LECTURE NOTES on K-THEORY 1. L 1.1. Paracompact
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[Math.DG] 1 Feb 2002
The BIC of a conical fibration. Martintxo Saralegi-Aranguren∗ Robert Wolak† Universit´ed’Artois Uniwersytet Jagiellonski November 14, 2018 Abstract In the paper we introduce the notions of a singular fibration and a singular Seifert fi- bration. These notions are natural generalizations of the notion of a locally trivial fibration to the category of stratified pseudomanifolds. For singular foliations defined by such fibra- tions we prove a de Rham type theorem for the basic intersection cohomology introduced the authors in a recent paper. One of important examples of such a structure is the natural projection onto the leaf space for the singular Riemannian foliation defined by an action of a compact Lie group on a compact smooth manifold. The failure of the Poincar´eduality for the homology and cohomology of some singular spaces led Goresky and MacPherson to introduce a new homology theory called intersection homology which took into account the properties of the singularities of the considered space (cf. [10]). This homology is defined for stratified pseudomanifolds. The initial idea was generalized in several ways. The theories of simplicial and singular homologies were developed as well as weaker notions of the perversity were proposed (cf. [11, 12]). Several versions of the Poincar´eduality were proved taking into account the notion of dual perversities. Finally, the deRham intersection cohomology was defined by Goresky and MacPherson for Thom-Mather stratified spaces. The first written reference is the paper by J.L. Brylinski, cf. [6]. This led to the search for the ”de Rham - type” theorem for the intersection cohomology. The first author in his thesis and several subsequent publications, (cf. -
The Jouanolou-Thomason Homotopy Lemma
The Jouanolou-Thomason homotopy lemma Aravind Asok February 9, 2009 1 Introduction The goal of this note is to prove what is now known as the Jouanolou-Thomason homotopy lemma or simply \Jouanolou's trick." Our main reason for discussing this here is that i) most statements (that I have seen) assume unncessary quasi-projectivity hypotheses, and ii) most applications of the result that I know (e.g., in homotopy K-theory) appeal to the result as merely a \black box," while the proof indicates that the construction is quite geometric and relatively explicit. For simplicity, throughout the word scheme means separated Noetherian scheme. Theorem 1.1 (Jouanolou-Thomason homotopy lemma). Given a smooth scheme X over a regular Noetherian base ring k, there exists a pair (X;~ π), where X~ is an affine scheme, smooth over k, and π : X~ ! X is a Zariski locally trivial smooth morphism with fibers isomorphic to affine spaces. 1 Remark 1.2. In terms of an A -homotopy category of smooth schemes over k (e.g., H(k) or H´et(k); see [MV99, x3]), the map π is an A1-weak equivalence (use [MV99, x3 Example 2.4]. Thus, up to A1-weak equivalence, any smooth k-scheme is an affine scheme smooth over k. 2 An explicit algebraic form Let An denote affine space over Spec Z. Let An n 0 denote the scheme quasi-affine and smooth over 2m Spec Z obtained by removing the fiber over 0. Let Q2m−1 denote the closed subscheme of A (with coordinates x1; : : : ; x2m) defined by the equation X xixm+i = 1: i Consider the following simple situation. -
Vector Bundles on Projective Space
Vector Bundles on Projective Space Takumi Murayama December 1, 2013 1 Preliminaries on vector bundles Let X be a (quasi-projective) variety over k. We follow [Sha13, Chap. 6, x1.2]. Definition. A family of vector spaces over X is a morphism of varieties π : E ! X −1 such that for each x 2 X, the fiber Ex := π (x) is isomorphic to a vector space r 0 0 Ak(x).A morphism of a family of vector spaces π : E ! X and π : E ! X is a morphism f : E ! E0 such that the following diagram commutes: f E E0 π π0 X 0 and the map fx : Ex ! Ex is linear over k(x). f is an isomorphism if fx is an isomorphism for all x. A vector bundle is a family of vector spaces that is locally trivial, i.e., for each x 2 X, there exists a neighborhood U 3 x such that there is an isomorphism ': π−1(U) !∼ U × Ar that is an isomorphism of families of vector spaces by the following diagram: −1 ∼ r π (U) ' U × A (1.1) π pr1 U −1 where pr1 denotes the first projection. We call π (U) ! U the restriction of the vector bundle π : E ! X onto U, denoted by EjU . r is locally constant, hence is constant on every irreducible component of X. If it is constant everywhere on X, we call r the rank of the vector bundle. 1 The following lemma tells us how local trivializations of a vector bundle glue together on the entire space X. -
Paracompactness and C2 -Paracompactness
Turkish Journal of Mathematics Turk J Math (2019) 43: 9 – 20 http://journals.tubitak.gov.tr/math/ © TÜBİTAK Research Article doi:10.3906/mat-1804-54 C -Paracompactness and C2 -paracompactness Maha Mohammed SAEED∗,, Lutfi KALANTAN,, Hala ALZUMI, Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia Received: 17.04.2018 • Accepted/Published Online: 29.08.2018 • Final Version: 18.01.2019 Abstract: A topological space X is called C -paracompact if there exist a paracompact space Y and a bijective function f : X −! Y such that the restriction fjA : A −! f(A) is a homeomorphism for each compact subspace A ⊆ X .A topological space X is called C2 -paracompact if there exist a Hausdorff paracompact space Y and a bijective function f : X −! Y such that the restriction fjA : A −! f(A) is a homeomorphism for each compact subspace A ⊆ X . We investigate these two properties and produce some examples to illustrate the relationship between them and C -normality, minimal Hausdorff, and other properties. Key words: Normal, paracompact, C -paracompact, C2 -paracompact, C -normal, epinormal, mildly normal, minimal Hausdorff, Fréchet, Urysohn 1. Introduction We introduce two new topological properties, C -paracompactness and C2 -paracompactness. They were defined by Arhangel’skiĭ. The purpose of this paper is to investigate these two properties. Throughout this paper, we denote an ordered pair by hx; yi, the set of positive integers by N, the rational numbers by Q, the irrational numbers by P, and the set of real numbers by R. T2 denotes the Hausdorff property. A T4 space is a T1 normal space and a Tychonoff space (T 1 ) is a T1 completely regular space. -
Base-Paracompact Spaces
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 128 (2003) 145–156 www.elsevier.com/locate/topol Base-paracompact spaces John E. Porter Department of Mathematics and Statistics, Faculty Hall Room 6C, Murray State University, Murray, KY 42071-3341, USA Received 14 June 2001; received in revised form 21 March 2002 Abstract A topological space is said to be totally paracompact if every open base of it has a locally finite subcover. It turns out this property is very restrictive. In fact, the irrationals are not totally paracompact. Here we give a generalization, base-paracompact, to the notion of totally paracompactness, and study which paracompact spaces satisfy this generalization. 2002 Elsevier Science B.V. All rights reserved. Keywords: Paracompact; Base-paracompact; Totally paracompact 1. Introduction A topological space X is paracompact if every open cover of X has a locally finite open refinement. A topological space is said to be totally paracompact if every open base of it has a locally finite subcover. Corson, McNinn, Michael, and Nagata announced [3] that the irrationals have a basis which does not have a locally finite subcover, and hence are not totally paracompact. Ford [6] was the first to give the definition of totally paracompact spaces. He used this property to show the equivalence of the small and large inductive dimension in metric spaces. Ford also showed paracompact, locally compact spaces are totally paracompact. Lelek [9] studied totally paracompact and totally metacompact separable metric spaces. He showed that these two properties are equivalent in separable metric spaces. -
Differential Geometry: Curvature and Holonomy Austin Christian
University of Texas at Tyler Scholar Works at UT Tyler Math Theses Math Spring 5-5-2015 Differential Geometry: Curvature and Holonomy Austin Christian Follow this and additional works at: https://scholarworks.uttyler.edu/math_grad Part of the Mathematics Commons Recommended Citation Christian, Austin, "Differential Geometry: Curvature and Holonomy" (2015). Math Theses. Paper 5. http://hdl.handle.net/10950/266 This Thesis is brought to you for free and open access by the Math at Scholar Works at UT Tyler. It has been accepted for inclusion in Math Theses by an authorized administrator of Scholar Works at UT Tyler. For more information, please contact [email protected]. DIFFERENTIAL GEOMETRY: CURVATURE AND HOLONOMY by AUSTIN CHRISTIAN A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Mathematics David Milan, Ph.D., Committee Chair College of Arts and Sciences The University of Texas at Tyler May 2015 c Copyright by Austin Christian 2015 All rights reserved Acknowledgments There are a number of people that have contributed to this project, whether or not they were aware of their contribution. For taking me on as a student and learning differential geometry with me, I am deeply indebted to my advisor, David Milan. Without himself being a geometer, he has helped me to develop an invaluable intuition for the field, and the freedom he has afforded me to study things that I find interesting has given me ample room to grow. For introducing me to differential geometry in the first place, I owe a great deal of thanks to my undergraduate advisor, Robert Huff; our many fruitful conversations, mathematical and otherwise, con- tinue to affect my approach to mathematics. -
Recent Developments in Representation Theory
RECENT DEVELOPMENTS IN REPRESENTATION THEORY Wilfried 5chmid* Department of Mathematics Harvard University Cambridge, MA 02138 For the purposes of this lecture, "representation theory" means representation theory of Lie groups, and more specifically, of semisimple Lie groups. I am interpreting my assignment to give a survey rather loosely: while I shall touch upon various major advances in the subject, I am concentrating on a single development. Both order and emphasis of my presentation are motivated by expository considerations, and do not reflect my view of the relative importance of various topics. Initially G shall denote a locally compact topological group which is unimodular -- i.e., left and right Haar measure coincide -- and H C G a closed unimodular subgroup. The quotient space G/H then carries a G- invariant measure, so G acts unitarily on the Hilbert space L2(G/H). In essence, the fundamental problem of harmonic analysis is to decompose L2(G/H) into a direct "sum" of irreducibles. The quotation marks allude to the fact that the decomposition typically involves the continuous ana- logue of a sum, namely a direct integral, as happens already for non- compact Abelian groups. If G is of type I -- loosely speaking, if the unitary representations of G behave reasonably -- the abstract Plan- cherel theorem [12] asserts the existence of such a decomposition. This existence theorem raises as many questions as it answers: to make the decomposition useful, one wants to know it explicitly and, most impor- tantly, one wants to understand the structure of the irreducible sum- mands. In principle, any irreducible unitary representation of G can occur as a constituent of L2(G/H), for some H C G. -
DEFINITIONS and THEOREMS in GENERAL TOPOLOGY 1. Basic
DEFINITIONS AND THEOREMS IN GENERAL TOPOLOGY 1. Basic definitions. A topology on a set X is defined by a family O of subsets of X, the open sets of the topology, satisfying the axioms: (i) ; and X are in O; (ii) the intersection of finitely many sets in O is in O; (iii) arbitrary unions of sets in O are in O. Alternatively, a topology may be defined by the neighborhoods U(p) of an arbitrary point p 2 X, where p 2 U(p) and, in addition: (i) If U1;U2 are neighborhoods of p, there exists U3 neighborhood of p, such that U3 ⊂ U1 \ U2; (ii) If U is a neighborhood of p and q 2 U, there exists a neighborhood V of q so that V ⊂ U. A topology is Hausdorff if any distinct points p 6= q admit disjoint neigh- borhoods. This is almost always assumed. A set C ⊂ X is closed if its complement is open. The closure A¯ of a set A ⊂ X is the intersection of all closed sets containing X. A subset A ⊂ X is dense in X if A¯ = X. A point x 2 X is a cluster point of a subset A ⊂ X if any neighborhood of x contains a point of A distinct from x. If A0 denotes the set of cluster points, then A¯ = A [ A0: A map f : X ! Y of topological spaces is continuous at p 2 X if for any open neighborhood V ⊂ Y of f(p), there exists an open neighborhood U ⊂ X of p so that f(U) ⊂ V . -
4. Coherent Sheaves Definition 4.1. If (X,O X) Is a Locally Ringed Space
4. Coherent Sheaves Definition 4.1. If (X; OX ) is a locally ringed space, then we say that an OX -module F is locally free if there is an open affine cover fUig of X such that FjUi is isomorphic to a direct sum of copies of OUi . If the number of copies r is finite and constant, then F is called locally free of rank r (aka a vector bundle). If F is locally free of rank one then we way say that F is invertible (aka a line bundle). The group of all invertible sheaves under tensor product, denoted Pic(X), is called the Picard group of X. A sheaf of ideals I is any OX -submodule of OX . Definition 4.2. Let X = Spec A be an affine scheme and let M be an A-module. M~ is the sheaf which assigns to every open subset U ⊂ X, the set of functions a s: U −! Mp; p2U which can be locally represented at p as a=g, a 2 M, g 2 R, p 2= Ug ⊂ U. Lemma 4.3. Let A be a ring and let M be an A-module. Let X = Spec A. ~ (1) M is a OX -module. ~ (2) If p 2 X then Mp is isomorphic to Mp. ~ (3) If f 2 A then M(Uf ) is isomorphic to Mf . Proof. (1) is clear and the rest is proved mutatis mutandis as for the structure sheaf. Definition 4.4. An OX -module F on a scheme X is called quasi- coherent if there is an open cover fUi = Spec Aig by affines and ~ isomorphisms FjUi ' Mi, where Mi is an Ai-module. -
256B Algebraic Geometry
256B Algebraic Geometry David Nadler Notes by Qiaochu Yuan Spring 2013 1 Vector bundles on the projective line This semester we will be focusing on coherent sheaves on smooth projective complex varieties. The organizing framework for this class will be a 2-dimensional topological field theory called the B-model. Topics will include 1. Vector bundles and coherent sheaves 2. Cohomology, derived categories, and derived functors (in the differential graded setting) 3. Grothendieck-Serre duality 4. Reconstruction theorems (Bondal-Orlov, Tannaka, Gabriel) 5. Hochschild homology, Chern classes, Grothendieck-Riemann-Roch For now we'll introduce enough background to talk about vector bundles on P1. We'll regard varieties as subsets of PN for some N. Projective will mean that we look at closed subsets (with respect to the Zariski topology). The reason is that if p : X ! pt is the unique map from such a subset X to a point, then we can (derived) push forward a bounded complex of coherent sheaves M on X to a bounded complex of coherent sheaves on a point Rp∗(M). Smooth will mean the following. If x 2 X is a point, then locally x is cut out by 2 a maximal ideal mx of functions vanishing on x. Smooth means that dim mx=mx = dim X. (In general it may be bigger.) Intuitively it means that locally at x the variety X looks like a manifold, and one way to make this precise is that the completion of the local ring at x is isomorphic to a power series ring C[[x1; :::xn]]; this is the ring where Taylor series expansions live. -
Notes on Principal Bundles and Classifying Spaces
Notes on principal bundles and classifying spaces Stephen A. Mitchell August 2001 1 Introduction Consider a real n-plane bundle ξ with Euclidean metric. Associated to ξ are a number of auxiliary bundles: disc bundle, sphere bundle, projective bundle, k-frame bundle, etc. Here “bundle” simply means a local product with the indicated fibre. In each case one can show, by easy but repetitive arguments, that the projection map in question is indeed a local product; furthermore, the transition functions are always linear in the sense that they are induced in an obvious way from the linear transition functions of ξ. It turns out that all of this data can be subsumed in a single object: the “principal O(n)-bundle” Pξ, which is just the bundle of orthonormal n-frames. The fact that the transition functions of the various associated bundles are linear can then be formalized in the notion “fibre bundle with structure group O(n)”. If we do not want to consider a Euclidean metric, there is an analogous notion of principal GLnR-bundle; this is the bundle of linearly independent n-frames. More generally, if G is any topological group, a principal G-bundle is a locally trivial free G-space with orbit space B (see below for the precise definition). For example, if G is discrete then a principal G-bundle with connected total space is the same thing as a regular covering map with G as group of deck transformations. Under mild hypotheses there exists a classifying space BG, such that isomorphism classes of principal G-bundles over X are in natural bijective correspondence with [X, BG]. -
D-Modules on Flag Varieties and Localization of G-Modules
D-modules on flag varieties and localization of g-modules. Jos´eSimental. October 7, 2013 1 (Twisted) Differential operators and D-modules. 1.1 Differential Operators. Let X be an algebraic variety. In this text, we assume all algebraic varieties are over C. Let R be a sheaf of rings on X. For R-modules F and G, define the sheaf HomR(F; G) as follows. For any open set U ⊂ X, the sections of HomR(F; G) on U are HomRjU (FjU ; GjU ). For any algebraic variety X, let CX be the constant sheaf of complex numbers C. Now assume that X is a smooth complex algebraic variety. Let OX be its sheaf of regular functions and VX be its sheaf of vector fields, that is, VX := DerC(OX ) := fθ 2 EndCX (OX ): θ(fg) = fθ(g) + θ(f)g; f; g 2 OX g. The sheaf DX of differential operators on X, is, by definition, the subsheaf (of associative rings) of EndCX (OX ) generated by OX and VX . Theorem 1.1 In the setup of the previous paragraph, if dim X = n, then for every point p 2 X there exists an affine open neighborhood U of p, x1; : : : ; xn 2 OX (U) and @1;:::;@n 2 VX (U) such that [@i;@j] = 0, @i(xj) = δij and VX (U) is free over OX (U) with basis @1;:::;@n. Proof. Let mp be the maximal ideal of OX;p. Since X is smooth of dimension n, there exist functions x1; : : : ; xn generating mp. Then, dx1; : : : ; dxn is a basis of ΩX;p, where ΩX is the cotangent sheaf of X.