive Maps 13 5. Extensions of the Banach Theorem 15 6. Miscellaneous Results and Examples 17 7. Notes and Comments 23 §2. Order-Theoretic Results 25 1. The Knaster-Tarski Theorem 25 2. Order and Completeness. Theorem of Bishop-Phelps 26 3. Fixed Points for Set-Valued Contractive Maps 28 4. Applications to Geometry of Banach Spaces 29 5. Applications to the Theory of Critical Points 30 6. Miscellaneous Results and Examples 31 7. Notes and Comments 34 X Contents §3. Results Based on Convexity 37 1. KKM-Maps and the Geometric KKM-Principle 37 2. Theorem of von Neumann and Systems of Inequalities 40 3. Fixed Points of Affine Maps. Markoff-Kakutani Theorem 42 4. Fixed Points for Families of Maps. Theorem of Kakutani 44 5. Miscellaneous Results and Examples 46 6. Notes and Comments 48 §4. Further Results and Applications 51 1. Nonexpansive Maps in Hilbert Space 51 2. Applications of the Banach Principle to Integral and Differential Equations 55 3. Applications of the Elementary Domain Invariance 57 4. Elementary KKM-Principle and its Applications 64 5. Theorems of Mazur-Orlicz and Hahn-Banach 70 6. -
1 CW Complex, Cellular Homology/Cohomology
Our goal is to develop a method to compute cohomology algebra and rational homotopy group of fiber bundles. 1 CW complex, cellular homology/cohomology Definition 1. (Attaching space with maps) Given topological spaces X; Y , closed subset A ⊂ X, and continuous map f : A ! y. We define X [f Y , X t Y / ∼ n n n−1 n where x ∼ y if x 2 A and f(x) = y. In the case X = D , A = @D = S , D [f X is said to be obtained by attaching to X the cell (Dn; f). n−1 n n Proposition 1. If f; g : S ! X are homotopic, then D [f X and D [g X are homotopic. Proof. Let F : Sn−1 × I ! X be the homotopy between f; g. Then in fact n n n D [f X ∼ (D × I) [F X ∼ D [g X Definition 2. (Cell space, cell complex, cellular map) 1. A cell space is a topological space obtained from a finite set of points by iterating the procedure of attaching cells of arbitrary dimension, with the condition that only finitely many cells of each dimension are attached. 2. If each cell is attached to cells of lower dimension, then the cell space X is called a cell complex. Define the n−skeleton of X to be the subcomplex consisting of cells of dimension less than n, denoted by Xn. 3. A continuous map f between cell complexes X; Y is called cellular if it sends Xk to Yk for all k. Proposition 2. 1. Every cell space is homotopic to a cell complex. -
Local Homology of Abstract Simplicial Complexes
Local homology of abstract simplicial complexes Michael Robinson Chris Capraro Cliff Joslyn Emilie Purvine Brenda Praggastis Stephen Ranshous Arun Sathanur May 30, 2018 Abstract This survey describes some useful properties of the local homology of abstract simplicial complexes. Although the existing literature on local homology is somewhat dispersed, it is largely dedicated to the study of manifolds, submanifolds, or samplings thereof. While this is a vital per- spective, the focus of this survey is squarely on the local homology of abstract simplicial complexes. Our motivation comes from the needs of the analysis of hypergraphs and graphs. In addition to presenting many classical facts in a unified way, this survey presents a few new results about how local homology generalizes useful tools from graph theory. The survey ends with a statistical comparison of graph invariants with local homology. Contents 1 Introduction 2 2 Historical discussion 4 3 Theoretical groundwork 6 3.1 Representing data with simplicial complexes . .7 3.2 Relative simplicial homology . .8 3.3 Local homology . .9 arXiv:1805.11547v1 [math.AT] 29 May 2018 4 Local homology of graphs 12 4.1 Basic graph definitions . 14 4.2 Local homology at a vertex . 15 5 Local homology of general complexes 18 5.1 Stratification detection . 19 5.2 Neighborhood filtration . 24 6 Computational considerations 25 1 7 Statistical comparison with graph invariants 26 7.1 Graph invariants used in our comparison . 27 7.2 Comparison methodology . 27 7.3 Dataset description . 28 7.4 The Karate graph . 28 7.5 The Erd}os-R´enyi graph . 30 7.6 The Barabasi-Albert graph . -
Zuoqin Wang Time: March 25, 2021 the QUOTIENT TOPOLOGY 1. The
Topology (H) Lecture 6 Lecturer: Zuoqin Wang Time: March 25, 2021 THE QUOTIENT TOPOLOGY 1. The quotient topology { The quotient topology. Last time we introduced several abstract methods to construct topologies on ab- stract spaces (which is widely used in point-set topology and analysis). Today we will introduce another way to construct topological spaces: the quotient topology. In fact the quotient topology is not a brand new method to construct topology. It is merely a simple special case of the co-induced topology that we introduced last time. However, since it is very concrete and \visible", it is widely used in geometry and algebraic topology. Here is the definition: Definition 1.1 (The quotient topology). (1) Let (X; TX ) be a topological space, Y be a set, and p : X ! Y be a surjective map. The co-induced topology on Y induced by the map p is called the quotient topology on Y . In other words, −1 a set V ⊂ Y is open if and only if p (V ) is open in (X; TX ). (2) A continuous surjective map p :(X; TX ) ! (Y; TY ) is called a quotient map, and Y is called the quotient space of X if TY coincides with the quotient topology on Y induced by p. (3) Given a quotient map p, we call p−1(y) the fiber of p over the point y 2 Y . Note: by definition, the composition of two quotient maps is again a quotient map. Here is a typical way to construct quotient maps/quotient topology: Start with a topological space (X; TX ), and define an equivalent relation ∼ on X. -
Simplicial Complexes and Lefschetz Fixed-Point Theorem
Helsingin Yliopisto Matemaattis-luonnontieteellinen tiedekunta Matematiikan ja tilastotieteen osasto Pro gradu -tutkielma Simplicial complexes and Lefschetz fixed-point theorem Aku Siekkinen Ohjaaja: Marja Kankaanrinta October 19, 2019 HELSINGIN YLIOPISTO — HELSINGFORS UNIVERSITET — UNIVERSITY OF HELSINKI Tiedekunta/Osasto — Fakultet/Sektion — Faculty Laitos — Institution — Department Matemaattis-luonnontieteellinen Matematiikan ja tilastotieteen laitos Tekijä — Författare — Author Aku Siekkinen Työn nimi — Arbetets titel — Title Simplicial complexes and Lefschetz fixed-point theorem Oppiaine — Läroämne — Subject Matematiikka Työn laji — Arbetets art — Level Aika — Datum — Month and year Sivumäärä — Sidoantal — Number of pages Pro gradu -tutkielma Lokakuu 2019 82 s. Tiivistelmä — Referat — Abstract We study a subcategory of topological spaces called polyhedrons. In particular, the work focuses on simplicial complexes out of which polyhedrons are constructed. With simplicial complexes we can calculate the homology groups of polyhedrons. These are computationally easier to handle compared to singular homology groups. We start by introducing simplicial complexes and simplicial maps. We show how polyhedrons and simplicial complexes are related. Simplicial maps are certain maps between simplicial complexes. These can be transformed to piecewise linear maps between polyhedrons. We prove the simplicial approximation theorem which states that for any continuous function between polyhedrons we can find a piecewise linear map which is homotopic to the continuous function. In section 4 we study simplicial homology groups. We prove that on polyhedrons the simplicial homology groups coincide with singular homology groups. Next we give an algorithm for calculating the homology groups from matrix presentations of boundary homomorphisms. Also examples of these calculations are given for some polyhedrons. In the last section, we assign an integer called the Lefschetz number for continuous maps from a polyhedron to itself. -
Introduction to Homology
Introduction to Homology Matthew Lerner-Brecher and Koh Yamakawa March 28, 2019 Contents 1 Homology 1 1.1 Simplices: a Review . .2 1.2 ∆ Simplices: not a Review . .2 1.3 Boundary Operator . .3 1.4 Simplicial Homology: DEF not a Review . .4 1.5 Singular Homology . .5 2 Higher Homotopy Groups and Hurweicz Theorem 5 3 Exact Sequences 5 3.1 Key Definitions . .5 3.2 Recreating Groups From Exact Sequences . .6 4 Long Exact Homology Sequences 7 4.1 Exact Sequences of Chain Complexes . .7 4.2 Relative Homology Groups . .8 4.3 The Excision Theorems . .8 4.4 Mayer-Vietoris Sequence . .9 4.5 Application . .9 1 Homology What is Homology? To put it simply, we use Homology to count the number of n dimensional holes in a topological space! In general, our approach will be to add a structure on a space or object ( and thus a topology ) and figure out what subsets of the space are cycles, then sort through those subsets that are holes. Of course, as many properties we care about in topology, this property is invariant under homotopy equivalence. This is the slightly weaker than homeomorphism which we before said gave us the same fundamental group. 1 Figure 1: Hatcher p.100 Just for reference to you, I will simply define the nth Homology of a topological space X. Hn(X) = ker @n=Im@n−1 which, as we have said before, is the group of n-holes. 1.1 Simplices: a Review k+1 Just for your sake, we review what standard K simplices are, as embedded inside ( or living in ) R ( n ) k X X ∆ = [v0; : : : ; vk] = xivi such that xk = 1 i=0 For example, the 0 simplex is a point, the 1 simplex is a line, the 2 simplex is a triangle, the 3 simplex is a tetrahedron. -
Homology Groups of Homeomorphic Topological Spaces
An Introduction to Homology Prerna Nadathur August 16, 2007 Abstract This paper explores the basic ideas of simplicial structures that lead to simplicial homology theory, and introduces singular homology in order to demonstrate the equivalence of homology groups of homeomorphic topological spaces. It concludes with a proof of the equivalence of simplicial and singular homology groups. Contents 1 Simplices and Simplicial Complexes 1 2 Homology Groups 2 3 Singular Homology 8 4 Chain Complexes, Exact Sequences, and Relative Homology Groups 9 ∆ 5 The Equivalence of H n and Hn 13 1 Simplices and Simplicial Complexes Definition 1.1. The n-simplex, ∆n, is the simplest geometric figure determined by a collection of n n + 1 points in Euclidean space R . Geometrically, it can be thought of as the complete graph on (n + 1) vertices, which is solid in n dimensions. Figure 1: Some simplices Extrapolating from Figure 1, we see that the 3-simplex is a tetrahedron. Note: The n-simplex is topologically equivalent to Dn, the n-ball. Definition 1.2. An n-face of a simplex is a subset of the set of vertices of the simplex with order n + 1. The faces of an n-simplex with dimension less than n are called its proper faces. 1 Two simplices are said to be properly situated if their intersection is either empty or a face of both simplices (i.e., a simplex itself). By \gluing" (identifying) simplices along entire faces, we get what are known as simplicial complexes. More formally: Definition 1.3. A simplicial complex K is a finite set of simplices satisfying the following condi- tions: 1 For all simplices A 2 K with α a face of A, we have α 2 K. -
Arxiv:1812.07604V1 [Math.AT] 18 Dec 2018 Ons Hsvledosa H Ieo H Oe of Model the of Size the As Drops Value This Points
THE TOPOLOGICAL COMPLEXITY OF FINITE MODELS OF SPHERES SHELLEY KANDOLA Abstract. In [2], Farber defined topological complexity (TC) to be the mini- mal number of continuous motion planning rules required to navigate between any two points in a topological space. Papers by [4] and [3] define notions of topological complexity for simplicial complexes. In [9], Tanaka defines a notion of topological complexity, called combinatorial complexity, for finite topologi- cal spaces. As is common with papers discussing topological complexity, each includes a computation of the TC of some sort of circle. In this paper, we compare the TC of models of S1 across each definition, exhibiting some of the nuances of TC that become apparent in the finite setting. In particular, we show that the TC of finite models of S1 can be 3 or 4 and that the TC of the minimal finite model of any n-sphere is equal to 4. Furthermore, we exhibit spaces weakly homotopy equivalent to a wedge of circles with arbitrarily high TC. 1. Introduction Farber introduced the notion of topological complexity in [2] as it relates to motion planning in robotics. Informally, the topological complexity of a robot’s space of configurations represents the minimal number of continuous motion plan- ning rules required to instruct that robot to move from one position into another position. Although topological complexity was originally defined for robots with a smooth, infinite range of motion (e.g. products of spheres or real projective space), it makes sense to consider the topological complexity of finite topological spaces. For example, one could determine the topological complexity of a finite state machine or a robot powered by stepper motors. -
Regular Cell Complexes in Total Positivity
REGULAR CELL COMPLEXES IN TOTAL POSITIVITY PATRICIA HERSH Abstract. Fomin and Shapiro conjectured that the link of the identity in the Bruhat stratification of the totally nonnegative real part of the unipotent radical of a Borel subgroup in a semisimple, simply connected algebraic group defined and split over R is a reg- ular CW complex homeomorphic to a ball. The main result of this paper is a proof of this conjecture. This completes the solution of the question of Bernstein of identifying regular CW complexes arising naturally from representation theory having the (lower) in- tervals of Bruhat order as their closure posets. A key ingredient is a new criterion for determining whether a finite CW complex is regular with respect to a choice of characteristic maps; it most naturally applies to images of maps from regular CW complexes and is based on an interplay of combinatorics of the closure poset with codimension one topology. 1. Introduction In this paper, the following conjecture of Sergey Fomin and Michael Shapiro from [12] is proven. Conjecture 1.1. Let Y be the link of the identity in the totally non- negative real part of the unipotent radical of a Borel subgroup B in a semisimple, simply connected algebraic group defined and split over R. − − Let Bu = B uB for u in the Weyl group W . Then the stratification of Y into Bruhat cells Y \ Bu is a regular CW decomposition. More- over, for each w 2 W , Yw = [u≤wY \ Bu is a regular CW complex homeomorphic to a ball, as is the link of each of its cells. -
A Radius Sphere Theorem
Invent. math. 112, 577-583 (1993) Inventiones mathematicae Springer-Verlag1993 A radius sphere theorem Karsten Grove 1'* and Peter Petersen 2'** 1 Department of Mathematics, University of Maryland, College Park, MD 20742, USA 2 Department of Mathematics, University of California, Los Angeles, CA 90024-1555, USA Oblatum 9-X-1992 Introduction The purpose of this paper is to present an optimal sphere theorem for metric spaces analogous to the celebrated Rauch-Berger-Klingenberg Sphere Theorem and the Diameter Sphere Theorem in Riemannian geometry. There has lately been considerable interest in studying spaces which are more singular than Riemannian manifolds. A natural reason for doing this is because Gromov-Hausdorff limits of Riemannian manifolds are almost never Riemannian manifolds, but usually only inner metric spaces with various nice properties. The kind of spaces we wish to study here are the so-called Alexandrov spaces. Alexan- drov spaces are finite Hausdorff dimensional inner metric spaces with a lower curvature bound in the distance comparison sense. The structure of Alexandrov spaces was studied in [BGP], [P1] and [P]. We point out that the curvature assumption implies that the Hausdorff dimension is equal to the topological dimension. Moreover, if X is an Alexandrov space and peX then the space of directions Zp at p is an Alexandrov space of one less dimension and with curvature > 1. Furthermore a neighborhood of p in X is homeomorphic to the linear cone over Zv- One of the important implications of this is that the local structure of n-dimensional Alexandrov spaces is determined by the structure of (n - 1)-dimen- sional Alexandrov spaces with curvature > 1.