DOCSLIB.ORG

Fibre Bundles

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Fibre Bundles Andrew Kobin Spring 2017 / Spring 2018 Contents Contents Contents 0 Introduction 1 1 Fibre Bundles 2 1.1 Locally Trivial Fibre Bundles . .2 1.2 Principal Bundles . .5 1.3 Classifying Spaces and Universal Bundles . 10 1.4 Reduction of the Structure Group . 16 1.5 Gauge Groups . 19 2 Vector Bundles 22 2.1 Vector and Tangent Bundles . 22 2.2 Sections . 25 2.3 The Induced Bundle . 28 2.4 Bundle Operations . 30 3 Characteristic Classes 34 3.1 Metrics on Vector Bundles . 34 3.2 Stiefel-Whitney Classes . 35 3.3 The Universal Bundle . 44 3.4 Construction of Stiefel-Whitney Classes . 51 3.5 Oriented Bundles . 53 3.6 Chern Classes . 55 3.7 Pontrjagin Classes . 57 3.8 General Characteristic Classes . 58 4 K-Theory 70 4.1 K-Theory and Reduced K-Theory . 71 4.2 External Products in K-theory . 73 4.3 Bott Periodicity . 75 4.4 Characteristic Classes in K-Theory . 82 5 Chern-Weil Theory 85 5.1 Connections on a Vector Bundle . 86 5.2 Characteristic Classes from Curvature . 92 5.3 The Levi-Civita Connection . 100 6 Index Theory 103 6.1 Differential Operators . 103 6.2 Hodge Decomposition . 108 6.3 The Index Theorem . 110 6.4 Dolbeault Cohomology . 113 i 0 Introduction 0 Introduction This document follows two courses in fibre bundles taught respectively by Dr. Slava Krushkal in Spring 2017 and Dr. Thomas Mark in Spring 2018 at the University of Virginia. The main concepts covered are: Locally trivial fibre bundles Principal bundles and classifying spaces Vector bundles and their important properties Characteristic classes, including Stiefel-Whitney, Chern and Pontryagin classes Orientability Euler class Elements of K-theory Connections and Chern-Weil theory The Atiyah-Singer index theorem. Standard references include: Milnor-Stasheff's Characteristic Classes, Hatcher's Vector Bun- dles and K-Theory and Husem¨oller's Fibre Bundles. 1 1 Fibre Bundles 1 Fibre Bundles 1.1 Locally Trivial Fibre Bundles Definition. A bundle is a triple (E; B; p) consisting of topological spaces E; B and a con- tinuous map between them, p : E ! B. We call E the total space of the bundle and B the base space. Definition. A morphism of bundles (E1;B1; p1) ! (E2;B2; p2) is a pair of maps f : ~ B1 ! B2; f : E1 ! E2, making the diagram f~ E1 E2 p1 p2 f B1 B2 −1 ~ −1 commute and preserving fibres, that is, for any x 2 B1 and y 2 p1 (x), f(y) 2 p2 (f(x)). Example 1.1.1. When f is the identity B ! B, we say the bundles and morphisms of bundles are over B. p Example 1.1.2. Let E −! B be a bundle and A ⊆ B be a subspace. Then the restriction −1 pjA : EjA := p (A) ! A is a bundle over A and the natural inclusion EjA ,! E induces a bundle morphism EjA E pjA p A B Example 1.1.3. Let F be any space. The trivial bundle over B with fibre F is the canonical projection p : B × F ! B. Definition. A locally trivial fibre bundle over B with fibre F , hereafter shortened to a p fibre bundle, is a bundle E −! B that is locally isomorphic to a trivial bundle with fibre F . That is, there exists a collection of open sets fUig together with bundle isomorphisms ∼ 'i : EjUi = Ui × F , called local trivializations. p Given a fibre bundle E −! B with locally trivial open sets fUig, there is a choice of −1 isomorphism 'i : EjUi ! Ui for each Ui. For each x 2 Ui, p (x) is homeomorphic to F via −1 'i, so when x 2 Ui \Uj, there are two different ways to identify p (x) with F . In particular, −1 'j ◦ 'i :(Ui \ Uj) × F ! (Ui \ Uj) × F is a homeomorphism and induces a homeomorphism F ! F . In other words, for each pair of open sets Ui;Uj, there is a function gij : Ui \ Uj −! Homeo(F ) 2 1.1 Locally Trivial Fibre Bundles 1 Fibre Bundles called a transition map. One can show that the gij are continuous. Note that the gij satisfy a few nice conditions, namely that gii = 1, gijgji = 1 and gijgjkgki = 1 pointwise on F . In fact, the data of a covering fUig and continuous transition maps fgijg satisfying these conditions are enough to determine a fibre bundle. Proposition 1.1.4. Let B and F be spaces and suppose there is an open cover fUig of B and continuous maps gij : Ui \ Uj ! Homeo(F ) satisfying: (a) For all x 2 Ui, gii(x) = 1, where 1 is the identity map F ! F . (b) For all x 2 Ui \ Uj, gij(x)gji(x) = 1. (c) For all x 2 Ui \ Uj \ Uk, gij(x)gjk(x)gki(x) = 1. Then there exists a fibre bundle over B with fibre F which is trivial over the Ui and has the gij as its transition maps. Moreover, such a bundle is unique up to isomorphism of bundles over B. Proof. (Sketch) Define the total space of the bundle by Y E := (Ui × F )= ∼ i where ∼ is the equivalence relation identifying (x; y) 2 Ui ×F and (x; gij(x)(y)) 2 Uj ×F for any x 2 Ui \ Uj and y 2 F . The bundle map p : E ! B is given by the natural projection on each disjoint piece, which is well-defined on all of E by (a) { (c) and is continuous since p the gij are continuous. It is straightforward to check that E −! B is a locally trivial bundle with fibre F having transition maps gij. p~ For uniqueness, suppose Ee −! B is another bundle with gij as its transition maps. For −1 each Ui, there is a fibre-preserving homeomorphismp ~ (Ui) ! Ui × F . Define a bundle map −1 Ee ! E by sending a pointx ~ 2 p (Ui) to the corresponding point in Ui × F ⊆ E. One now checks that this is a bundle isomorphism. p1 p2 Given two bundles E1 −! B and E2 −! B with fibre F and local trivializations f(Ui;'i)g and f(Vj; j)g, respectively, when can we tell if they are isomorphic? First, we may refine the covers fUig and fVjg by taking intersections until each bundle is trivial over both collections. Thus we may assume both bundles are trivial over fUig. Suppose f : E1 ! E2 is an isomorphism of bundles over B. Then each Ui corresponds to a local isomorphism of bundles −1 fi := i ◦ f ◦ 'i : Ui × F ! E1jUi ! E2jUi ! Ui × F: In fact, this gives us a necessary and sufficient condition for f to induce a bundle isomorphism. Lemma 1.1.5. The map f : E1 ! E2 is an isomorphism of bundles over B if and only if there exist local bundle isomorphisms fi : Ui × F ! Ui × F that are compatible with the −1 −1 transition maps 'j ◦ 'i and j ◦ i , i.e. the following diagram commutes: 3 1.1 Locally Trivial Fibre Bundles 1 Fibre Bundles fi (Ui \ Uj) × F (Ui \ Uj) × F −1 −1 'j ◦ 'i j ◦ i fj (Ui \ Uj) × F (Ui \ Uj) × F Example 1.1.6. Let M ⊆ RN be a smooth manifold and define the tangent bundle to M by N N TM = f(x; v) 2 R × R j x 2 M; v 2 TxMg: The projection p : TM ! M; (x; v) 7! x gives a fibre bundle structure on TM with fibre −1 ∼ n p (x) = TxM = R if M is an n-manifold. If f(Ui;'i)g is an atlas of coordinate charts for the manifold structure on M, then the derivatives of their transition maps, −1 gij : x 7−! d'i(x)('j ◦ 'i ); n define the transition maps gij : Ui \ Uj ! Homeo(R ) of the tangent bundle. Since these n derivatives are linear, the gij in fact have target GLn(R) ⊂ Homeo(R ). This reflects the fact that TM is a vector bundle (see Chapter 2). For example, let M = S2 be the 2-sphere. We may view S2 = CP 1, the complex projective line, which has a covering by two coordinate charts U0 = f[z0; z0] j z0 6= 0g and U1 = f[z0; z1] j z1 6= 0g z1 z0 with homeomorphisms '0 : U0 ! ; [z0; z1] 7! and '1 : U1 ! ; [z0; z1] 7! . Then C z0 C z1 −1 1 the transition maps 'j ◦ 'i : C ! C are both given by λ 7! λ . Thus the transition maps 2 1 g01; g10 for the tangent bundle TS are each given by λ 7! − λ2 , viewed as a 2×2 matrix with × × × real entries in GL2(R), or a scalar element of GL1(C) = C . Notice that g01 : C ! C is homotopic to a map S1 ! S1 of degree −2. It is no accident that S2 has Euler characteristic 2 { we will see this connection explicitly borne out by the Euler class (see Section 3.5). Let G be a topological group and F a space on which G acts by homeomorphisms, i.e. such that there is a continuous homomorphism ρ : G ! Homeo(F ). Definition. A fibre bundle with fibre F has structure group G if it has transition functions gij : Ui \ Uj ! G. Example 1.1.7. Fibre bundles with fibre F = Rn (resp. F = Cn) and structure group G = GLn(R) (resp. GLn(C)) are called real (resp. complex) vector bundles (see Chapter 2). Example 1.1.8. A bundle with fibre F = Sn and structure group G = O(n + 1) is called an (orthogonal) sphere bundle.
Recommended publications
  • Diagonal Lifts of Metrics to Coframe Bundle

    Diagonal Lifts of Metrics to Coframe Bundle

    Proceedings of the Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan Volume 44, Number 2, 2018, Pages 328{337 DIAGONAL LIFTS OF METRICS TO COFRAME BUNDLE HABIL FATTAYEV AND ARIF SALIMOV Abstract. In this paper the diagonal lift Dg of a Riemannian metric g ∗ of a manifold Mn to the coframe bundle F (Mn) is defined, Levi-Civita connection, Killing vector fields with respect to the metric Dg and also an almost paracomplex structures in the coframe bundle are studied. 1. Introduction The Riemannian metrics in the tangent bundle firstly has been investigated by the Sasaki [14]. Tondeur [16] and Sato [15] have constructed Riemannian metrics on the cotangent bundle, the construction being the analogue of the metric Sasaki for the tangent bundle. Mok [7] has defined so-called the diagonal lift of metric to the linear frame bundle, which is a Riemannian metric resembles the Sasaki metric of tangent bundle. Some properties and applications for the Riemannian metrics of the tangent, cotangent, linear frame and tensor bundles are given in [1-4,7-9,12,13]. This paper is devoted to the investigation of Riemannian metrics in the coframe bundle. In 2 we briefly describe the definitions and results that are needed later, after which the diagonal lift Dg of a Riemannian metric g is constructed in 3. The Levi-Civita connection of the metric Dg is determined in In 4. In 5 we consider Killing vector fields in coframe bundle with respect to Riemannian metric Dg. An almost paracomplex structures in the coframe bundle equipped with metric Dg are studied in 6.
  • Orientability of Real Parts and Spin Structures

    Orientability of Real Parts and Spin Structures

    JP Jour. Geometry & Topology 7 (2007) 159-174 ORIENTABILITY OF REAL PARTS AND SPIN STRUCTURES SHUGUANG WANG Abstract. We establish the orientability and orientations of vector bundles that arise as the real parts of real structures by utilizing spin structures. 1. Introduction. Unlike complex algebraic varieties, real algebraic varieties are in general nonorientable, the simplest example being the real projective plane RP2. Even if they are orientable, there may not be canonical orientations. It has been an important problem to resolve the orientability and orientation issues in real algebraic geometry. In 1974, Rokhlin introduced the complex orientation for dividing real algebraic curves in RP2, which was then extended around 1982 by Viro to the so-called type-I real algebraic surfaces. A detailed historic count was presented in the lucid survey by Viro [10], where he also made some speculations on higher dimensional varieties. In this short note, we investigate the following more general situation. We take σ : X → X to be a smooth involution on a smooth manifold of an arbitrary dimension. (It is possible to consider involutions on topological manifolds with appropriate modifications.) Henceforth, we will assume that X is connected for certainty. In view of the motivation above, let us denote the fixed point set by XR, which in general is disconnected and will be assumed to be non-empty throughout the paper. Suppose E → X is a complex vector bundle and assume σ has an involutional lifting σE on E that is conjugate linear fiberwise. We call σE a real structure on E and its fixed point set ER a real part.
  • Realizing Cohomology Classes As Euler Classes 1

    Realizing Cohomology Classes As Euler Classes 1

    Ó DOI: 10.2478/s12175-012-0057-2 Math. Slovaca 62 (2012), No. 5, 949–966 REALIZING COHOMOLOGY CLASSES AS EULER CLASSES Aniruddha C Naolekar (Communicated by J´ulius Korbaˇs ) s ◦ ABSTRACT. For a space X,letEk(X), Ek(X)andEk (X) denote respectively the set of Euler classes of oriented k-plane bundles over X, the set of Euler classes of stably trivial k-plane bundles over X and the spherical classes in Hk(X; Z). s ◦ We prove some general facts about the sets Ek(X), Ek(X)andEk (X). We also compute these sets in the cases where X is a projective space, the Dold manifold P (m, 1) and obtain partial computations in the case that X is a product of spheres. c 2012 Mathematical Institute Slovak Academy of Sciences 1. Introduction Given a topological space X, we address the question of which classes x ∈ Hk(X; Z) can be realized as the Euler class e(ξ)ofanorientedk-plane bundle ξ over X. We also look at the question of which integral cohomology classes are spherical. It is convenient to make the following definition. s º ≥ ÒØÓÒ 1.1 For a space X and an integer k 1, the sets Ek(X), Ek(X) ◦ and Ek (X) are defined to be k Ek(X)= e(ξ) ∈ H (X; Z) | ξ is an oriented k-plane bundle over X s k Ek(X)= e(ξ) ∈ H (X; Z) | ξ is a stably trivial k-plane bundle over X ◦ k k ∗ Ek (X)= x ∈ H (X; Z) | there exists f : S −→ X with f (x) =0 .
  • The Jouanolou-Thomason Homotopy Lemma

    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.
  • Horizontal Holonomy and Foliated Manifolds Yacine Chitour, Erlend Grong, Frédéric Jean, Petri Kokkonen

    Horizontal Holonomy and Foliated Manifolds Yacine Chitour, Erlend Grong, Frédéric Jean, Petri Kokkonen

    Horizontal holonomy and foliated manifolds Yacine Chitour, Erlend Grong, Frédéric Jean, Petri Kokkonen To cite this version: Yacine Chitour, Erlend Grong, Frédéric Jean, Petri Kokkonen. Horizontal holonomy and foliated manifolds. Annales de l’Institut Fourier, Association des Annales de l’Institut Fourier, 2019, 69 (3), pp.1047-1086. 10.5802/aif.3265. hal-01268119 HAL Id: hal-01268119 https://hal-ensta-paris.archives-ouvertes.fr//hal-01268119 Submitted on 8 Mar 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. HORIZONTAL HOLONOMY AND FOLIATED MANIFOLDS YACINE CHITOUR, ERLEND GRONG, FRED´ ERIC´ JEAN AND PETRI KOKKONEN Abstract. We introduce horizontal holonomy groups, which are groups de- fined using parallel transport only along curves tangent to a given subbundle D of the tangent bundle. We provide explicit means of computing these holo- nomy groups by deriving analogues of Ambrose-Singer's and Ozeki's theorems. We then give necessary and sufficient conditions in terms of the horizontal ho- lonomy groups for existence of solutions of two problems on foliated manifolds: determining when a foliation can be either (a) totally geodesic or (b) endowed with a principal bundle structure.
  • Arxiv:Gr-Qc/0611154 V1 30 Nov 2006 Otx Fcra Geometry

    Arxiv:Gr-Qc/0611154 V1 30 Nov 2006 Otx Fcra Geometry

    MacDowell–Mansouri Gravity and Cartan Geometry Derek K. Wise Department of Mathematics University of California Riverside, CA 92521, USA email: [email protected] November 29, 2006 Abstract The geometric content of the MacDowell–Mansouri formulation of general relativity is best understood in terms of Cartan geometry. In particular, Cartan geometry gives clear geomet- ric meaning to the MacDowell–Mansouri trick of combining the Levi–Civita connection and coframe field, or soldering form, into a single physical field. The Cartan perspective allows us to view physical spacetime as tangentially approximated by an arbitrary homogeneous ‘model spacetime’, including not only the flat Minkowski model, as is implicitly used in standard general relativity, but also de Sitter, anti de Sitter, or other models. A ‘Cartan connection’ gives a prescription for parallel transport from one ‘tangent model spacetime’ to another, along any path, giving a natural interpretation of the MacDowell–Mansouri connection as ‘rolling’ the model spacetime along physical spacetime. I explain Cartan geometry, and ‘Cartan gauge theory’, in which the gauge field is replaced by a Cartan connection. In particular, I discuss MacDowell–Mansouri gravity, as well as its recent reformulation in terms of BF theory, in the arXiv:gr-qc/0611154 v1 30 Nov 2006 context of Cartan geometry. 1 Contents 1 Introduction 3 2 Homogeneous spacetimes and Klein geometry 8 2.1 Kleingeometry ................................... 8 2.2 MetricKleingeometry ............................. 10 2.3 Homogeneousmodelspacetimes. ..... 11 3 Cartan geometry 13 3.1 Ehresmannconnections . .. .. .. .. .. .. .. .. 13 3.2 Definition of Cartan geometry . ..... 14 3.3 Geometric interpretation: rolling Klein geometries . .............. 15 3.4 ReductiveCartangeometry . 17 4 Cartan-type gauge theory 20 4.1 Asequenceofbundles .............................
  • Vector Bundles on Projective Space

    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.
  • Notes on the Atiyah-Singer Index Theorem Liviu I. Nicolaescu

    Notes on the Atiyah-Singer Index Theorem Liviu I. Nicolaescu

    Notes on the Atiyah-Singer Index Theorem Liviu I. Nicolaescu Notes for a topics in topology course, University of Notre Dame, Spring 2004, Spring 2013. Last revision: November 15, 2013 i The Atiyah-Singer Index Theorem This is arguably one of the deepest and most beautiful results in modern geometry, and in my view is a must know for any geometer/topologist. It has to do with elliptic partial differential opera- tors on a compact manifold, namely those operators P with the property that dim ker P; dim coker P < 1. In general these integers are very difficult to compute without some very precise information about P . Remarkably, their difference, called the index of P , is a “soft” quantity in the sense that its determination can be carried out relying only on topological tools. You should compare this with the following elementary situation. m n Suppose we are given a linear operator A : C ! C . From this information alone we cannot compute the dimension of its kernel or of its cokernel. We can however compute their difference which, according to the rank-nullity theorem for n×m matrices must be dim ker A−dim coker A = m − n. Michael Atiyah and Isadore Singer have shown in the 1960s that the index of an elliptic operator is determined by certain cohomology classes on the background manifold. These cohomology classes are in turn topological invariants of the vector bundles on which the differential operator acts and the homotopy class of the principal symbol of the operator. Moreover, they proved that in order to understand the index problem for an arbitrary elliptic operator it suffices to understand the index problem for a very special class of first order elliptic operators, namely the Dirac type elliptic operators.
  • Stable Isomorphism Vs Isomorphism of Vector Bundles: an Application to Quantum Systems

    Stable Isomorphism Vs Isomorphism of Vector Bundles: an Application to Quantum Systems

    Alma Mater Studiorum · Universita` di Bologna Scuola di Scienze Corso di Laurea in Matematica STABLE ISOMORPHISM VS ISOMORPHISM OF VECTOR BUNDLES: AN APPLICATION TO QUANTUM SYSTEMS Tesi di Laurea in Geometria Differenziale Relatrice: Presentata da: Prof.ssa CLARA PUNZI ALESSIA CATTABRIGA Correlatore: Chiar.mo Prof. RALF MEYER VI Sessione Anno Accademico 2017/2018 Abstract La classificazione dei materiali sulla base delle fasi topologiche della materia porta allo studio di particolari fibrati vettoriali sul d-toro con alcune strut- ture aggiuntive. Solitamente, tale classificazione si fonda sulla nozione di isomorfismo tra fibrati vettoriali; tuttavia, quando il sistema soddisfa alcune assunzioni e ha dimensione abbastanza elevata, alcuni autori ritengono in- vece sufficiente utilizzare come relazione d0equivalenza quella meno fine di isomorfismo stabile. Scopo di questa tesi `efissare le condizioni per le quali la relazione di isomorfismo stabile pu`osostituire quella di isomorfismo senza generare inesattezze. Ci`onei particolari casi in cui il sistema fisico quantis- tico studiato non ha simmetrie oppure `edotato della simmetria discreta di inversione temporale. Contents Introduction 3 1 The background of the non-equivariant problem 5 1.1 CW-complexes . .5 1.2 Bundles . 11 1.3 Vector bundles . 15 2 Stability properties of vector bundles 21 2.1 Homotopy properties of vector bundles . 21 2.2 Stability . 23 3 Quantum mechanical systems 28 3.1 The single-particle model . 28 3.2 Topological phases and Bloch bundles . 32 4 The equivariant problem 36 4.1 Involution spaces and general G-spaces . 36 4.2 G-CW-complexes . 39 4.3 \Real" vector bundles . 41 4.4 \Quaternionic" vector bundles .
  • LECTURE 6: FIBER BUNDLES in This Section We Will Introduce The

    LECTURE 6: FIBER BUNDLES in This Section We Will Introduce The

    LECTURE 6: FIBER BUNDLES In this section we will introduce the interesting class of fibrations given by fiber bundles. Fiber bundles play an important role in many geometric contexts. For example, the Grassmaniann varieties and certain fiber bundles associated to Stiefel varieties are central in the classification of vector bundles over (nice) spaces. The fact that fiber bundles are examples of Serre fibrations follows from Theorem ?? which states that being a Serre fibration is a local property. 1. Fiber bundles and principal bundles Definition 6.1. A fiber bundle with fiber F is a map p: E ! X with the following property: every ∼ −1 point x 2 X has a neighborhood U ⊆ X for which there is a homeomorphism φU : U × F = p (U) such that the following diagram commutes in which π1 : U × F ! U is the projection on the first factor: φ U × F U / p−1(U) ∼= π1 p * U t Remark 6.2. The projection X × F ! X is an example of a fiber bundle: it is called the trivial bundle over X with fiber F . By definition, a fiber bundle is a map which is `locally' homeomorphic to a trivial bundle. The homeomorphism φU in the definition is a local trivialization of the bundle, or a trivialization over U. Let us begin with an interesting subclass. A fiber bundle whose fiber F is a discrete space is (by definition) a covering projection (with fiber F ). For example, the exponential map R ! S1 is a covering projection with fiber Z. Suppose X is a space which is path-connected and locally simply connected (in fact, the weaker condition of being semi-locally simply connected would be enough for the following construction).
  • K-Theoryand Characteristic Classes

    K-Theoryand Characteristic Classes

    MATH 6530: K-THEORY AND CHARACTERISTIC CLASSES Taught by Inna Zakharevich Notes by David Mehrle [email protected] Cornell University Fall 2017 Last updated November 8, 2018. The latest version is online here. Contents 1 Vector bundles................................ 3 1.1 Grassmannians ............................ 8 1.2 Classification of Vector bundles.................... 11 2 Cohomology and Characteristic Classes................. 15 2.1 Cohomology of Grassmannians................... 18 2.2 Characteristic Classes......................... 22 2.3 Axioms for Stiefel-Whitney classes................. 26 2.4 Some computations.......................... 29 3 Cobordism.................................. 35 3.1 Stiefel-Whitney Numbers ...................... 35 3.2 Cobordism Groups.......................... 37 3.3 Geometry of Thom Spaces...................... 38 3.4 L-equivalence and Transversality.................. 42 3.5 Characteristic Numbers and Boundaries.............. 47 4 K-Theory................................... 49 4.1 Bott Periodicity ............................ 49 4.2 The K-theory spectrum........................ 56 4.3 Some properties of K-theory..................... 58 4.4 An example: K-theory of S2 ..................... 59 4.5 Power Operations............................ 61 4.6 When is the Hopf Invariant one?.................. 64 4.7 The Splitting Principle........................ 66 5 Where do we go from here?........................ 68 5.1 The J-homomorphism ........................ 70 5.2 The Chern Character and e invariant...............
  • Isometry Types of Frame Bundles

    Isometry Types of Frame Bundles

    Pacific Journal of Mathematics ISOMETRY TYPES OF FRAME BUNDLES WOUTER VAN LIMBEEK Volume 285 No. 2 December 2016 PACIFIC JOURNAL OF MATHEMATICS Vol. 285, No. 2, 2016 dx.doi.org/10.2140/pjm.2016.285.393 ISOMETRY TYPES OF FRAME BUNDLES WOUTER VAN LIMBEEK We consider the oriented orthonormal frame bundle SO.M/ of an oriented Riemannian manifold M. The Riemannian metric on M induces a canon- ical Riemannian metric on SO.M/. We prove that for two closed oriented Riemannian n-manifolds M and N, the frame bundles SO.M/ and SO.N/ are isometric if and only if M and N are isometric, except possibly in di- mensions 3, 4, and 8. This answers a question of Benson Farb except in dimensions 3, 4, and 8. 1. Introduction 393 2. Preliminaries 396 3. High dimensional isometry groups of manifolds 400 4. Geometric characterization of the fibers of SO.M/ ! M 403 5. Proof for M with positive constant curvature 415 6. Proof of the main theorem for surfaces 422 Acknowledgements 425 References 425 1. Introduction Let M be an oriented Riemannian manifold, and let X VD SO.M/ be the oriented orthonormal frame bundle of M. The Riemannian structure g on M induces in a canonical way a Riemannian metric gSO on SO.M/. This construction was first carried out by O’Neill[1966] and independently by Mok[1978], and is very similar to Sasaki’s[1958; 1962] construction of a metric on the unit tangent bundle of M, so we will henceforth refer to gSO as the Sasaki–Mok–O’Neill metric on SO.M/.