Covering Spaces
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(Pro-) Étale Cohomology 3. Exercise Sheet
(Pro-) Étale Cohomology 3. Exercise Sheet Department of Mathematics Winter Semester 18/19 Prof. Dr. Torsten Wedhorn 2nd November 2018 Timo Henkel Homework Exercise H9 (Clopen subschemes) (12 points) Let X be a scheme. We define Clopen(X ) := Z X Z open and closed subscheme of X . f ⊆ j g Recall that Clopen(X ) is in bijection to the set of idempotent elements of X (X ). Let X S be a morphism of schemes. We consider the functor FX =S fromO the category of S-schemes to the category of sets, given! by FX =S(T S) = Clopen(X S T). ! × Now assume that X S is a finite locally free morphism of schemes. Show that FX =S is representable by an affine étale S-scheme which is of! finite presentation over S. Exercise H10 (Lifting criteria) (12 points) Let f : X S be a morphism of schemes which is locally of finite presentation. Consider the following diagram of S-schemes:! T0 / X (1) f T / S Let be a class of morphisms of S-schemes. We say that satisfies the 1-lifting property (resp. !-lifting property) ≤ withC respect to f , if for all morphisms T0 T in andC for all diagrams9 of the form (1) there exists9 at most (resp. exactly) one morphism of S-schemes T X!which makesC the diagram commutative. Let ! 1 := f : T0 T closed immersion of S-schemes f is given by a locally nilpotent ideal C f ! j g 2 := f : T0 T closed immersion of S-schemes T is affine and T0 is given by a nilpotent ideal C f ! j g 2 3 := f : T0 T closed immersion of S-schemes T is the spectrum of a local ring and T0 is given by an ideal I with I = 0 C f ! j g Show that the following assertions are equivalent: (i) 1 satisfies the 1-lifting property (resp. -
1.6 Covering Spaces 1300Y Geometry and Topology
1.6 Covering spaces 1300Y Geometry and Topology 1.6 Covering spaces Consider the fundamental group π1(X; x0) of a pointed space. It is natural to expect that the group theory of π1(X; x0) might be understood geometrically. For example, subgroups may correspond to images of induced maps ι∗π1(Y; y0) −! π1(X; x0) from continuous maps of pointed spaces (Y; y0) −! (X; x0). For this induced map to be an injection we would need to be able to lift homotopies in X to homotopies in Y . Rather than consider a huge category of possible spaces mapping to X, we restrict ourselves to a category of covering spaces, and we show that under some mild conditions on X, this category completely encodes the group theory of the fundamental group. Definition 9. A covering map of topological spaces p : X~ −! X is a continuous map such that there S −1 exists an open cover X = α Uα such that p (Uα) is a disjoint union of open sets (called sheets), each homeomorphic via p with Uα. We then refer to (X;~ p) (or simply X~, abusing notation) as a covering space of X. Let (X~i; pi); i = 1; 2 be covering spaces of X. A morphism of covering spaces is a covering map φ : X~1 −! X~2 such that the diagram commutes: φ X~1 / X~2 AA } AA }} p1 AA }}p2 A ~}} X We will be considering covering maps of pointed spaces p :(X;~ x~0) −! (X; x0), and pointed morphisms between them, which are defined in the obvious fashion. -
Symplectic Group Actions and Covering Spaces
Symplectic group actions and covering spaces Montaldi, James and Ortega, Juan-Pablo 2009 MIMS EPrint: 2007.84 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 Symplectic Group Actions and Covering Spaces James Montaldi & Juan-Pablo Ortega January 2009 Abstract For symplectic group actions which are not Hamiltonian there are two ways to define reduction. Firstly using the cylinder-valued momentum map and secondly lifting the action to any Hamiltonian cover (such as the universal cover), and then performing symplectic reduction in the usual way. We show that provided the action is free and proper, and the Hamiltonian holonomy associated to the action is closed, the natural projection from the latter to the former is a symplectic cover. At the same time we give a classification of all Hamiltonian covers of a given symplectic group action. The main properties of the lifting of a group action to a cover are studied. Keywords: lifted group action, symplectic reduction, universal cover, Hamiltonian holonomy, momentum map MSC2000: 53D20, 37J15. Introduction There are many instances of symplectic group actions which are not Hamiltonian—ie, for which there is no momentum map. These can occur both in applications [13] as well as in fundamental studies of symplectic geometry [1, 2, 5]. In such cases it is possible to define a “cylinder valued momentum map” [3], and then perform symplectic reduction with respect to this map [16, 17]. -
On Modeling Homotopy Type Theory in Higher Toposes
Review: model categories for type theory Left exact localizations Injective fibrations On modeling homotopy type theory in higher toposes Mike Shulman1 1(University of San Diego) Midwest homotopy type theory seminar Indiana University Bloomington March 9, 2019 Review: model categories for type theory Left exact localizations Injective fibrations Here we go Theorem Every Grothendieck (1; 1)-topos can be presented by a model category that interprets \Book" Homotopy Type Theory with: • Σ-types, a unit type, Π-types with function extensionality, and identity types. • Strict universes, closed under all the above type formers, and satisfying univalence and the propositional resizing axiom. Review: model categories for type theory Left exact localizations Injective fibrations Here we go Theorem Every Grothendieck (1; 1)-topos can be presented by a model category that interprets \Book" Homotopy Type Theory with: • Σ-types, a unit type, Π-types with function extensionality, and identity types. • Strict universes, closed under all the above type formers, and satisfying univalence and the propositional resizing axiom. Review: model categories for type theory Left exact localizations Injective fibrations Some caveats 1 Classical metatheory: ZFC with inaccessible cardinals. 2 Classical homotopy theory: simplicial sets. (It's not clear which cubical sets can even model the (1; 1)-topos of 1-groupoids.) 3 Will not mention \elementary (1; 1)-toposes" (though we can deduce partial results about them by Yoneda embedding). 4 Not the full \internal language hypothesis" that some \homotopy theory of type theories" is equivalent to the homotopy theory of some kind of (1; 1)-category. Only a unidirectional interpretation | in the useful direction! 5 We assume the initiality hypothesis: a \model of type theory" means a CwF. -
A Godefroy-Kalton Principle for Free Banach Lattices 3
A GODEFROY-KALTON PRINCIPLE FOR FREE BANACH LATTICES ANTONIO AVILES,´ GONZALO MART´INEZ-CERVANTES, JOSE´ RODR´IGUEZ, AND PEDRO TRADACETE Abstract. Motivated by the Lipschitz-lifting property of Banach spaces introduced by Godefroy and Kalton, we consider the lattice-lifting property, which is an analogous notion within the category of Ba- nach lattices and lattice homomorphisms. Namely, a Banach lattice X satisfies the lattice-lifting property if every lattice homomorphism to X having a bounded linear right-inverse must have a lattice homomor- phism right-inverse. In terms of free Banach lattices, this can be rephrased into the following question: which Banach lattices embed into the free Banach lattice which they generate as a lattice-complemented sublattice? We will provide necessary conditions for a Banach lattice to have the lattice-lifting property, and show that this property is shared by Banach spaces with a 1-unconditional basis as well as free Banach lattices. The case of C(K) spaces will also be analyzed. 1. Introduction In a fundamental paper concerning the Lipschitz structure of Banach spaces, G. Godefroy and N.J. Kalton introduced the Lipschitz-lifting property of a Banach space. In order to properly introduce this notion, and as a motivation for our work, let us recall the basic ingredients for this construction (see [9] for details). Given a Banach space E, let Lip0(E) denote the Banach space of all real-valued Lipschitz functions on E which vanish at 0, equipped with the norm |f(x) − f(y)| kfk = sup : x, y ∈ E, x 6= y . Lip0(E) kx − yk E The Lipschitz-free space over E, denoted by F(E), is the canonical predual of Lip0(E), that is the closed ∗ linear span of the evaluation functionals δ(x) ∈ Lip0(E) given by hδ(x),fi = f(x) for all x ∈ E and all f ∈ Lip0(E). -
Lecture 11 Last Lecture We Defined Connections on Fiber Bundles. This Lecture We Will Attempt to Explain What a Connection Does
Lecture 11 Last lecture we defined connections on fiber bundles. This lecture we will attempt to explain what a connection does for differential ge- ometers. We start with two basic concepts from algebraic topology. The first is that of homotopy lifting property. Let us recall: Definition 1. Let π : E → B be a continuous map and X a space. We say that (X, π) satisfies homotopy lifting property if for every H : X × I → B and map f : X → E with π ◦ f = H( , 0) there exists a homotopy F : X × I → E starting at f such that π ◦ F = H. If ({∗}, π) satisfies this property then we say that π has the homotopy lifting property for a point. Now, as is well known and easily proven, every fiber bundle satisfies the homotopy lifting property. A much stronger property is the notion of a covering space (which we will not recall with precision). Here we have the homotopy lifting property for a point plus uniqueness or the unique path lifting property. I.e. there is exactly one lift of a given path in the base starting at a specified point in the total space. The point of a connection is to take the ambiguity out of the homotopy lifting property for X = {∗} and force a fiber bundle to satisfy the unique path lifting property. We will see many conceptual advantages of this uniqueness, but let us start by making this a bit clearer via parallel transport. Definition 2. Let B be a topological space and PB the category whose objects are points of B and whose morphisms are defined via {γ : → B : ∃ c, d ∈ s.t. -
Fibrations II - the Fundamental Lifting Property
Fibrations II - The Fundamental Lifting Property Tyrone Cutler July 13, 2020 Contents 1 The Fundamental Lifting Property 1 2 Spaces Over B 3 2.1 Homotopy in T op=B ............................... 7 3 The Homotopy Theorem 8 3.1 Implications . 12 4 Transport 14 4.1 Implications . 16 5 Proof of the Fundamental Lifting Property Completed. 19 6 The Mutal Characterisation of Cofibrations and Fibrations 20 6.1 Implications . 22 1 The Fundamental Lifting Property This section is devoted to stating Theorem 1.1 and beginning its proof. We prove only the first of its two statements here. This part of the theorem will then be used in the sequel, and the proof of the second statement will follow from the results obtained in the next section. We will be careful to avoid circular reasoning. The utility of the theorem will soon become obvious as we repeatedly use its statement to produce maps having very specific properties. However, the true power of the theorem is not unveiled until x 5, where we show how it leads to a mutual characterisation of cofibrations and fibrations in terms of an orthogonality relation. Theorem 1.1 Let j : A,! X be a closed cofibration and p : E ! B a fibration. Assume given the solid part of the following strictly commutative diagram f A / E |> j h | p (1.1) | | X g / B: 1 Then the dotted filler can be completed so as to make the whole diagram commute if either of the following two conditions are met • j is a homotopy equivalence. • p is a homotopy equivalence. -
1 the Spin Homomorphism SL2(C) → SO1,3(R) a Summary from Multiple Sources Written by Y
1 The spin homomorphism SL2(C) ! SO1;3(R) A summary from multiple sources written by Y. Feng and Katherine E. Stange Abstract We will discuss the spin homomorphism SL2(C) ! SO1;3(R) in three manners. Firstly we interpret SL2(C) as acting on the Minkowski 1;3 spacetime R ; secondly by viewing the quadratic form as a twisted 1 1 P × P ; and finally using Clifford groups. 1.1 Introduction The spin homomorphism SL2(C) ! SO1;3(R) is a homomorphism of classical matrix Lie groups. The lefthand group con- sists of 2 × 2 complex matrices with determinant 1. The righthand group consists of 4 × 4 real matrices with determinant 1 which preserve some fixed real quadratic form Q of signature (1; 3). This map is alternately called the spinor map and variations. The image of this map is the identity component + of SO1;3(R), denoted SO1;3(R). The kernel is {±Ig. Therefore, we obtain an isomorphism + PSL2(C) = SL2(C)= ± I ' SO1;3(R): This is one of a family of isomorphisms of Lie groups called exceptional iso- morphisms. In Section 1.3, we give the spin homomorphism explicitly, al- though these formulae are unenlightening by themselves. In Section 1.4 we describe O1;3(R) in greater detail as the group of Lorentz transformations. This document describes this homomorphism from three distinct per- spectives. The first is very concrete, and constructs, using the language of Minkowski space, Lorentz transformations and Hermitian matrices, an ex- 4 plicit action of SL2(C) on R preserving Q (Section 1.5). -
Algebraic Topology (Math 592): Problem Set 6
Algebraic topology (Math 592): Problem set 6 Bhargav Bhatt All spaces are assumed to be Hausdorff and locally path-connected, i.e., there exists a basis of path-connected open subsets of X. 1. Let f : X ! Y be a covering space. Assume that there exists s : Y ! X such that f ◦ s = id. Show that s(Y ) ⊂ X is clopen. Moreover, also check that (X − s(Y )) ! Y is a covering space, provided X 6= s(Y ) and Y is path-connected. 2. Let α : Z ! Y be a map of spaces. Fix a base point z 2 Z, and assume that Z and Y are path-connected, and that Y admits a universal cover. Show that α∗ : π1(Z; z) ! π1(Y; α(z)) is surjective if and only if for every connected covering space X ! Y , the pullback X ×Y Z ! Z is connected. All spaces are now further assumed to be path-connected unless otherwise specified. 3. Let G be a topological group, and let p : H ! G be a covering space. Fix a base point h 2 H living over the identity element e 2 G. Using the lifting property, show that H admits a unique topological group structure with identity element h such that p is a group homomorphism. Using a previous problem set, show that the kernel of p is abelian. 4. Let f : X ! Y be a covering space. Assume that there exists y 2 Y such that #f −1(y) = d is finite. Show that #f −1(y0) = d for all y0 2 Y . -
Lecture 34: Hadamaard's Theorem and Covering Spaces
Lecture 34: Hadamaard’s Theorem and Covering spaces In the next few lectures, we’ll prove the following statement: Theorem 34.1. Let (M,g) be a complete Riemannian manifold. Assume that for every p M and for every x, y T M, we have ∈ ∈ p K(x, y) 0. ≤ Then for any p M,theexponentialmap ∈ exp : T M M p p → is a local diffeomorphism, and a covering map. 1. Complete Riemannian manifolds Recall that a connected Riemannian manifold (M,g)iscomplete if the geodesic equation has a solution for all time t.Thisprecludesexamplessuch as R2 0 with the standard Euclidean metric. −{ } 2. Covering spaces This, not everybody is familiar with, so let me give a quick introduction to covering spaces. Definition 34.2. Let p : M˜ M be a continuous map between topological spaces. p is said to be a covering→ map if (1) p is a local homeomorphism, and (2) p is evenly covered—that is, for every x M,thereexistsaneighbor- 1 ∈ hood U M so that p− (U)isadisjointunionofspacesU˜ for which ⊂ α p ˜ is a homeomorphism onto U. |Uα Example 34.3. Here are some standard examples: (1) The identity map M M is a covering map. (2) The obvious map M →... M M is a covering map. → (3) The exponential map t exp(it)fromR to S1 is a covering map. First, it is clearly a local → homeomorphism—a small enough open in- terval in R is mapped homeomorphically onto an open interval in S1. 113 As for the evenly covered property—if you choose a proper open in- terval inside S1, then its preimage under the exponential map is a disjoint union of open intervals in R,allhomeomorphictotheopen interval in S1 via the exponential map. -
The Quillen Model Category of Topological Spaces 3
THE QUILLEN MODEL CATEGORY OF TOPOLOGICAL SPACES PHILIP S. HIRSCHHORN Abstract. We give a complete and careful proof of Quillen’s theorem on the existence of the standard model category structure on the category of topological spaces. We do not assume any familiarity with model categories. Contents 1. Introduction 2 2. Definitions and the main theorem 2 3. Lifting 4 3.1. Pushouts, pullbacks, and lifting 5 3.2. Coproducts, transfinite composition, and lifting 5 3.3. Retracts and lifting 6 4. Relative cell complexes 7 4.1. Compact subsets of relative cell complexes 9 4.2. Relative cell complexes and lifting 10 5. The small object argument 11 5.1. The first factorization 11 5.2. The second factorization 13 6. The factorization axiom 15 6.1. Cofibration and trivial fibration 15 6.2. Trivial cofibration and fibration 16 7. Homotopygroupsandmapsofdisks 16 7.1. Difference maps 17 7.2. Lifting maps of disks 18 arXiv:1508.01942v3 [math.AT] 22 Oct 2017 8. The lifting axiom 20 8.1. Cofibrations and trivial fibrations 20 8.2. Trivial cofibrations and fibrations 21 9. The proof of Theorem 2.5 21 References 22 Date: September 11, 2017. 2010 Mathematics Subject Classification. Primary 18G55, 55U35. Key words and phrases. model category, small object argument, relative cell complex, cell complex, difference map. c 2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/. 1 2 PHILIP S. HIRSCHHORN 1. Introduction Quillen defined model categories (see Definition 2.2) in [5, 6] to apply the tech- niques of homotopy theory to categories other than topological spaces or simplicial sets. -
COVERING SPACES, GRAPHS, and GROUPS Contents 1. Covering
COVERING SPACES, GRAPHS, AND GROUPS CARSON COLLINS Abstract. We introduce the theory of covering spaces, with emphasis on explaining the Galois correspondence of covering spaces and the deck trans- formation group. We focus especially on the topological properties of Cayley graphs and the information these can give us about their corresponding groups. At the end of the paper, we apply our results in topology to prove a difficult theorem on free groups. Contents 1. Covering Spaces 1 2. The Fundamental Group 5 3. Lifts 6 4. The Universal Covering Space 9 5. The Deck Transformation Group 12 6. The Fundamental Group of a Cayley Graph 16 7. The Nielsen-Schreier Theorem 19 Acknowledgments 20 References 20 1. Covering Spaces The aim of this paper is to introduce the theory of covering spaces in algebraic topology and demonstrate a few of its applications to group theory using graphs. The exposition assumes that the reader is already familiar with basic topological terms, and roughly follows Chapter 1 of Allen Hatcher's Algebraic Topology. We begin by introducing the covering space, which will be the main focus of this paper. Definition 1.1. A covering space of X is a space Xe (also called the covering space) equipped with a continuous, surjective map p : Xe ! X (called the covering map) which is a local homeomorphism. Specifically, for every point x 2 X there is some open neighborhood U of x such that p−1(U) is the union of disjoint open subsets Vλ of Xe, such that the restriction pjVλ for each Vλ is a homeomorphism onto U.