Configuration Spaces and Braid Groups
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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. -
Nonlinear Elasticity in a Deforming Ambient Space Contents 1
Nonlinear Elasticity in a Deforming Ambient Space∗ Arash Yavari† Arkadas Ozakin‡ Souhayl Sadik§ 27 May 2016 Abstract In this paper we formulate a nonlinear elasticity theory in which the ambient space is evolving. For a continuum moving in an evolving ambient space, we model time dependency of the metric by a time- dependent embedding of the ambient space in a larger manifold with a fixed background metric. We derive both the tangential and the normal governing equations. We then reduce the standard energy balance written in the larger ambient space to that in the evolving ambient space. We consider quasi-static deformations of the ambient space and show that a quasi-static deformation of the ambient space results in stresses, in general. We linearize the nonlinear theory about a reference motion and show that variation of the spatial metric corresponds to an effective field of body forces. Keywords: Geometric mechanics, nonlinear elasticity, deforming ambient space. Contents 1 Introduction 1 2 Motion of an Elastic Body in an Evolving Ambient Space3 2.1 Lagrangian Field Theory of Elasticity in an Evolving Ambient Space................3 2.2 Conservation of Mass for Motion in an Evolving Ambient Space................... 10 2.3 Energy Balance in Nonlinear Elasticity in an Evolving Ambient Space................ 11 3 Quasi-Static Deformations of the Ambient Space Metric 14 3.1 Examples of elastic bodies in evolving ambient spaces and the induced stresses.......... 14 3.2 Elastic deformations due to linear perturbations of the ambient space metric............ 20 A Geometry of Riemannian Submanifolds 24 B An Alternative Derivation of the Tangent Balance of Linear Momentum 27 1 Introduction In the geometric theory of elasticity, an elastic body is represented by a material manifold B , which defines the natural, stress-free state of the body. -
Persistence Stability for Geometric Complexes Arxiv:1207.3885V3
Persistence stability for geometric complexes Fr´ed´ericChazal∗, Vin de Silvay, Steve Oudotz November 11, 2013 Abstract In this paper we study the properties of the homology of different geometric filtered complexes (such as Vietoris{Rips, Cechˇ and witness complexes) built on top of totally bounded metric spaces. Using recent developments in the theory of topological persis- tence, we provide simple and natural proofs of the stability of the persistent homology of such complexes with respect to the Gromov–Hausdorff distance. We also exhibit a few noteworthy properties of the homology of the Rips and Cechˇ complexes built on top of compact spaces. 1 Introduction The inference of topological properties of metric spaces from approximations is a problem that has attracted special attention in computational topology in recent years. Given a metric space (Y; dY ) approximating an unknown metric space (X; dX ), the aim is to build a simplicial complex on the vertex set Y whose homology or homotopy type is the same as X. Note that, although Y is finite in many applications, finiteness is not a requirement a priori. Among the many geometric complexes available to us, the Vietoris{Rips complex (or simply `Rips complex') is particularly useful, being easy to compute and having good approximation arXiv:1207.3885v3 [math.AT] 15 Nov 2013 properties. We recall the definition. Let (X; dx) be a metric space and α a real parameter (the `scale'). Then Rips(X; α) is the simplical complex on X whose simplices are the finite subsets of X with diameter at most α: σ = [x0; x1; : : : ; xk] Rips(X; α) dX (xi; xj) α for all i; j 2 , ≤ ∗[email protected] [email protected] [email protected] 1 When (X; dX ) is a closed Riemannian manifold, J.-C. -
On Orbit Configuration Spaces of Spheres
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Topology and its Applications 118 (2002) 85–102 On orbit configuration spaces of spheres Eva Maria Feichtner a,∗, Günter M. Ziegler b,∗ a Departement Mathematik, ETH Zürich, 8092 Zürich, Switzerland b Fachbereich Mathematik, MA 6-2, TU Berlin, 10623 Berlin, Germany Received 26 January 2000; received in revised form 10 July 2000 Abstract k The orbit configuration space FZ2 (S ,n) is the space of all ordered n-tuples of points on the k k-sphere such that no two of them are identical or antipodal. The cohomology algebra of FZ2 (S ,n), with integer coefficients, is here determined completely, and described in terms of generators, bases and relations. To this end, we analyze the cohomology spectral sequence of a fibration k → k FZ2 (S ,n) S , where the fiber—in contrast to the situation for the classical configuration space F(Sk,n)—is not the complement of a linear subspace arrangement. Analogies to the arrangement case, however, are crucial for getting a complete description of its cohomology. 2002 Elsevier Science B.V. All rights reserved. Keywords: Orbit configuration spaces; Integral cohomology algebra; Subspace arrangements; Leray–Serre spectral sequence 1. Introduction The ordered configuration spaces n F(M, n) := (x1,x2,...,xn) ∈ M : xi = xj for all i = j of a manifold M were first studied by Fadell and Neuwirth [15] in 1962. The cohomology of such spaces has received a lot of attention, at least since the work by Arnol’d [1] related to Hilbert’s 13th problem. -
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. -
Appendix a Topological Groups and Lie Groups
Appendix A Topological Groups and Lie Groups This appendix studies topological groups, and also Lie groups which are special topological groups as well as manifolds with some compatibility conditions. The concept of a topological group arose through the work of Felix Klein (1849–1925) and Marius Sophus Lie (1842–1899). One of the concrete concepts of the the- ory of topological groups is the concept of Lie groups named after Sophus Lie. The concept of Lie groups arose in mathematics through the study of continuous transformations, which constitute in a natural way topological manifolds. Topo- logical groups occupy a vast territory in topology and geometry. The theory of topological groups first arose in the theory of Lie groups which carry differential structures and they form the most important class of topological groups. For exam- ple, GL (n, R), GL (n, C), GL (n, H), SL (n, R), SL (n, C), O(n, R), U(n, C), SL (n, H) are some important classical Lie Groups. Sophus Lie first systematically investigated groups of transformations and developed his theory of transformation groups to solve his integration problems. David Hilbert (1862–1943) presented to the International Congress of Mathe- maticians, 1900 (ICM 1900) in Paris a series of 23 research projects. He stated in this lecture that his Fifth Problem is linked to Sophus Lie theory of transformation groups, i.e., Lie groups act as groups of transformations on manifolds. A translation of Hilbert’s fifth problem says “It is well-known that Lie with the aid of the concept of continuous groups of transformations, had set up a system of geometrical axioms and, from the standpoint of his theory of groups has proved that this system of axioms suffices for geometry”. -
Geometric Meaning of Isoparametric Hypersurfaces in a Real Space Form
Canad. Math. Bull. Vol. 43 (1), 2000 pp. 74–78 Geometric Meaning of Isoparametric Hypersurfaces in a Real Space Form Makoto Kimura and Sadahiro Maeda Abstract. We shall provide a characterization of all isoparametric hypersurfaces M’s in a real space form M˜ (c) by observing the extrinsic shape of geodesics of M in the ambient manifold M˜ (c). 0 Introduction In differential geometry it is interesting to know the shape of a Riemannian submanifold by observing the extrinsic shape of geodesics of the submanifold. For example: A hypersurface Mn isometrically immersed into a real space form Mn+1(c)ofconstantcurvaturec (that is, Mn+1(c) = Rn+1, Sn+1(c)orHn+1(c) according as the curvature c is zero, positive, or negative) is totally umbilic in Mn+1(c) if and only if every geodesic of M,consideredasa curve in the ambient space Mn+1(c), is a circle. Here we recall the definition of circles in Riemannian geometry. A smooth curve γ : R −→ M in a complete Riemannian manifold M is called a circle of curvature κ( 0) if it is parametrized by its arclength s and it satisfies the following equation: 2 ∇γ˙ ∇γ˙ γ˙(s) = −κ γ˙(s), where κ is constant and ∇γ˙ denotes the covariant differentiation along γ with respect to the Riemannian connection ∇ of M.Since∇γ˙ γ˙ = κ,thisequationisequivalenttothe equation of geodesics, when κ = 0. So we treat a geodesic as a circle of null curvature. In general, a circle in a Riemannian manifold is not closed. Of course, any circles of positive curvature in Euclidean m-space Rm are closed. -
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. -
Exotic Aspherical Manifolds a Quick Tour Through Davis's Zoo of Aspherical Manifolds
Exotic Aspherical Manifolds A quick tour through Davis's zoo of aspherical manifolds Clara Loh Spring 2009 In a series of three talks, we give an introduction to Davis’s construction of peculiar aspherical manifolds and give a quick tour through the resulting zoo of aspherical manifolds. These notes contain a primer of the subject (describing the basic tools and results), a schedule for the three talks, as well as some illustrative material. 1 A quick tour through the quick tour The class of aspherical manifolds plays an important rˆolein various fields of topology. For instance, according to the Borel conjecture, aspherical manifolds should be very rigid topological objects. Two natural perspectives on aspherical manifolds are established by the questions “Which manifolds have a contractible universal covering?” and “Which groups admit a model for the classifying space that happens to be a manifold?”. The condition for a topological space to be both a manifold and aspherical is very restrictive and for a long time basically the only known examples were the ones given by non-positively curved Riemannian manifolds and fibre bundle constructions. The perception of aspherical manifolds changed drastically when Davis in his landmark paper [2] provided the first construction of exotic aspherical manifolds – more specifically, of closed, aspherical manifolds whose universal covering is not homeomorphic to Euclidean space(!). Subsequently, Davis and others refined this technique to construct closed, aspherical manifolds with exotic fundamental groups and exotic geometric properties [3, 4]. 2 1 A quick tour through the quick tour Mirror, mirror on the wall . The key idea underlying Davis’s construction is to glue copies of an aspher- ical space along certain subspaces, called mirrors, according to the com- binatorics of an abstract reflection group – when the input space and the reflection group are chosen appropriately, the glued space is an aspherical manifold; taking quotients then leads to closed, aspherical manifolds.