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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. -
Floer Homology, Gauge Theory, and Low-Dimensional Topology
Floer Homology, Gauge Theory, and Low-Dimensional Topology Clay Mathematics Proceedings Volume 5 Floer Homology, Gauge Theory, and Low-Dimensional Topology Proceedings of the Clay Mathematics Institute 2004 Summer School Alfréd Rényi Institute of Mathematics Budapest, Hungary June 5–26, 2004 David A. Ellwood Peter S. Ozsváth András I. Stipsicz Zoltán Szabó Editors American Mathematical Society Clay Mathematics Institute 2000 Mathematics Subject Classification. Primary 57R17, 57R55, 57R57, 57R58, 53D05, 53D40, 57M27, 14J26. The cover illustrates a Kinoshita-Terasaka knot (a knot with trivial Alexander polyno- mial), and two Kauffman states. These states represent the two generators of the Heegaard Floer homology of the knot in its topmost filtration level. The fact that these elements are homologically non-trivial can be used to show that the Seifert genus of this knot is two, a result first proved by David Gabai. Library of Congress Cataloging-in-Publication Data Clay Mathematics Institute. Summer School (2004 : Budapest, Hungary) Floer homology, gauge theory, and low-dimensional topology : proceedings of the Clay Mathe- matics Institute 2004 Summer School, Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary, June 5–26, 2004 / David A. Ellwood ...[et al.], editors. p. cm. — (Clay mathematics proceedings, ISSN 1534-6455 ; v. 5) ISBN 0-8218-3845-8 (alk. paper) 1. Low-dimensional topology—Congresses. 2. Symplectic geometry—Congresses. 3. Homol- ogy theory—Congresses. 4. Gauge fields (Physics)—Congresses. I. Ellwood, D. (David), 1966– II. Title. III. Series. QA612.14.C55 2004 514.22—dc22 2006042815 Copying and reprinting. Material in this book may be reproduced by any means for educa- tional and scientific purposes without fee or permission with the exception of reproduction by ser- vices that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. -
Topology - Wikipedia, the Free Encyclopedia Page 1 of 7
Topology - Wikipedia, the free encyclopedia Page 1 of 7 Topology From Wikipedia, the free encyclopedia Topology (from the Greek τόπος , “place”, and λόγος , “study”) is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation. Ideas that are now classified as topological were expressed as early as 1736. Toward the end of the 19th century, a distinct A Möbius strip, an object with only one discipline developed, which was referred to in Latin as the surface and one edge. Such shapes are an geometria situs (“geometry of place”) or analysis situs object of study in topology. (Greek-Latin for “picking apart of place”). This later acquired the modern name of topology. By the middle of the 20 th century, topology had become an important area of study within mathematics. The word topology is used both for the mathematical discipline and for a family of sets with certain properties that are used to define a topological space, a basic object of topology. Of particular importance are homeomorphisms , which can be defined as continuous functions with a continuous inverse. For instance, the function y = x3 is a homeomorphism of the real line. Topology includes many subfields. The most basic and traditional division within topology is point-set topology , which establishes the foundational aspects of topology and investigates concepts inherent to topological spaces (basic examples include compactness and connectedness); algebraic topology , which generally tries to measure degrees of connectivity using algebraic constructs such as homotopy groups and homology; and geometric topology , which primarily studies manifolds and their embeddings (placements) in other manifolds. -
Homotopy Theory of Infinite Dimensional Manifolds?
Topo/ogy Vol. 5. pp. I-16. Pergamon Press, 1966. Printed in Great Britain HOMOTOPY THEORY OF INFINITE DIMENSIONAL MANIFOLDS? RICHARD S. PALAIS (Received 23 Augrlsf 1965) IN THE PAST several years there has been considerable interest in the theory of infinite dimensional differentiable manifolds. While most of the developments have quite properly stressed the differentiable structure, it is nevertheless true that the results and techniques are in large part homotopy theoretic in nature. By and large homotopy theoretic results have been brought in on an ad hoc basis in the proper degree of generality appropriate for the application immediately at hand. The result has been a number of overlapping lemmas of greater or lesser generality scattered through the published and unpublished literature. The present paper grew out of the author’s belief that it would serve a useful purpose to collect some of these results and prove them in as general a setting as is presently possible. $1. DEFINITIONS AND STATEMENT OF RESULTS Let V be a locally convex real topological vector space (abbreviated LCTVS). If V is metrizable we shall say it is an MLCTVS and if it admits a complete metric then we shall say that it is a CMLCTVS. A half-space in V is a subset of the form {v E V(l(u) L 0} where 1 is a continuous linear functional on V. A chart for a topological space X is a map cp : 0 + V where 0 is open in X, V is a LCTVS, and cp maps 0 homeomorphically onto either an open set of V or an open set of a half space of V. -
Local Homotopy Theory, Iwasawa Theory and Algebraic K–Theory
K(1){local homotopy theory, Iwasawa theory and algebraic K{theory Stephen A. Mitchell? University of Washington, Seattle, WA 98195, USA 1 Introduction The Iwasawa algebra Λ is a power series ring Z`[[T ]], ` a fixed prime. It arises in number theory as the pro-group ring of a certain Galois group, and in homotopy theory as a ring of operations in `-adic complex K{theory. Fur- thermore, these two incarnations of Λ are connected in an interesting way by algebraic K{theory. The main goal of this paper is to explore this connection, concentrating on the ideas and omitting most proofs. Let F be a number field. Fix a prime ` and let F denote the `-adic cyclotomic tower { that is, the extension field formed by1 adjoining the `n-th roots of unity for all n 1. The central strategy of Iwasawa theory is to study number-theoretic invariants≥ associated to F by analyzing how these invariants change as one moves up the cyclotomic tower. Number theorists would in fact consider more general towers, but we will be concerned exclusively with the cyclotomic case. This case can be viewed as analogous to the following geometric picture: Let X be a curve over a finite field F, and form a tower of curves over X by extending scalars to the algebraic closure of F, or perhaps just to the `-adic cyclotomic tower of F. This analogy was first considered by Iwasawa, and has been the source of many fruitful conjectures. As an example of a number-theoretic invariant, consider the norm inverse limit A of the `-torsion part of the class groups in the tower. -
1 Homotopy Groups
Math 752 Week 13 1 Homotopy Groups Definition 1. For n ≥ 0 and X a topological space with x0 2 X, define n n πn(X) = ff :(I ; @I ) ! (X; x0)g= ∼ where ∼ is the usual homotopy of maps. Then we have the following diagram of sets: f (In; @In) / (X; x ) 6 0 g (In=@In; @In=@In) n n n n n Now we have (I =@I ; @I =@I ) ' (S ; s0). So we can also define n πn(X; x0) = fg :(S ; s0) ! (X; x0)g= ∼ where g is as above. 0 Remark 1. If n = 0, then π0(X) is the set of connected components of X. Indeed, we have I = pt 0 and @I = ?, so π0(X) consists of homotopy classes of maps from a point into the space X. Now we will prove several results analogous to the n = 1 case. Proposition 1. If n ≥ 1, then πn(X; x0) is a group with respect to the following operation +: 8 > 1 <f(2s1; s2; : : : ; sn) 0 ≤ s1 ≤ 2 (f + g)(s1; s2; : : : ; sn) = > 1 :g(2s1 − 1; s2; : : : ; sn) 2 ≤ s1 ≤ 1 (Note that if n = 1, this is the usual concatenation of paths/loops.) Proof. First note that since only the first coordinate is involved in this operation, the same argument used to prove π1 is a group is valid here as well. Then the identity element is the constant map n taking all of I to x0 and inverses are given by −f = f(1 − s1; s2; : : : ; sn). 1 2 Proposition 2. If n ≥ 2, then πn(X; x0) is abelian. -
HOMOTOPY THEORY for BEGINNERS Contents 1. Notation
HOMOTOPY THEORY FOR BEGINNERS JESPER M. MØLLER Abstract. This note contains comments to Chapter 0 in Allan Hatcher's book [5]. Contents 1. Notation and some standard spaces and constructions1 1.1. Standard topological spaces1 1.2. The quotient topology 2 1.3. The category of topological spaces and continuous maps3 2. Homotopy 4 2.1. Relative homotopy 5 2.2. Retracts and deformation retracts5 3. Constructions on topological spaces6 4. CW-complexes 9 4.1. Topological properties of CW-complexes 11 4.2. Subcomplexes 12 4.3. Products of CW-complexes 12 5. The Homotopy Extension Property 14 5.1. What is the HEP good for? 14 5.2. Are there any pairs of spaces that have the HEP? 16 References 21 1. Notation and some standard spaces and constructions In this section we fix some notation and recollect some standard facts from general topology. 1.1. Standard topological spaces. We will often refer to these standard spaces: • R is the real line and Rn = R × · · · × R is the n-dimensional real vector space • C is the field of complex numbers and Cn = C × · · · × C is the n-dimensional complex vector space • H is the (skew-)field of quaternions and Hn = H × · · · × H is the n-dimensional quaternion vector space • Sn = fx 2 Rn+1 j jxj = 1g is the unit n-sphere in Rn+1 • Dn = fx 2 Rn j jxj ≤ 1g is the unit n-disc in Rn • I = [0; 1] ⊂ R is the unit interval • RP n, CP n, HP n is the topological space of 1-dimensional linear subspaces of Rn+1, Cn+1, Hn+1. -
Homotopy Equivalence Between Diagrams of Spaces
Journal of Pure and Applied Algebra 41 (1986) 169-182 169 North-Holland HOMOTOPY EQUIVALENCE BETWEEN DIAGRAMS OF SPACES E. DROR FARJOUN Department of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel A. ZABRODSKY* Department of Mathematics, University of Washington, Seattle, WA 98195, USA Communicated by J. Stasheff Received 12 April 1984 Revised 24 November and 18 March 1985 O. Introduction In the present note we would like to generalize certain basic concepts and results from the case of a discrete group action on a topological space to that of quite general diagrams of spaces - by which one means a functor X: D---,Top from a small category D to topological spaces. In particular, we prove a generalization of Bredon's fundamental theorem about G-homotopy equivalences [1]. That result says that a G-map f:X~Y between two G-CW-complexes is a G-homotopy equivalence if and only if it induces a homotopy equivalence fl-t. X H ~ yH on the fixed point sets for each subgroup HC G. After introducing a notion of D-CW-complex and D-orbit type we prove a generalization of Bredon's theorem that can be applied, in particular, to any diagram of simplicial complexes and simplicial maps between them. Main results In Section 1 we introduce D-CW-complexes. These are built as homotopy push- outs out of D-orbits: A D-orbit is any D-space T (i.e. a D-diagram of spaces) with dir limDT= (pt); i.e. for which the direct limit over D consists of one point. -
Arxiv:Math/0205027V2
HYPERCOVERS AND SIMPLICIAL PRESHEAVES DANIEL DUGGER, SHARON HOLLANDER, AND DANIEL C. ISAKSEN Abstract. We use hypercovers to study the homotopy theory of simpli- cial presheaves. The main result says that model structures for simplicial presheaves involving local weak equivalences can be constructed by localizing at the hypercovers. One consequence is that the fibrant objects can be explic- itly described in terms of a hypercover descent condition, and the fibrations can be described by a relative descent condition. We give a few applications for this new description of the homotopy theory of simplicial presheaves. Contents 1. Introduction 1 2. Model structures on simplicial presheaves 5 3. Local weak equivalences and local lifting properties 6 4. Background on hypercovers 8 5. Hypercovers and lifting problems 12 6. Hypercovers and localizations 16 7. Fibrations and descent conditions 19 8. Other applications 23 9. Verdier sites 29 10. Internal hypercovers 32 Appendix A. Cechˇ localizations 33 References 42 1. Introduction This paper is concerned with the subject of homotopical sheaf theory, as it has developed over time in the articles [I, B, BG, Th, Jo, J1, J2, J3, J4]. Given a fixed arXiv:math/0205027v2 [math.AT] 20 Jun 2003 Grothendieck site C, one wants to consider contravariant functors F defined on C whose values have a homotopy type associated to them. The most basic question is: what should it mean for F to be a sheaf? The desire is for some kind of local-to-global property—also called a descent property—where the value of F on an object X can be recovered by homotopical methods from the values on a cover. -
A Topological Manifold Is Homotopy Equivalent to Some CW-Complex
A topological manifold is homotopy equivalent to some CW-complex Aasa Feragen Supervisor: Erik Elfving December 17, 2004 Contents 1 Introduction 3 1.1 Thanks............................... 3 1.2 Theproblem............................ 3 1.3 Notationandterminology . 3 1.4 Continuity of combined maps . 4 1.5 Paracompactspaces........................ 5 1.6 Properties of normal and fully normal spaces . 13 2 Retracts 16 2.1 ExtensorsandRetracts. 16 2.2 Polytopes ............................. 18 2.3 Dugundji’s extension theorem . 28 2.4 The Eilenberg-Wojdyslawski theorem . 37 2.5 ANE versus ANR . 39 2.6 Dominatingspaces ........................ 41 2.7 ManifoldsandlocalANRs . 48 3 Homotopy theory 55 3.1 Higherhomotopygroups . 55 3.2 The exact homotopy sequence of a pair of spaces . 58 3.3 Adjunction spaces and the method of adjoining cells . 62 3.4 CW-complexes .......................... 69 3.5 Weak homotopy equivalence . 79 3.6 A metrizable ANR is homotopy equivalent to a CW complex . 85 2 Chapter 1 Introduction 1.1 Thanks First of all, I would like to thank my supervisor Erik Elfving for suggesting the topic and for giving valuable feedback while I was writing the thesis. 1.2 The problem The goal of this Pro Gradu thesis is to show that a topological manifold has the same homotopy type as some CW complex. This will be shown in several ”parts”: A) A metrizable ANR has the same homotopy type as some CW complex. i) For any ANR Y there exists a dominating space X of Y which is a CW complex. ii) A space which is dominated by a CW complex is homotopy equiv- alent to a CW complex. -
Homotopy and the Fundamental Group
CHAPTER 2 Homotopy and the Fundamental Group 1. Homotopy Denote I = [0; 1]. Let f; g : X ! Y be maps of topological spaces. A homotopy from f to g is a map H : X×I ! Y such that for all x 2 X, H(x; 0) = f(x) and H(x; 1) = g(x). Thus, on the `bottom' edge, H agrees with f and on the `top' edge it agrees with g. The intermediate maps H(−; t) for 0 < t < 1 may be thought of a 1-parameter family of maps through which f is continuously deformed into g. We say f is homotopic to g (f ∼ g) if there is a homotopy from f to g. Example 2.1. Let X = S1 and Y = S1 × I (a cylinder of radius 1 and height 1). Define H(x; t) = (x; t). (What are f and g?) Example 2.2. Let X = S1 and Y = D2. Let f be the inclusion of S1 in D2 and let g map S1 to the center of D2. Then H(x; t) = (1−t)x defines a homotopy of f to g. Example 2.3. Let X = Y = Rn. Let f be the identity map, and let g be defined by g(x) = 0 for all x. Define H(x; t) = (1 − t)x. Note that the same argument would work for any point with a slightly different H. If the identity map of a space is homotopic to a constant map (as in Example 2.3), we say the space is contractible. It is also useful to have a relative version of this definition. -
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.