DOCSLIB.ORG

Differential Topology

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Differential Topology Differential Topology Andrew Kobin Fall 2015 Contents Contents Contents 0 Introduction 1 0.1 Review of General Topology . .1 1 Smooth Manifolds 4 1.1 Smooth Maps and the Tangent Space . .5 1.2 Submanifolds . .7 1.3 Abstract Manifolds . .8 1.4 Manifolds With Boundary . 13 2 Regular Values 15 2.1 Regular and Critical Values . 15 2.2 Sard's Theorem . 22 2.3 Homotopy and Degree mod 2 . 24 2.4 Orientation . 27 2.5 Brouwer Degree . 28 3 Transversality and Embeddings 34 3.1 Transversality, Immersion and Submersion . 34 3.2 Embedding Theorems . 38 4 Vector Bundles 41 4.1 Vector and Tangent Bundles . 41 4.2 Sections . 44 4.3 Constructing New Vector Bundles . 47 i 0 Introduction 0 Introduction These notes cover a course in differential topology taught by Dr. Slava Krushkal in the fall of 2015 at the University of Virginia. The main references for the course are Wallace's Differ- ential Topology: First Steps and Milnor's Princeton lectures, Topology from the differentiable viewpoint. The topics covered in the course are: Smooth maps and manifolds Tangent spaces Submanifolds of Rn and the embedding theorem Transversality Regular and critical points Sard's theorem and Brouwer's fixed point theorem Degree of maps Vector fields and the Euler characteristic An introduction to vector bundles and differential forms. 0.1 Review of General Topology The formal objects in general topology were only laid out about a century ago, but the concepts they represent go back much further in time. Definition. A topology T on a set X is a collection of subsets of X which satisfy (1) ?;X 2 T . S (2) For every arbitrary collection Uα 2 T , α Uα 2 T . Tk (3) For every finite collection U1;:::;Uk 2 T , i=1 Ui 2 T . The pair (X; T ) is called a topological space, although this is often abbreviated by X when the topology is understood. One of the most important topologies is the one generated by a metric. Definition. A metric on a space X is a function d : X × X ! R such that (1) (Positive semidefinite) d(x; y) ≥ 0 for all x; y 2 X and d(x; y) = 0 if and only if x = y. (2) (Symmetric) d(x; y) = d(y; x) for all x; y 2 X. (3) (Triangle inequality) d(x; y) + d(y; z) ≤ d(x; z) for all x; y; z 2 X. 1 0.1 Review of General Topology 0 Introduction The sets that make up a topology are by convention called open sets or neighborhoods, while their complements are called closed sets. Definition. A point p 2 X is a limit point of a set A ⊂ X if every neighborhood U of p intersects A in a point other than p. Definition. A point p 2 X lies in the closure of A, denoted A, if every neighborhood U of p intersects A nontrivially. It is easy to see from these two definitions that a set is closed if and only if it contains all of its limit points. For this reason it's common to write the closure as A = A [ L, where L is the set of limit points of A. One of the most important conceits in mathematics is the idea of comparison. We generally accomplish this by constructing maps between objects (categorically, maps are called morphisms) that preserve a desired structure. In topology, the objects are spaces and the natural maps between them are continuous functions. Definition. A function f : X ! Y between topological spaces is continuous at a point p 2 X if for every open set U containing f(p), f −1(U) ⊂ X is also open. The function is continuous (on X) if it is continuous at every point in X. Definition. A continuous map f : X ! Y is called a homeomorphism if it is one-to-one and onto (a bijection) and has a continuous inverse. In this case, X and Y are said to be homeomorphic, denoted X ∼= Y . One of the principal themes in topology is the study of topological properties, or properties preserved under homeomorphism. We recall several key topological properties next. Definition. Given a topological space X, a subset A ⊂ X is compact if every open cover of A has a finite subcover. Proposition 0.1.1. Closed subspaces of compact spaces are compact. The Separation Axioms are additional assumptions we make on a topological space to ensure there are \enough" open sets in X to do certain things. The first two are listed below. Definition. A space X is T1 if every point set is closed. Definition. A space X is Hausdorff (also called T2) if for every pair of distinct x; y 2 X, there exist disjoint open sets U and V such that x 2 U and y 2 V . Proposition 0.1.2. If X is Hausdorff and A ⊂ X is compact, then A is closed. The converse of this statement is not true in general, and even in nice spaces (metric spaces) there are additional conditions (e.g. boundedness) to imply compactness. Definition. A topological space X is said to be disconnected provided there exist nonempty, disjoint open sets U and V whose union is X. If no such sets exist, we say X is connected. 2 0.1 Review of General Topology 0 Introduction In this course, we will implicitly assume that every map is smooth, i.e. infinitely differ- entiable on an open subset U ⊂ Rn. However, we need to define the notion of a smooth map on any subset of Rn, not just the open ones. Definition. For an arbitrary subset S ⊂ Rn, a function f : S ! Rm is smooth on S if for every x 2 S, there exists a neighborhood W ⊂ n containing x and an extension R f~ : W ! Rm which is smooth on W and satisfies f~ = f. W \S In differential topology, the fundamental notion of equivalence is even stronger than homeomorphism. Definition. A diffeomorphism is a map f : X ! Y that is a homeomorphism (continuous and invertible, with continuous inverse) that is smooth and whose inverse is also smooth. In the case that f is a diffeomorphism, we say X and Y are diffeomorphic. An important theorem going forward is the Inverse Function Theorem from analysis, which we recall here. Theorem 0.1.3 (Inverse Function Theorem). Suppose f : U ! Rn is a smooth map on an open set U ⊂ n, and for a point x 2 U, assume the differential D f is non-singular, or R x equivalently, the Jacobian @fi (x) is invertible. Then there exists a neighborhood V ⊂ U @xj containing x such that fjV : V ! f(V ) is a diffeomorphism. 3 1 Smooth Manifolds 1 Smooth Manifolds The fundamental object in differential topology is the smooth manifold. Imprecisely, a manifold is a topological space which is locally Euclidean. In this section we will present two different definitions of a smooth manifold, but ultimately we will see that they are equivalent. Definition. A smooth n-dimensional manifold is a subspace M ⊂ Rk, k ≥ n, such that for any point x 2 M, there is a neighborhood U ⊂ M containing x which is diffeomorphic to an open subset of Rn. We will often write M n to convey that M is an n-dimensional manifold. Example 1.0.4. The unit circle S1 ⊂ R2 is a smooth one-dimensional manifold. R2 ( x ) Given a point x 2 S1, there are many good choices of a diffeomorphism, including Coordinate projection: f(x1; x2) = x1. Stereographic projection from N = (0; 1). Polar coordinates: (cos θ; sin θ) 7! θ for θ 2 (0; 2π). Each of these will be a diffeomorphism on a different neighborhood of x. However notice that there's no global diffeomorphism S1 ! R since these spaces aren't homeomorphic to begin with. n Definition. A collection of diffeomorphisms gx : Ux ! Vx ⊂ R , one for each point x 2 M, is called an atlas of M. Each Ux is called a coordinate chart (or coordinate system) −1 around x, and the inverse map gx is called a parametrization about x. Note that when dim M ≥ 1, around any point x 2 M, there are uncountably many choices of parametrizations and coordinate systems. 4 1.1 Smooth Maps and the Tangent Space 1 Smooth Manifolds 1.1 Smooth Maps and the Tangent Space Loosely, the tangent space to a manifold M at a point x 2 M is the set of all tangent vectors to M at x. This generalizes the concepts of tangent lines and tangent planes from multivariable calculus: tangent plane M tangent line f(x) Definition. The tangent space to a smooth manifold M at a point x 2 M is the set of tangent vectors to all parametrized differentiable curves lying in M and passing through x: 0 d TxM := α (0) = dt α(t)jt=0 α :(−"; ") ! M is smooth and α(0) = x : Remarks. α 1 A curve α :(−"; ") ! M is smooth if the composition (−"; ") −! M,! Rk is smooth, which in turn means the composition with every coordinate projection is smooth: α mi (−"; ") −! M −! R: 2 It is not immediately clear that TxM is closed under addition { however, scalar multi- ples are easily obtained by reparametrization of the curve. Suppose M m and N n are smooth manifolds, possibly of different dimensions. What does it mean for a map f : M ! N to be smooth? Definition. If M ⊂ Rk and N ⊂ R`, then f is smooth if for every x 2 M there is an open k ~ ` ~ neighborhood W ⊂ R of x and a smooth map f : W ! R such that fjW \M = fjW \M .
Load more

Recommended publications
  • Codimension Zero Laminations Are Inverse Limits 3
    CODIMENSION ZERO LAMINATIONS ARE INVERSE LIMITS ALVARO´ LOZANO ROJO Abstract. The aim of the paper is to investigate the relation between inverse limit of branched manifolds and codimension zero laminations. We give necessary and sufficient conditions for such an inverse limit to be a lamination. We also show that codimension zero laminations are inverse limits of branched manifolds. The inverse limit structure allows us to show that equicontinuous codimension zero laminations preserves a distance function on transver- sals. 1. Introduction Consider the circle S1 = { z ∈ C | |z| = 1 } and the cover of degree 2 of it 2 p2(z)= z . Define the inverse limit 1 1 2 S2 = lim(S ,p2)= (zk) ∈ k≥0 S zk = zk−1 . ←− Q This space has a natural foliated structure given by the flow Φt(zk) = 2πit/2k (e zk). The set X = { (zk) ∈ S2 | z0 = 1 } is a complete transversal for the flow homeomorphic to the Cantor set. This space is called solenoid. 1 This construction can be generalized replacing S and p2 by a sequence of compact p-manifolds and submersions between them. The spaces obtained this way are compact laminations with 0 dimensional transversals. This construction appears naturally in the study of dynamical systems. In [17, 18] R.F. Williams proves that an expanding attractor of a diffeomor- phism of a manifold is homeomorphic to the inverse limit f f f S ←− S ←− S ←−· · · where f is a surjective immersion of a branched manifold S on itself. A branched manifold is, roughly speaking, a CW-complex with tangent space arXiv:1204.6439v2 [math.DS] 20 Nov 2012 at each point.
    [Show full text]
  • Part III : the Differential
    77 Part III : The differential 1 Submersions In this section we will introduce topological and PL submersions and we will prove that each closed submersion with compact fibres is a locally trivial fibra- tion. We will use Γ to stand for either Top or PL and we will suppose that we are in the category of Γ–manifolds without boundary. 1.1 A Γ–map p: Ek → Xl betweenΓ–manifoldsisaΓ–submersion if p is locally the projection Rk−→πl R l on the first l–coordinates. More precisely, p: E → X is a Γ–submersion if there exists a commutative diagram p / E / X O O φy φx Uy Ux ∩ ∩ πl / Rk / Rl k l where x = p(y), Uy and Ux are open sets in R and R respectively and ϕy , ϕx are charts around x and y respectively. It follows from the definition that, for each x ∈ X ,thefibre p−1(x)isaΓ– manifold. 1.2 The link between the notion of submersions and that of bundles is very straightforward. A Γ–map p: E → X is a trivial Γ–bundle if there exists a Γ– manifold Y and a Γ–isomorphism f : Y × X → E , such that pf = π2 ,where π2 is the projection on X . More generally, p: E → X is a locally trivial Γ–bundle if each point x ∈ X has an open neighbourhood restricted to which p is a trivial Γ–bundle. Even more generally, p: E → X is a Γ–submersion if each point y of E has an open neighbourhood A, such that p(A)isopeninXand the restriction A → p(A) is a trivial Γ–bundle.
    [Show full text]
  • DIFFERENTIAL GEOMETRY Contents 1. Introduction 2 2. Differentiation 3
    DIFFERENTIAL GEOMETRY FINNUR LARUSSON´ Lecture notes for an honours course at the University of Adelaide Contents 1. Introduction 2 2. Differentiation 3 2.1. Review of the basics 3 2.2. The inverse function theorem 4 3. Smooth manifolds 7 3.1. Charts and atlases 7 3.2. Submanifolds and embeddings 8 3.3. Partitions of unity and Whitney’s embedding theorem 10 4. Tangent spaces 11 4.1. Germs, derivations, and equivalence classes of paths 11 4.2. The derivative of a smooth map 14 5. Differential forms and integration on manifolds 16 5.1. Introduction 16 5.2. A little multilinear algebra 17 5.3. Differential forms and the exterior derivative 18 5.4. Integration of differential forms on oriented manifolds 20 6. Stokes’ theorem 22 6.1. Manifolds with boundary 22 6.2. Statement and proof of Stokes’ theorem 24 6.3. Topological applications of Stokes’ theorem 26 7. Cohomology 28 7.1. De Rham cohomology 28 7.2. Cohomology calculations 30 7.3. Cechˇ cohomology and de Rham’s theorem 34 8. Exercises 36 9. References 42 Last change: 26 September 2008. These notes were originally written in 2007. They have been classroom-tested twice. Address: School of Mathematical Sciences, University of Adelaide, Adelaide SA 5005, Australia. E-mail address: [email protected] Copyright c Finnur L´arusson 2007. 1 1. Introduction The goal of this course is to acquire familiarity with the concept of a smooth manifold. Roughly speaking, a smooth manifold is a space on which we can do calculus. Manifolds arise in various areas of mathematics; they have a rich and deep theory with many applications, for example in physics.
    [Show full text]
  • Hirschdx.Pdf
    130 5. Degrees, Intersection Numbers, and the Euler Characteristic. 2. Intersection Numbers and the Euler Characteristic M c: Jr+1 Exercises 8. Let be a compad lHlimensionaJ submanifold without boundary. Two points x, y E R"+' - M are separated by M if and only if lkl(x,Y}.MI * O. lSee 1. A complex polynomial of degree n defines a map of the Riemann sphere to itself Exercise 7.) of degree n. What is the degree of the map defined by a rational function p(z)!q(z)? 9. The Hopfinvariant ofa map f:5' ~ 5' is defined to be the linking number Hlf) ~ 2. (a) Let M, N, P be compact connected oriented n·manifolds without boundaries Lk(g-l(a),g-l(b)) (see Exercise 7) where 9 is a c~ map homotopic 10 f and a, b are and M .!, N 1. P continuous maps. Then deg(fg) ~ (deg g)(deg f). The same holds distinct regular values of g. The linking number is computed in mod 2 if M, N, P are not oriented. (b) The degree of a homeomorphism or homotopy equivalence is ± 1. f(c) * a, b. *3. Let IDl. be the category whose objects are compact connected n-manifolds and whose (a) H(f) is a well-defined homotopy invariant off which vanishes iff is nuD homo- topic. morphisms are homotopy classes (f] of maps f:M ~ N. For an object M let 7t"(M) (b) If g:5' ~ 5' has degree p then H(fg) ~ pH(f). be the set of homotopy classes M ~ 5".
    [Show full text]
  • 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.
    [Show full text]
  • Lecture Notes on Foliation Theory
    INDIAN INSTITUTE OF TECHNOLOGY BOMBAY Department of Mathematics Seminar Lectures on Foliation Theory 1 : FALL 2008 Lecture 1 Basic requirements for this Seminar Series: Familiarity with the notion of differential manifold, submersion, vector bundles. 1 Some Examples Let us begin with some examples: m d m−d (1) Write R = R × R . As we know this is one of the several cartesian product m decomposition of R . Via the second projection, this can also be thought of as a ‘trivial m−d vector bundle’ of rank d over R . This also gives the trivial example of a codim. d- n d foliation of R , as a decomposition into d-dimensional leaves R × {y} as y varies over m−d R . (2) A little more generally, we may consider any two manifolds M, N and a submersion f : M → N. Here M can be written as a disjoint union of fibres of f each one is a submanifold of dimension equal to dim M − dim N = d. We say f is a submersion of M of codimension d. The manifold structure for the fibres comes from an atlas for M via the surjective form of implicit function theorem since dfp : TpM → Tf(p)N is surjective at every point of M. We would like to consider this description also as a codim d foliation. However, this is also too simple minded one and hence we would call them simple foliations. If the fibres of the submersion are connected as well, then we call it strictly simple. (3) Kronecker Foliation of a Torus Let us now consider something non trivial.
    [Show full text]
  • Lecture 10: Tubular Neighborhood Theorem
    LECTURE 10: TUBULAR NEIGHBORHOOD THEOREM 1. Generalized Inverse Function Theorem We will start with Theorem 1.1 (Generalized Inverse Function Theorem, compact version). Let f : M ! N be a smooth map that is one-to-one on a compact submanifold X of M. Moreover, suppose dfx : TxM ! Tf(x)N is a linear diffeomorphism for each x 2 X. Then f maps a neighborhood U of X in M diffeomorphically onto a neighborhood V of f(X) in N. Note that if we take X = fxg, i.e. a one-point set, then is the inverse function theorem we discussed in Lecture 2 and Lecture 6. According to that theorem, one can easily construct neighborhoods U of X and V of f(X) so that f is a local diffeomor- phism from U to V . To get a global diffeomorphism from a local diffeomorphism, we will need the following useful lemma: Lemma 1.2. Suppose f : M ! N is a local diffeomorphism near every x 2 M. If f is invertible, then f is a global diffeomorphism. Proof. We need to show f −1 is smooth. Fix any y = f(x). The smoothness of f −1 at y depends only on the behaviour of f −1 near y. Since f is a diffeomorphism from a −1 neighborhood of x onto a neighborhood of y, f is smooth at y. Proof of Generalized IFT, compact version. According to Lemma 1.2, it is enough to prove that f is one-to-one in a neighborhood of X. We may embed M into RK , and consider the \"-neighborhood" of X in M: X" = fx 2 M j d(x; X) < "g; where d(·; ·) is the Euclidean distance.
    [Show full text]
  • New Ideas in Algebraic Topology (K-Theory and Its Applications)
    NEW IDEAS IN ALGEBRAIC TOPOLOGY (K-THEORY AND ITS APPLICATIONS) S.P. NOVIKOV Contents Introduction 1 Chapter I. CLASSICAL CONCEPTS AND RESULTS 2 § 1. The concept of a fibre bundle 2 § 2. A general description of fibre bundles 4 § 3. Operations on fibre bundles 5 Chapter II. CHARACTERISTIC CLASSES AND COBORDISMS 5 § 4. The cohomological invariants of a fibre bundle. The characteristic classes of Stiefel–Whitney, Pontryagin and Chern 5 § 5. The characteristic numbers of Pontryagin, Chern and Stiefel. Cobordisms 7 § 6. The Hirzebruch genera. Theorems of Riemann–Roch type 8 § 7. Bott periodicity 9 § 8. Thom complexes 10 § 9. Notes on the invariance of the classes 10 Chapter III. GENERALIZED COHOMOLOGIES. THE K-FUNCTOR AND THE THEORY OF BORDISMS. MICROBUNDLES. 11 § 10. Generalized cohomologies. Examples. 11 Chapter IV. SOME APPLICATIONS OF THE K- AND J-FUNCTORS AND BORDISM THEORIES 16 § 11. Strict application of K-theory 16 § 12. Simultaneous applications of the K- and J-functors. Cohomology operation in K-theory 17 § 13. Bordism theory 19 APPENDIX 21 The Hirzebruch formula and coverings 21 Some pointers to the literature 22 References 22 Introduction In recent years there has been a widespread development in topology of the so-called generalized homology theories. Of these perhaps the most striking are K-theory and the bordism and cobordism theories. The term homology theory is used here, because these objects, often very different in their geometric meaning, Russian Math. Surveys. Volume 20, Number 3, May–June 1965. Translated by I.R. Porteous. 1 2 S.P. NOVIKOV share many of the properties of ordinary homology and cohomology, the analogy being extremely useful in solving concrete problems.
    [Show full text]
  • On the Homology of Configuration Spaces
    TopologyVol. 28, No. I. pp. I II-123, 1989 Lmo-9383 89 53 al+ 00 Pm&d I” Great Bntam fy 1989 Pcrgamon Press plc ON THE HOMOLOGY OF CONFIGURATION SPACES C.-F. B~DIGHEIMER,~ F. COHEN$ and L. TAYLOR: (Receiued 27 July 1987) 1. INTRODUCTION 1.1 BY THE k-th configuration space of a manifold M we understand the space C”(M) of subsets of M with catdinality k. If c’(M) denotes the space of k-tuples of distinct points in M, i.e. Ck(M) = {(z,, . , zk)eMklzi # zj for i #j>, then C’(M) is the orbit space of C’(M) under the permutation action of the symmetric group Ik, c’(M) -+ c’(M)/E, = Ck(M). Configuration spaces appear in various contexts such as algebraic geometry, knot theory, differential topology or homotopy theory. Although intensively studied their homology is unknown except for special cases, see for example [ 1, 2, 7, 8, 9, 12, 13, 14, 18, 261 where different terminology and notation is used. In this article we study the Betti numbers of Ck(M) for homology with coefficients in a field IF. For IF = IF, the rank of H,(C’(M); IF,) is determined by the IF,-Betti numbers of M. the dimension of M, and k. Similar results were obtained by Ldffler-Milgram [ 171 for closed manifolds. For [F = ff, or a field of characteristic zero the corresponding result holds in the case of odd-dimensional manifolds; it is no longer true for even-dimensional manifolds, not even for surfaces, see [S], [6], or 5.5 here.
    [Show full text]
  • Floer Homology, Gauge Theory, and Low-Dimensional Topology
    Floer Homology, Gauge Theory, and Low-Dimensional Topology Clay Mathematics Proceedings Volume 5 Floer Homology, Gauge Theory, and Low-Dimensional Topology Proceedings of the Clay Mathematics Institute 2004 Summer School Alfréd Rényi Institute of Mathematics Budapest, Hungary June 5–26, 2004 David A. Ellwood Peter S. Ozsváth András I. Stipsicz Zoltán Szabó Editors American Mathematical Society Clay Mathematics Institute 2000 Mathematics Subject Classification. Primary 57R17, 57R55, 57R57, 57R58, 53D05, 53D40, 57M27, 14J26. The cover illustrates a Kinoshita-Terasaka knot (a knot with trivial Alexander polyno- mial), and two Kauffman states. These states represent the two generators of the Heegaard Floer homology of the knot in its topmost filtration level. The fact that these elements are homologically non-trivial can be used to show that the Seifert genus of this knot is two, a result first proved by David Gabai. Library of Congress Cataloging-in-Publication Data Clay Mathematics Institute. Summer School (2004 : Budapest, Hungary) Floer homology, gauge theory, and low-dimensional topology : proceedings of the Clay Mathe- matics Institute 2004 Summer School, Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary, June 5–26, 2004 / David A. Ellwood ...[et al.], editors. p. cm. — (Clay mathematics proceedings, ISSN 1534-6455 ; v. 5) ISBN 0-8218-3845-8 (alk. paper) 1. Low-dimensional topology—Congresses. 2. Symplectic geometry—Congresses. 3. Homol- ogy theory—Congresses. 4. Gauge fields (Physics)—Congresses. I. Ellwood, D. (David), 1966– II. Title. III. Series. QA612.14.C55 2004 514.22—dc22 2006042815 Copying and reprinting. Material in this book may be reproduced by any means for educa- tional and scientific purposes without fee or permission with the exception of reproduction by ser- vices that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given.
    [Show full text]
  • LOCAL PROPERTIES of SMOOTH MAPS 1. Submersions And
    LECTURE 6: LOCAL PROPERTIES OF SMOOTH MAPS 1. Submersions and Immersions Last time we showed that if f : M ! N is a diffeomorphism, then dfp : TpM ! Tf(p)N is a linear isomorphism. As in the Euclidean case (see Lecture 2), we can prove the following partial converse: Theorem 1.1 (Inverse Mapping Theorem). Let f : M ! N be a smooth map such that dfp : TpM ! Tf(p)N is a linear isomorphism, then f is a local diffeomorphism near p, i.e. it maps a neighborhood U1 of p diffeomorphically to a neighborhood X1 of q = f(p). Proof. Take a chart f'; U; V g near p and a chart f ; X; Y g near f(p) so that f(U) ⊂ X (This is always possible by shrinking U and V ). Since ' : U ! V and : X ! Y are diffeomorphisms, −1 −1 n n d( ◦ f ◦ ' )'(p) = d q ◦ dfp ◦ d''(p) : T'(p)V = R ! T (q)Y = R is a linear isomorphism. It follows from the inverse function theorem in Lecture 2 −1 that there exist neighborhoods V1 of '(p) and Y1 of (q) so that ◦ f ◦ ' is a −1 −1 diffeomorphism from V1 to Y1. Take U1 = ' (V1) and X1 = (Y1). Then f = −1 ◦ ( ◦ f ◦ '−1) ◦ ' is a diffeomorphism from U1 to X1. Again we cannot conclude global diffeomorphism even if dfp is a linear isomorphism everywhere, since f might not be invertible. In fact, now we can construct an example which is much simpler than the example we constructed in Lecture 2: Let f : S1 ! S1 be given by f(eiθ) = e2iθ.
    [Show full text]
  • Complete Connections on Fiber Bundles
    Complete connections on fiber bundles Matias del Hoyo IMPA, Rio de Janeiro, Brazil. Abstract Every smooth fiber bundle admits a complete (Ehresmann) connection. This result appears in several references, with a proof on which we have found a gap, that does not seem possible to remedy. In this note we provide a definite proof for this fact, explain the problem with the previous one, and illustrate with examples. We also establish a version of the theorem involving Riemannian submersions. 1 Introduction: A rather tricky exercise An (Ehresmann) connection on a submersion p : E → B is a smooth distribution H ⊂ T E that is complementary to the kernel of the differential, namely T E = H ⊕ ker dp. The distributions H and ker dp are called horizontal and vertical, respectively, and a curve on E is called horizontal (resp. vertical) if its speed only takes values in H (resp. ker dp). Every submersion admits a connection: we can take for instance a Riemannian metric ηE on E and set H as the distribution orthogonal to the fibers. Given p : E → B a submersion and H ⊂ T E a connection, a smooth curve γ : I → B, t0 ∈ I, locally defines a horizontal lift γ˜e : J → E, t0 ∈ J ⊂ I,γ ˜e(t0)= e, for e an arbitrary point in the fiber. This lift is unique if we require J to be maximal, and depends smoothly on e. The connection H is said to be complete if for every γ its horizontal lifts can be defined in the whole domain. In that case, a curve γ induces diffeomorphisms between the fibers by parallel transport.
    [Show full text]