HOMOTOPY THEORY for BEGINNERS Contents 1. Notation
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
TOPOLOGY QUALIFYING EXAM May 14, 2016 Examiners: Dr
Department of Mathematics and Statistics University of South Florida TOPOLOGY QUALIFYING EXAM May 14, 2016 Examiners: Dr. M. Elhamdadi, Dr. M. Saito Instructions: For Ph.D. level, complete at least seven problems, at least three problems from each section. For Master’s level, complete at least five problems, at least one problem from each section. State all theorems or lemmas you use. Section I: POINT SET TOPOLOGY 1. Let K = 1/n n N R.Let K = (a, b), (a, b) K a, b R,a<b . { | 2 }⇢ B { \ | 2 } Show that K forms a basis of a topology, RK ,onR,andthatRK is strictly finer than B the standard topology of R. 2. Recall that a space X is countably compact if any countable open covering of X contains afinitesubcovering.ProvethatX is countably compact if and only if every nested sequence C C of closed nonempty subsets of X has a nonempty intersection. 1 ⊃ 2 ⊃··· 3. Assume that all singletons are closed in X.ShowthatifX is regular, then every pair of points of X has neighborhoods whose closures are disjoint. 4. Let X = R2 (0,n) n Z (integer points on the y-axis are removed) and Y = \{ | 2 } R2 (0,y) y R (the y-axis is removed) be subspaces of R2 with the standard topology.\{ Prove| 2 or disprove:} There exists a continuous surjection X Y . ! 5. Let X and Y be metrizable spaces with metrics dX and dY respectively. Let f : X Y be a function. Prove that f is continuous if and only if given x X and ✏>0, there! exists δ>0suchthatd (x, y) <δ d (f(x),f(y)) <✏. -
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. -
On Contractible J-Saces
IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1867 On Contractible J-Saces Narjis A. Dawood [email protected] Dept. of Mathematics / College of Education for Pure Science/ Ibn Al – Haitham- University of Baghdad Suaad G. Gasim [email protected] Dept. of Mathematics / College of Education for Pure Science/ Ibn Al – Haitham- University of Baghdad Abstract Jordan curve theorem is one of the classical theorems of mathematics, it states the following : If C is a graph of a simple closed curve in the complex plane the complement of C is the union of two regions, C being the common boundary of the two regions. One of the region is bounded and the other is unbounded. We introduced in this paper one of Jordan's theorem generalizations. A new type of space is discussed with some properties and new examples. This new space called Contractible J-space. Key words : Contractible J- space, compact space and contractible map. For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/ www.ihsciconf.org Mathematics |330 IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1867 1. Introduction Recall the Jordan curve theorem which states that, if C is a simple closed curve in the plane ℝ, then ℝ\C is disconnected and consists of two components with C as their common boundary, exactly one of these components is bounded (see, [1]). Many generalizations of Jordan curve theorem are discussed by many researchers, for example not limited, we recall some of these generalizations. -
Sheaves and Homotopy Theory
SHEAVES AND HOMOTOPY THEORY DANIEL DUGGER The purpose of this note is to describe the homotopy-theoretic version of sheaf theory developed in the work of Thomason [14] and Jardine [7, 8, 9]; a few enhancements are provided here and there, but the bulk of the material should be credited to them. Their work is the foundation from which Morel and Voevodsky build their homotopy theory for schemes [12], and it is our hope that this exposition will be useful to those striving to understand that material. Our motivating examples will center on these applications to algebraic geometry. Some history: The machinery in question was invented by Thomason as the main tool in his proof of the Lichtenbaum-Quillen conjecture for Bott-periodic algebraic K-theory. He termed his constructions `hypercohomology spectra', and a detailed examination of their basic properties can be found in the first section of [14]. Jardine later showed how these ideas can be elegantly rephrased in terms of model categories (cf. [8], [9]). In this setting the hypercohomology construction is just a certain fibrant replacement functor. His papers convincingly demonstrate how many questions concerning algebraic K-theory or ´etale homotopy theory can be most naturally understood using the model category language. In this paper we set ourselves the specific task of developing some kind of homotopy theory for schemes. The hope is to demonstrate how Thomason's and Jardine's machinery can be built, step-by-step, so that it is precisely what is needed to solve the problems we encounter. The papers mentioned above all assume a familiarity with Grothendieck topologies and sheaf theory, and proceed to develop the homotopy-theoretic situation as a generalization of the classical case. -
Properties of Digital N-Manifolds and Simply Connected Spaces
The Jordan-Brouwer theorem for the digital normal n-space Zn. Alexander V. Evako. [email protected] Abstract. In this paper we investigate properties of digital spaces which are represented by graphs. We find conditions for digital spaces to be digital n-manifolds and n-spheres. We study properties of partitions of digital spaces and prove a digital analog of the Jordan-Brouwer theorem for the normal digital n-space Zn. Key Words: axiomatic digital topology, graph, normal space, partition, Jordan surface theorem 1. Introduction. A digital approach to geometry and topology plays an important role in analyzing n-dimensional digitized images arising in computer graphics as well as in many areas of science including neuroscience and medical imaging. Concepts and results of the digital approach are used to specify and justify some important low- level image processing algorithms, including algorithms for thinning, boundary extraction, object counting, and contour filling. To study properties of digital images, it is desirable to construct a convenient digital n-space modeling Euclidean n-space. For 2D and 3D grids, a solution was offered first by A. Rosenfeld [18] and developed in a number of papers, just mention [1-2,13,15-16], but it can be seen from the vast amount of literature that generalization to higher dimensions is not easy. Even more, as T. Y. Kong indicates in [14], there are problems even for the 2D and 3D grids. For example, different adjacency relations are used to define connectedness on a set of grid points and its complement. Another approach to overcome connectivity paradoxes and some other apparent contradictions is to build a digital space by using basic structures from algebraic topology such as cell complexes or simplicial complexes, see e.g. -
The Boundary Manifold of a Complex Line Arrangement
Geometry & Topology Monographs 13 (2008) 105–146 105 The boundary manifold of a complex line arrangement DANIEL CCOHEN ALEXANDER ISUCIU We study the topology of the boundary manifold of a line arrangement in CP2 , with emphasis on the fundamental group G and associated invariants. We determine the Alexander polynomial .G/, and more generally, the twisted Alexander polynomial associated to the abelianization of G and an arbitrary complex representation. We give an explicit description of the unit ball in the Alexander norm, and use it to analyze certain Bieri–Neumann–Strebel invariants of G . From the Alexander polynomial, we also obtain a complete description of the first characteristic variety of G . Comparing this with the corresponding resonance variety of the cohomology ring of G enables us to characterize those arrangements for which the boundary manifold is formal. 32S22; 57M27 For Fred Cohen on the occasion of his sixtieth birthday 1 Introduction 1.1 The boundary manifold Let A be an arrangement of hyperplanes in the complex projective space CPm , m > 1. Denote by V S H the corresponding hypersurface, and by X CPm V its D H A D n complement. Among2 the origins of the topological study of arrangements are seminal results of Arnol’d[2] and Cohen[8], who independently computed the cohomology of the configuration space of n ordered points in C, the complement of the braid arrangement. The cohomology ring of the complement of an arbitrary arrangement A is by now well known. It is isomorphic to the Orlik–Solomon algebra of A, see Orlik and Terao[34] as a general reference. -
Graph Manifolds and Taut Foliations Mark Brittenham1, Ramin Naimi, and Rachel Roberts2 Introduction
Graph manifolds and taut foliations 1 2 Mark Brittenham , Ramin Naimi, and Rachel Roberts UniversityofTexas at Austin Abstract. We examine the existence of foliations without Reeb comp onents, taut 1 1 foliations, and foliations with no S S -leaves, among graph manifolds. We show that each condition is strictly stronger than its predecessors, in the strongest p os- sible sense; there are manifolds admitting foliations of eachtyp e which do not admit foliations of the succeeding typ es. x0 Introduction Taut foliations have b een increasingly useful in understanding the top ology of 3-manifolds, thanks largely to the work of David Gabai [Ga1]. Many 3-manifolds admit taut foliations [Ro1],[De],[Na1], although some do not [Br1],[Cl]. To date, however, there are no adequate necessary or sucient conditions for a manifold to admit a taut foliation. This pap er seeks to add to this confusion. In this pap er we study the existence of taut foliations and various re nements, among graph manifolds. What we show is that there are many graph manifolds which admit foliations that are as re ned as wecho ose, but which do not admit foliations admitting any further re nements. For example, we nd manifolds which admit foliations without Reeb comp onents, but no taut foliations. We also nd 0 manifolds admitting C foliations with no compact leaves, but which do not admit 2 any C such foliations. These results p oint to the subtle nature b ehind b oth top ological and analytical assumptions when dealing with foliations. A principal motivation for this work came from a particularly interesting exam- ple; the manifold M obtained by 37/2 Dehn surgery on the 2,3,7 pretzel knot K . -
Local Topological Properties of Asymptotic Cones of Groups
Local topological properties of asymptotic cones of groups Greg Conner and Curt Kent October 8, 2018 Abstract We define a local analogue to Gromov’s loop division property which is use to give a sufficient condition for an asymptotic cone of a complete geodesic metric space to have uncountable fundamental group. As well, this property is used to understand the local topological structure of asymptotic cones of many groups currently in the literature. Contents 1 Introduction 1 1.1 Definitions...................................... 3 2 Coarse Loop Division Property 5 2.1 Absolutely non-divisible sequences . .......... 10 2.2 Simplyconnectedcones . .. .. .. .. .. .. .. 11 3 Examples 15 3.1 An example of a group with locally simply connected cones which is not simply connected ........................................ 17 1 Introduction Gromov [14, Section 5.F] was first to notice a connection between the homotopic properties of asymptotic cones of a finitely generated group and algorithmic properties of the group: if arXiv:1210.4411v1 [math.GR] 16 Oct 2012 all asymptotic cones of a finitely generated group are simply connected, then the group is finitely presented, its Dehn function is bounded by a polynomial (hence its word problem is in NP) and its isodiametric function is linear. A version of that result for higher homotopy groups was proved by Riley [25]. The converse statement does not hold: there are finitely presented groups with non-simply connected asymptotic cones and polynomial Dehn functions [1], [26], and even with polynomial Dehn functions and linear isodiametric functions [21]. A partial converse statement was proved by Papasoglu [23]: a group with quadratic Dehn function has all asymptotic cones simply connected (for groups with subquadratic Dehn functions, i.e. -
MTH 304: General Topology Semester 2, 2017-2018
MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds.......................... -
Compact Non-Hausdorff Manifolds
A.G.M. Hommelberg Compact non-Hausdorff Manifolds Bachelor Thesis Supervisor: dr. D. Holmes Date Bachelor Exam: June 5th, 2014 Mathematisch Instituut, Universiteit Leiden 1 Contents 1 Introduction 3 2 The First Question: \Can every compact manifold be obtained by glueing together a finite number of compact Hausdorff manifolds?" 4 2.1 Construction of (Z; TZ ).................................5 2.2 The space (Z; TZ ) is a Compact Manifold . .6 2.3 A Negative Answer to our Question . .8 3 The Second Question: \If (R; TR) is a compact manifold, is there always a surjective ´etale map from a compact Hausdorff manifold to R?" 10 3.1 Construction of (R; TR)................................. 10 3.2 Etale´ maps and Covering Spaces . 12 3.3 A Negative Answer to our Question . 14 4 The Fundamental Groups of (R; TR) and (Z; TZ ) 16 4.1 Seifert - Van Kampen's Theorem for Fundamental Groupoids . 16 4.2 The Fundamental Group of (R; TR)........................... 19 4.3 The Fundamental Group of (Z; TZ )........................... 23 2 Chapter 1 Introduction One of the simplest examples of a compact manifold (definitions 2.0.1 and 2.0.2) that is not Hausdorff (definition 2.0.3), is the space X obtained by glueing two circles together in all but one point. Since the circle is a compact Hausdorff manifold, it is easy to prove that X is indeed a compact manifold. It is also easy to show X is not Hausdorff. Many other examples of compact manifolds that are not Hausdorff can also be created by glueing together compact Hausdorff manifolds. -
On Stratifiable Spaces
Pacific Journal of Mathematics ON STRATIFIABLE SPACES CARLOS JORGE DO REGO BORGES Vol. 17, No. 1 January 1966 PACIFIC JOURNAL OF MATHEMATICS Vol. 17, No. 1, 1966 ON STRATIFIABLE SPACES CARLOS J. R. BORGES In the enclosed paper, it is shown that (a) the closed continuous image of a stratifiable space is stratifiable (b) the well-known extension theorem of Dugundji remains valid for stratifiable spaces (see Theorem 4.1, Pacific J. Math., 1 (1951), 353-367) (c) stratifiable spaces can be completely characterized in terms of continuous real-valued functions (d) the adjunction space of two stratifiable spaces is stratifiable (e) a topological space is stratifiable if and only if it is dominated by a collection of stratifiable subsets (f) a stratifiable space is metrizable if and only if it can be mapped to a metrizable space by a perfect map. In [4], J. G. Ceder studied various classes of topological spaces, called MΓspaces (ί = 1, 2, 3), obtaining excellent results, but leaving questions of major importance without satisfactory solutions. Here we propose to solve, in full generality, two of the most important questions to which he gave partial solutions (see Theorems 3.2 and 7.6 in [4]), as well as obtain new results.1 We will thus establish that Ceder's ikf3-spaces are important enough to deserve a better name and we propose to call them, henceforth, STRATIFIABLE spaces. Since we will exclusively work with stratifiable spaces, we now ex- hibit their definition. DEFINITION 1.1. A topological space X is a stratifiable space if X is T1 and, to each open UaX, one can assign a sequence {i7Λ}»=i of open subsets of X such that (a) U cU, (b) Un~=1Un=U, ( c ) Un c Vn whenever UczV. -
A TEXTBOOK of TOPOLOGY Lltld
SEIFERT AND THRELFALL: A TEXTBOOK OF TOPOLOGY lltld SEI FER T: 7'0PO 1.OG 1' 0 I.' 3- Dl M E N SI 0 N A I. FIRERED SPACES This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: SAMUELEILENBERG AND HYMANBASS A list of recent titles in this series appears at the end of this volunie. SEIFERT AND THRELFALL: A TEXTBOOK OF TOPOLOGY H. SEIFERT and W. THRELFALL Translated by Michael A. Goldman und S E I FE R T: TOPOLOGY OF 3-DIMENSIONAL FIBERED SPACES H. SEIFERT Translated by Wolfgang Heil Edited by Joan S. Birman and Julian Eisner @ 1980 ACADEMIC PRESS A Subsidiary of Harcourr Brace Jovanovich, Publishers NEW YORK LONDON TORONTO SYDNEY SAN FRANCISCO COPYRIGHT@ 1980, BY ACADEMICPRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. 11 1 Fifth Avenue, New York. New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NWI 7DX Mit Genehmigung des Verlager B. G. Teubner, Stuttgart, veranstaltete, akin autorisierte englische Ubersetzung, der deutschen Originalausgdbe. Library of Congress Cataloging in Publication Data Seifert, Herbert, 1897- Seifert and Threlfall: A textbook of topology. Seifert: Topology of 3-dimensional fibered spaces. (Pure and applied mathematics, a series of mono- graphs and textbooks ; ) Translation of Lehrbuch der Topologic. Bibliography: p. Includes index. 1.