DOCSLIB.ORG

Lecture 1: Connections on Principal Fibre Bundles

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

GT 2006 (jmf) 3 Lecture 1: Connections on principal fibre bundles The beauty and profundity of the geometry of fibre bundles were to a large extent brought forth by the (early) work of Chern. I must admit, however, that the appreciation of this beauty came to physicists only in recent years. — CN Yang, 1979 In this lecture we introduce the notion of a principal fibre bundle and of a connection on it, but we start with some motivation. 1.1 Motivation: the Dirac monopole It is only fitting to start this course, which takes place at the JCMB, with a solution of Maxwell’s equa- tions. The magnetic field B of a magnetic monopole sitting at the origin in R3 is given by x B(x) Æ , 4¼r 3 where r Æ jxj. This satisfies divB Æ 0 and hence is a solution of Maxwell’s equations in R3 \{0}. Maxwell’s equations are linear and hence this solution requires a source of magnetic field, namely the monopole which sits at the origin. We will see later in this course that there are other monopoles which do not require sources and which extend smoothly to the ‘origin’. In modern language, the vector field B is understood as the 2-form F 2 ­2(R3 \{0}) given by 1 ¡ ¢ F Æ x1dx2 ^ dx3 Å cyclic 2¼r 3 in the cartesian coordinates of R3. Maxwell’s equations say that dF Æ 0. This is perhaps more evident in spherical coordinates (x1 Æ r sinθcosÁ,...), where 1 F Æ sinθdθ ^ dÁ . 4¼ Since dF Æ 0 we may hope to find a one-form A such that F Æ dA. For example, 1 A Æ¡ cosθdÁ . 4¼ This is not regular over all of R3\{0}, however. This should not come as a surprise, since after all spherical coordinates are singular on the x3-axis. Rewriting A in cartesian coordinates x3 x2dx1 ¡ x1dx2 A Æ , 2 2 4¼r x1 Å x2 clearly displays the singularity at x1 Æ x2 Æ 0. In the old Physics literature this singularity is known as the “Dirac string,” a language we shall distance ourselves from in this course. This singularity will afflict any A we come up with. Indeed, notice that F restricts to the (normalised) area form on the unit sphere S ½ R3, whence Z F Æ 1 . S R If F Æ dA for a smooth one-form A, then Stokes’s theorem would have forced S F Æ 0. The principal aim of the first couple of lectures is to develop the geometric framework to which F (and A) belong: the theory of connections on principal fibre bundles, to which we now turn. GT 2006 (jmf) 4 1.2 Principal fibre bundles A principal fibre bundle consists of the following data: • a manifold P, called the total space; • a Lie group G acting freely on P on the right: P £ G ! P (p,g) 7! pg (or sometimes Rg p) where by a free action we mean that the stabilizer of every point is trivial, or paraphrasing, that every element G (except the identity) moves every point in P. We will also assume that the space of orbits M Æ P/G is a manifold (called the base) and the natural map ¼ : P ! M taking a point to its orbit is a smooth surjection. For every m 2 M, the submanifold ¼¡1(m) ½ P is called the fibre over m. Further, this data will be subject to the condition of local triviality: that M admits an open cover {U®} ¡1 and G-equivariant diffeomorphisms î : ¼ U® ! U® £ G such that the following diagram commutes à ¡1 ® ¼ U / U® £ G ®F x FF xx FF xx ¼ FF xx pr F" {xx 1 U® ¡1 This means that î(p) Æ (¼(p),g®(p)), for some G-equivariant map g® : ¼ U® ! G which is a fibrewise diffeomorphism. Equivariance means that g®(pg) Æ g®(p)g . One often abbreviates the above data by saying that P G ¡¡¡¡¡! P ? ? ? ? y¼ or y¼ is a principal G-bundle, M M but be aware that abuses of language are rife in this topic. Don’t be surprised by statements such as “Let P be a principal bundle...” We say that the bundle is trivial if there exists a diffeomorphism ª : P ! M £ G such that ª(p) Æ (¼(p),Ã(p)) and such that Ã(pg) Æ Ã(p)g. This last condition is simply the G-equivariance of ª. A section is a (smooth) map s : M ! P such that ¼ ± s Æ id. In other words, it is a smooth assignment to each point in the base of a point in the fibre over it. Sections are rare. Indeed, one has Done? ❑ Exercise 1.1. Show that a principal fibre bundle admits a section if and only if it is trivial. (This is in sharp contrast with, say, vector bundles, which always admit sections.) Nevertheless, since P is locally trivial, local sections do exist. In fact, there are local sections s® : U® ! ¡1 ¼ U® canonically associated to the trivialization, defined so that for every m 2 U®, î(s®(m)) Æ (m,e), where e 2 G is the identity element. In other words, g® ± s® : U® ! G is the constant function sending every point to the identity. Conversely, a local section s® allows us to identify the fibre over m with G. ¡1 Indeed, given any p 2 ¼ (m), there is a unique group element g®(p) 2 G such that p Æ s®(m)g®(p). On nonempty overlaps U®¯ :Æ U® \ U¯, we have two ways of trivialising the bundle: ï î U £ G o ¼¡1U / U £ G ®¯ J ®¯ ®¯ JJ tt JJ tt JJ ¼ tt pr1 JJ tt pr1 J$ ² ztt U®¯ GT 2006 (jmf) 5 ¡1 For m 2 U®¯ and p 2 ¼ (m), we have î(p) Æ (m,g®(p)) and ï(p) Æ (m,g¯(p)), whence there is g¯®¯(p) 2 G such that g®(p) Æ g¯®¯(p)g¯(p). In other words, ¡1 (1) g¯®¯(p) Æ g®(p)g¯(p) . In fact, g¯®¯(p) is constant on each fibre: ¡1 g¯®¯(pg) Æ g®(pg)g¯(pg) ¡1 (by equivariance of g®, g¯) Æ g®(p)g g g¯(p) Æ g¯®¯(p) . In other words, g¯®¯(p) Æ g®¯(¼(p)) for some function g®¯ : U®¯ ! G. From equation (1), it follows that these transition functions obey the following cocycle conditions: (2) g®¯(m)g¯®(m) Æ e for every m 2 U®¯, and (3) g®¯(m)gβγ(m)gγα(m) Æ e for every m 2 Uαβγ, where Uαβγ Æ U® \ U¯ \ Uγ. Done? ❑ Exercise 1.2. Show that on double overlaps, the canonical sections s® are related by s¯(m) Æ s®(m)g®¯(m) for every m 2 U®¯. One can reconstruct the bundle from an open cover {U®} and transition functions {g®¯} obeying the cocycle conditions (2) as follows: G . P Æ (U® £ G) » , ® where (m,g) » (m,g®¯(m)g) for all m 2 U®¯ and g 2 G. Notice that ¼ is induced by the projection onto the first factor and the action of G on P is induced by right multiplication on G, both of which are preserved by the equivalence relation, which uses left multiplication by the transition functions. (Asso- ciativity of group multiplication guarantees that right and left multiplications commute.) Example 1.1 (Möbius band). The boundary of the Möbius band is an example of a nontrivial principal 1 Z2-bundle. This can be described as follows. Let S ½ C denote the complex numbers of unit modulus and let ¼ : S1 ! S1 be the map defined by z 7! z2. Then the fibre ¼¡1(z2) Æ {§z} consists of two points. A global section would correspond to choosing a square-root function smoothly on the unit circle. This does not exist, however, since any definition of z1/2 always has a branch cut from the origin out to the point at infinity. Therefore the bundle is not trivial. In fact, if the bundle were trivial, the total space would be disconnected, being two disjoint copies of the circle. However building a paper model of the Möbius band one quickly sees that its boundary is connected. We can understand this bundle in terms of the local data as follows. Cover the circle by two overlap- ping open sets: U1 and U2. Their intersection is the disjoint union of two intervals in the circle: V1 tV2. Let gi : Vi ! Z2 denote the transition functions, which are actually constant since Vi are connected and Z2 is discrete, so we can think of gi 2 Z2. There are no triple overlaps, so the cocycle condition is vacu- ously satisfied. It is an easy exercise to check that the resulting bundle is trivial if and only if g1 Æ g2 and nontrivial otherwise. GT 2006 (jmf) 6 1.3 Connections The push-forward and the pull-back Let f : M ! N be a smooth map between manifolds. The push-forward f¤ : TM ! TN (also written T f when in a categorical mood) is the collection of fibre-wise linear maps f¤ : TmM ! Tf (m)N defined as follows. Let v 2 TmM be represented as the velocity of a curve t 7! γ(t) through m; that is, γ(0) Æ m and γ0(0) Æ v. 0 0 Then f¤(v) 2 Tf (m)N is the velocity at f (m) of the curve t 7! f (γ(t)); that is, f¤γ (0) Æ (f ±γ) (0).
Recommended publications
  • 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.
  • 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.
  • 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).
  • FOLIATIONS Introduction. the Study of Foliations on Manifolds Has a Long

    FOLIATIONS Introduction. the Study of Foliations on Manifolds Has a Long

    BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 80, Number 3, May 1974 FOLIATIONS BY H. BLAINE LAWSON, JR.1 TABLE OF CONTENTS 1. Definitions and general examples. 2. Foliations of dimension-one. 3. Higher dimensional foliations; integrability criteria. 4. Foliations of codimension-one; existence theorems. 5. Notions of equivalence; foliated cobordism groups. 6. The general theory; classifying spaces and characteristic classes for foliations. 7. Results on open manifolds; the classification theory of Gromov-Haefliger-Phillips. 8. Results on closed manifolds; questions of compact leaves and stability. Introduction. The study of foliations on manifolds has a long history in mathematics, even though it did not emerge as a distinct field until the appearance in the 1940's of the work of Ehresmann and Reeb. Since that time, the subject has enjoyed a rapid development, and, at the moment, it is the focus of a great deal of research activity. The purpose of this article is to provide an introduction to the subject and present a picture of the field as it is currently evolving. The treatment will by no means be exhaustive. My original objective was merely to summarize some recent developments in the specialized study of codimension-one foliations on compact manifolds. However, somewhere in the writing I succumbed to the temptation to continue on to interesting, related topics. The end product is essentially a general survey of new results in the field with, of course, the customary bias for areas of personal interest to the author. Since such articles are not written for the specialist, I have spent some time in introducing and motivating the subject.
  • Math 704: Part 1: Principal Bundles and Connections

    Math 704: Part 1: Principal Bundles and Connections

    MATH 704: PART 1: PRINCIPAL BUNDLES AND CONNECTIONS WEIMIN CHEN Contents 1. Lie Groups 1 2. Principal Bundles 3 3. Connections and curvature 6 4. Covariant derivatives 12 References 13 1. Lie Groups A Lie group G is a smooth manifold such that the multiplication map G × G ! G, (g; h) 7! gh, and the inverse map G ! G, g 7! g−1, are smooth maps. A Lie subgroup H of G is a subgroup of G which is at the same time an embedded submanifold. A Lie group homomorphism is a group homomorphism which is a smooth map between the Lie groups. The Lie algebra, denoted by Lie(G), of a Lie group G consists of the set of left-invariant vector fields on G, i.e., Lie(G) = fX 2 X (G)j(Lg)∗X = Xg, where Lg : G ! G is the left translation Lg(h) = gh. As a vector space, Lie(G) is naturally identified with the tangent space TeG via X 7! X(e). A Lie group homomorphism naturally induces a Lie algebra homomorphism between the associated Lie algebras. Finally, the universal cover of a connected Lie group is naturally a Lie group, which is in one to one correspondence with the corresponding Lie algebras. Example 1.1. Here are some important Lie groups in geometry and topology. • GL(n; R), GL(n; C), where GL(n; C) can be naturally identified as a Lie sub- group of GL(2n; R). • SL(n; R), O(n), SO(n) = O(n) \ SL(n; R), Lie subgroups of GL(n; R).
  • Notes on Principal Bundles and Classifying Spaces

    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].
  • WHAT IS a CONNECTION, and WHAT IS IT GOOD FOR? Contents 1. Introduction 2 2. the Search for a Good Directional Derivative 3 3. F

    WHAT IS a CONNECTION, and WHAT IS IT GOOD FOR? Contents 1. Introduction 2 2. the Search for a Good Directional Derivative 3 3. F

    WHAT IS A CONNECTION, AND WHAT IS IT GOOD FOR? TIMOTHY E. GOLDBERG Abstract. In the study of differentiable manifolds, there are several different objects that go by the name of \connection". I will describe some of these objects, and show how they are related to each other. The motivation for many notions of a connection is the search for a sufficiently nice directional derivative, and this will be my starting point as well. The story will by necessity include many supporting characters from differential geometry, all of whom will receive a brief but hopefully sufficient introduction. I apologize for my ungrammatical title. Contents 1. Introduction 2 2. The search for a good directional derivative 3 3. Fiber bundles and Ehresmann connections 7 4. A quick word about curvature 10 5. Principal bundles and principal bundle connections 11 6. Associated bundles 14 7. Vector bundles and Koszul connections 15 8. The tangent bundle 18 References 19 Date: 26 March 2008. 1 1. Introduction In the study of differentiable manifolds, there are several different objects that go by the name of \connection", and this has been confusing me for some time now. One solution to this dilemma was to promise myself that I would some day present a talk about connections in the Olivetti Club at Cornell University. That day has come, and this document contains my notes for this talk. In the interests of brevity, I do not include too many technical details, and instead refer the reader to some lovely references. My main references were [2], [4], and [5].
  • Unstable Bundles in Quantum Field Theory

    Unstable Bundles in Quantum Field Theory

    Contemporary Mathematics Volume 17, 1998 Unstable Bundles in Quantum Field Theory M. Asorey†, F. Falceto† and G. Luz´on‡ Abstract. The relation between connections on 2-dimensional manifolds and holomorphic bundles provides a new perspective on the role of classical gauge fields in quantum field theory in two, three and four dimensions. In particular we show that there is a close relation between unstable bundles and monopoles, sphalerons and instantons. Some of these classical configurations emerge as nodes of quantum vacuum states in non-confining phases of quantum field theory which suggests a relevant role for those configurations in the mechanism of quark confinement in QCD. 1. Introduction The role of principal bundles in the theory of gauge fields is now well established as the kinematical geometric background where classical and quantum fluctuations of gauge fields evolve. However, it is much less known that holomorphic bundles also play a fundamental role in the dynamics of the theory. There are two cases where the theory of holomorphic bundles proved to be very useful in the analysis of the classical dynamics of Yang-Mills fields. The first applica- tion arises in the construction of self-dual solutions of four-dimensional Yang-Mills equations (instantons), i.e. connections which minimize the Yang-Mills functional in non-trivial bundles. U(N) instantons in S4 are in one–to–one correspondence with holomorphic bundles of rank N on CP 3 which are trivial along the canonical U(1) fibres [29]. The second application occurs in 2-D Yang-Mills theory, where arXiv:hep-th/9812173v1 18 Dec 1998 the Yang-Mills functional becomes an almost equivariantly perfect Morse function in the space of 2-D connections [8].
  • Principal Bundles, Vector Bundles and Connections

    Principal Bundles, Vector Bundles and Connections

    Appendix A Principal Bundles, Vector Bundles and Connections Abstract The appendix defines fiber bundles, principal bundles and their associate vector bundles, recall the definitions of frame bundles, the orthonormal frame bun- dle, jet bundles, product bundles and the Whitney sums of bundles. Next, equivalent definitions of connections in principal bundles and in their associate vector bundles are presented and it is shown how these concepts are related to the concept of a covariant derivative in the base manifold of the bundle. Also, the concept of exterior covariant derivatives (crucial for the formulation of gauge theories) and the meaning of a curvature and torsion of a linear connection in a manifold is recalled. The concept of covariant derivative in vector bundles is also analyzed in details in a way which, in particular, is necessary for the presentation of the theory in Chap. 12. Propositions are in general presented without proofs, which can be found, e.g., in Choquet-Bruhat et al. (Analysis, Manifolds and Physics. North-Holland, Amsterdam, 1982), Frankel (The Geometry of Physics. Cambridge University Press, Cambridge, 1997), Kobayashi and Nomizu (Foundations of Differential Geometry. Interscience Publishers, New York, 1963), Naber (Topology, Geometry and Gauge Fields. Interactions. Applied Mathematical Sciences. Springer, New York, 2000), Nash and Sen (Topology and Geometry for Physicists. Academic, London, 1983), Nicolescu (Notes on Seiberg-Witten Theory. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2000), Osborn (Vector Bundles. Academic, New York, 1982), and Palais (The Geometrization of Physics. Lecture Notes from a Course at the National Tsing Hua University, Hsinchu, 1981). A.1 Fiber Bundles Definition A.1 A fiber bundle over M with Lie group G will be denoted by E D .E; M;;G; F/.
  • On Fibre Bundles and Differential Geometry

    On Fibre Bundles and Differential Geometry

    Lectures On Fibre Bundles and Differential Geometry By J.L. Koszul Notes by S. Ramanan No part of this book may be reproduced in any form by print, microfilm or any other means without written permission from the Tata Insti- tute of Fundamental Research, Apollo Pier Road, Bombay-1 Tata Institute of Fundamental Research, Bombay 1960 Contents 1 Differential Calculus 1 1.1 .............................. 1 1.2 Derivationlaws ...................... 2 1.3 Derivation laws in associated modules . 4 1.4 TheLiederivative... ..... ...... ..... .. 6 1.5 Lie derivation and exterior product . 8 1.6 Exterior differentiation . 8 1.7 Explicit formula for exterior differentiation . 10 1.8 Exterior differentiation and exterior product . 11 1.9 The curvature form . 12 1.10 Relations between different derivation laws . 16 1.11 Derivation law in C .................... 17 1.12 Connections in pseudo-Riemannian manifolds . 18 1.13 Formulae in local coordinates . 19 2 Differentiable Bundles 21 2.1 .............................. 21 2.2 Homomorphisms of bundles . 22 2.3 Trivial bundles . 23 2.4 Induced bundles . 24 2.5 Examples ......................... 25 2.6 Associated bundles . 26 2.7 Vector fields on manifolds . 29 2.8 Vector fields on differentiable principal bundles . 30 2.9 Projections vector fields . 31 iii iv Contents 3 Connections on Principal Bundles 35 3.1 .............................. 35 3.2 Horizontal vector fields . 37 3.3 Connection form . 39 3.4 Connection on Induced bundles . 39 3.5 Maurer-Cartan equations . 40 3.6 Curvatureforms. 43 3.7 Examples ......................... 46 4 Holonomy Groups 49 4.1 Integral paths . 49 4.2 Displacement along paths . 50 4.3 Holonomygroup ....................
  • Foliated Manifolds, Algebraic K-Theory, and a Secondary Invariant

    Foliated Manifolds, Algebraic K-Theory, and a Secondary Invariant

    Foliated manifolds, algebraic K-theory, and a secondary invariant Ulrich Bunke∗ June 25, 2018 Abstract We introduce a C=Z-valued invariant of a foliated manifold with a stable framing and with a partially flat vector bundle. This invariant can be expressed in terms of integration in differential K-theory, or alternatively, in terms of η-invariants of Dirac operators and local correction terms. Initially, the construction of the element in C=Z involves additional choices. But if the codimension of the foliation is sufficiently small, then this element is independent of these choices and therefore an invariant of the data listed above. We show that the invariant comprises various classical invariants like Adams' e- invariant, the ρ-invariant of twisted Dirac operators, or the Godbillon-Vey invariant from foliation theory. Using methods from differential cohomology theory we construct a regulator map from the algebraic K-theory of smooth functions on a manifold to its connective K-theory with C=Z coefficients. Our main result is a formula for the invariant in terms of this regulator and integration in algebraic and topological K-theory. arXiv:1507.06404v4 [math.KT] 22 Jun 2018 Contents 1 Introduction 2 2 Basic notions 4 2.1 Foliated manifolds . .4 2.2 Filtrations on the de Rham complex . .6 2.3 Vector bundles with flat partial connections . .8 2.4 Connections and characteristic forms . 10 2.5 Characteristic forms of real foliations . 12 2.6 Transgression . 14 ∗NWF I - Mathematik, Universit¨at Regensburg, 93040 Regensburg, GERMANY, [email protected] 1 3 Differential K-theory 18 3.1 Basic structures .
  • CHAPTER 4: PRINCIPAL BUNDLES 4.1 Lie Groups a Lie Group Is A

    CHAPTER 4: PRINCIPAL BUNDLES 4.1 Lie Groups a Lie Group Is A

    CHAPTER 4: PRINCIPAL BUNDLES 4.1 Lie groups A Lie group is a group G which is also a smooth manifold such that the mul- tiplication map G × G → G, (a, b) 7→ ab, and the inverse G → G, a 7→ a−1, are smooth. Actually, one can prove (but this is not easy) that it is sufficient to assume continuety, smoothness comes free. (This was one of the famous problems listed by David Hilbert in his address to the international congress of mathematicians in 1900. The result was proven by D. Montgomery and L. Zippin in 1952.) Examples The vector space Rn is a Lie group. The group multiplication is just the addition of vectors. The set GL(n, R) of invertible real n × n matrices is a Lie group with respect to the usual matrix multiplication. Theorem. Any closed subgroup of a Lie group is a Lie group. The proof is complicated. See for example S. Helgason: Differential Geometry, Lie Groups and Symmetric Spaces, section II.2. The theorem gives an additional set of examples of Lie groups: The group of real orthogonal matrices, the group of complex unitary matrices, the group of invertible upper triangular matrices .... For a fixed a ∈ G in a Lie group we can define a pair of smooth maps, the left translation la : G → G, la(g) = ag, and the right translation ra : G → G, ra(g) = ga. 1 We say that a vector field X ∈ D (G) is left (resp. right) invariant if (la)∗X = X (resp. (ra)∗X = X) for all a ∈ G.