Some Immersion Theorems for Manifolds
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The Extension Problem in Complex Analysis II; Embeddings with Positive Normal Bundle Author(S): Phillip A
The Extension Problem in Complex Analysis II; Embeddings with Positive Normal Bundle Author(s): Phillip A. Griffiths Source: American Journal of Mathematics, Vol. 88, No. 2 (Apr., 1966), pp. 366-446 Published by: The Johns Hopkins University Press Stable URL: http://www.jstor.org/stable/2373200 Accessed: 25/10/2010 12:00 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=jhup. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics. http://www.jstor.org THE EXTENSION PROBLEM IN COMPLEX ANALYSIS II; EMBEDDINGS WITH POSITIVE NORMAL BUNDLE. -
1.10 Partitions of Unity and Whitney Embedding 1300Y Geometry and Topology
1.10 Partitions of unity and Whitney embedding 1300Y Geometry and Topology Theorem 1.49 (noncompact Whitney embedding in R2n+1). Any smooth n-manifold may be embedded in R2n+1 (or immersed in R2n). Proof. We saw that any manifold may be written as a countable union of increasing compact sets M = [Ki, and that a regular covering f(Ui;k ⊃ Vi;k;'i;k)g of M can be chosen so that for fixed i, fVi;kgk is a finite ◦ ◦ cover of Ki+1nKi and each Ui;k is contained in Ki+2nKi−1. This means that we can express M as the union of 3 open sets W0;W1;W2, where [ Wj = ([kUi;k): i≡j(mod3) 2n+1 Each of the sets Ri = [kUi;k may be injectively immersed in R by the argument for compact manifolds, 2n+1 since they have a finite regular cover. Call these injective immersions Φi : Ri −! R . The image Φi(Ri) is bounded since all the charts are, by some radius ri. The open sets Ri; i ≡ j(mod3) for fixed j are disjoint, and by translating each Φi; i ≡ j(mod3) by an appropriate constant, we can ensure that their images in R2n+1 are disjoint as well. 0 −! 0 2n+1 Let Φi = Φi + (2(ri−1 + ri−2 + ··· ) + ri) e 1. Then Ψj = [i≡j(mod3)Φi : Wj −! R is an embedding. 2n+1 Now that we have injective immersions Ψ0; Ψ1; Ψ2 of W0;W1;W2 in R , we may use the original argument for compact manifolds: Take the partition of unity subordinate to Ui;k and resum it, obtaining a P P 3-element partition of unity ff1; f2; f3g, with fj = i≡j(mod3) k fi;k. -
Characteristic Classes and K-Theory Oscar Randal-Williams
Characteristic classes and K-theory Oscar Randal-Williams https://www.dpmms.cam.ac.uk/∼or257/teaching/notes/Kthy.pdf 1 Vector bundles 1 1.1 Vector bundles . 1 1.2 Inner products . 5 1.3 Embedding into trivial bundles . 6 1.4 Classification and concordance . 7 1.5 Clutching . 8 2 Characteristic classes 10 2.1 Recollections on Thom and Euler classes . 10 2.2 The projective bundle formula . 12 2.3 Chern classes . 14 2.4 Stiefel–Whitney classes . 16 2.5 Pontrjagin classes . 17 2.6 The splitting principle . 17 2.7 The Euler class revisited . 18 2.8 Examples . 18 2.9 Some tangent bundles . 20 2.10 Nonimmersions . 21 3 K-theory 23 3.1 The functor K ................................. 23 3.2 The fundamental product theorem . 26 3.3 Bott periodicity and the cohomological structure of K-theory . 28 3.4 The Mayer–Vietoris sequence . 36 3.5 The Fundamental Product Theorem for K−1 . 36 3.6 K-theory and degree . 38 4 Further structure of K-theory 39 4.1 The yoga of symmetric polynomials . 39 4.2 The Chern character . 41 n 4.3 K-theory of CP and the projective bundle formula . 44 4.4 K-theory Chern classes and exterior powers . 46 4.5 The K-theory Thom isomorphism, Euler class, and Gysin sequence . 47 n 4.6 K-theory of RP ................................ 49 4.7 Adams operations . 51 4.8 The Hopf invariant . 53 4.9 Correction classes . 55 4.10 Gysin maps and topological Grothendieck–Riemann–Roch . 58 Last updated May 22, 2018. -
Tight Immersions of Simplicial Surfaces in Three Space
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Pergamon Topology Vol. 35, No. 4, pp. 863-873, 19% Copyright Q 1996 Elsevier Sctence Ltd Printed in Great Britain. All tights reserved CO4&9383/96 $15.00 + 0.00 0040-9383(95)00047-X TIGHT IMMERSIONS OF SIMPLICIAL SURFACES IN THREE SPACE DAVIDE P. CERVONE (Received for publication 13 July 1995) 1. INTRODUCTION IN 1960,Kuiper [12,13] initiated the study of tight immersions of surfaces in three space, and showed that every compact surface admits a tight immersion except for the Klein bottle, the real projective plane, and possibly the real projective plane with one handle. The fate of the latter surface went unresolved for 30 years until Haab recently proved [9] that there is no smooth tight immersion of the projective plane with one handle. In light of this, a recently described simplicial tight immersion of this surface [6] came as quite a surprise, and its discovery completes Kuiper’s initial survey. Pinkall [17] broadened the question in 1985 by looking for tight immersions in each equivalence class of immersed surfaces under image homotopy. He first produced a complete description of these classes, and proceeded to generate tight examples in all but finitely many of them. In this paper, we improve Pinkall’s result by giving tight simplicial examples in all but two of the immersion classes under image homotopy for which tight immersions are possible, and in particular, we point out and resolve an error in one of his examples that would have left an infinite number of immersion classes with no tight examples. -
Symplectic Topology Math 705 Notes by Patrick Lei, Spring 2020
Symplectic Topology Math 705 Notes by Patrick Lei, Spring 2020 Lectures by R. Inanç˙ Baykur University of Massachusetts Amherst Disclaimer These notes were taken during lecture using the vimtex package of the editor neovim. Any errors are mine and not the instructor’s. In addition, my notes are picture-free (but will include commutative diagrams) and are a mix of my mathematical style (omit lengthy computations, use category theory) and that of the instructor. If you find any errors, please contact me at [email protected]. Contents Contents • 2 1 January 21 • 5 1.1 Course Description • 5 1.2 Organization • 5 1.2.1 Notational conventions•5 1.3 Basic Notions • 5 1.4 Symplectic Linear Algebra • 6 2 January 23 • 8 2.1 More Basic Linear Algebra • 8 2.2 Compatible Complex Structures and Inner Products • 9 3 January 28 • 10 3.1 A Big Theorem • 10 3.2 More Compatibility • 11 4 January 30 • 13 4.1 Homework Exercises • 13 4.2 Subspaces of Symplectic Vector Spaces • 14 5 February 4 • 16 5.1 Linear Algebra, Conclusion • 16 5.2 Symplectic Vector Bundles • 16 6 February 6 • 18 6.1 Proof of Theorem 5.11 • 18 6.2 Vector Bundles, Continued • 18 6.3 Compatible Triples on Manifolds • 19 7 February 11 • 21 7.1 Obtaining Compatible Triples • 21 7.2 Complex Structures • 21 8 February 13 • 23 2 3 8.1 Kähler Forms Continued • 23 8.2 Some Algebraic Geometry • 24 8.3 Stein Manifolds • 25 9 February 20 • 26 9.1 Stein Manifolds Continued • 26 9.2 Topological Properties of Kähler Manifolds • 26 9.3 Complex and Symplectic Structures on 4-Manifolds • 27 10 February 25 -
Topological K-Theory
TOPOLOGICAL K-THEORY ZACHARY KIRSCHE Abstract. The goal of this paper is to introduce some of the basic ideas sur- rounding the theory of vector bundles and topological K-theory. To motivate this, we will use K-theoretic methods to prove Adams' theorem about the non- existence of maps of Hopf invariant one in dimensions other than n = 1; 2; 4; 8. We will begin by developing some of the basics of the theory of vector bundles in order to properly explain the Hopf invariant one problem and its implica- tions. We will then introduce the theory of characteristic classes and see how Stiefel-Whitney classes can be used as a sort of partial solution to the problem. We will then develop the methods of topological K-theory in order to provide a full solution by constructing the Adams operations. Contents 1. Introduction and background 1 1.1. Preliminaries 1 1.2. Vector bundles 2 1.3. Classifying Spaces 4 2. The Hopf invariant 6 2.1. H-spaces, division algebras, and tangent bundles of spheres 6 3. Characteristic classes 8 3.1. Definition and construction 9 3.2. Real projective space 10 4. K-theory 11 4.1. The ring K(X) 11 4.2. Clutching functions and Bott periodicity 14 4.3. Adams operations 17 Acknowledgments 19 References 19 1. Introduction and background 1.1. Preliminaries. This paper will presume a fair understanding of the basic ideas of algebraic topology, including homotopy theory and (co)homology. As with most writings on algebraic topology, all maps are assumed to be continuous unless otherwise stated. -
MA4E0 Lie Groups
MA4E0 Lie Groups December 5, 2012 Contents 0 Topological Groups 1 1 Manifolds 2 2 Lie Groups 3 3 The Lie Algebra 6 4 The Tangent space 8 5 The Lie Algebra of a Lie Group 10 6 Integrating Vector fields and the exponential map 13 7 Lie Subgroups 17 8 Continuous homomorphisms are smooth 22 9 Distributions and Frobenius Theorem 23 10 Lie Group topics 24 These notes are based on the 2012 MA4E0 Lie Groups course, taught by John Rawnsley, typeset by Matthew Egginton. No guarantee is given that they are accurate or applicable, but hopefully they will assist your study. Please report any errors, factual or typographical, to [email protected] i MA4E0 Lie Groups Lecture Notes Autumn 2012 0 Topological Groups Let G be a group and then write the group structure in terms of maps; multiplication becomes m : G × G ! G −1 defined by m(g1; g2) = g1g2 and inversion becomes i : G ! G defined by i(g) = g . If we suppose that there is a topology on G as a set given by a subset T ⊂ P (G) with the usual rules. We give G × G the product topology. Then we require that m and i are continuous maps. Then G with this topology is a topological group Examples include 1. G any group with the discrete topology 2. Rn with the Euclidean topology and m(x; y) = x + y and i(x) = −x 1 2 2 2 3. S = f(x; y) 2 R : x + y = 1g = fz 2 C : jzj = 1g with m(z1; z2) = z1z2 and i(z) =z ¯. -
Some Very Basic Deformation Theory
Some very basic deformation theory Carlo Pagano W We warm ourselves up with some infinitesimal deformation tasks, to get a feeling of the type of question we are dealing with: 1 Infinitesimal Deformation of a sub-scheme of a scheme Take k a field, X a scheme defined over k, Y a closed subscheme of X. An 0 infinitesimal deformation of Y in X, for us, is any closed subscheme Y ⊆ X ×spec(k) 0 spec(k[]), flat over k[], such that Y ×Spec(k[]) Spec(k) = Y . We want to get a description of these deformations. Let's do it locally, working the affine case, and then we'll get the global one by patching. Translating the above conditions you get: classify ideals I0 ⊆ B0 := Bj], such that I0 reduces to I modulo and B0=I0 is flat over k[]. Flatness over k[] can be tested just looking at the sequence 0 ! () ! k[] ! k ! 0(observe that the first and the third are isomorphic as k[]-modules), being so equivalent to the exactness of the sequence 0 ! B=I ! B0=I0 ! B=I ! 0. So in this situation you see that if you have such an I0 then every element y 2 I lift uniquely to an element of B=I, giving 2 rise to a map of B − modules, 'I : I=I ! B=I, proceeding backwards from 0 0 ' 2 HomB(I; B=I), you gain an ideal I of the type wanted(to check this use again the particularly easy test of flatness over k[]). So we find that our set of 2 deformation is naturally parametrized by HomB(I=I ; B=I). -
LECTURE 1. Differentiable Manifolds, Differentiable Maps
LECTURE 1. Differentiable manifolds, differentiable maps Def: topological m-manifold. Locally euclidean, Hausdorff, second-countable space. At each point we have a local chart. (U; h), where h : U ! Rm is a topological embedding (homeomorphism onto its image) and h(U) is open in Rm. Def: differentiable structure. An atlas of class Cr (r ≥ 1 or r = 1 ) for a topological m-manifold M is a collection U of local charts (U; h) satisfying: (i) The domains U of the charts in U define an open cover of M; (ii) If two domains of charts (U; h); (V; k) in U overlap (U \V 6= ;) , then the transition map: k ◦ h−1 : h(U \ V ) ! k(U \ V ) is a Cr diffeomorphism of open sets in Rm. (iii) The atlas U is maximal for property (ii). Def: differentiable map between differentiable manifolds. f : M m ! N n (continuous) is differentiable at p if for any local charts (U; h); (V; k) at p, resp. f(p) with f(U) ⊂ V the map k ◦ f ◦ h−1 = F : h(U) ! k(V ) is m differentiable at x0 = h(p) (as a map from an open subset of R to an open subset of Rn.) f : M ! N (continuous) is of class Cr if for any charts (U; h); (V; k) on M resp. N with f(U) ⊂ V , the composition k ◦ f ◦ h−1 : h(U) ! k(V ) is of class Cr (between open subsets of Rm, resp. Rn.) f : M ! N of class Cr is an immersion if, for any local charts (U; h); (V; k) as above (for M, resp. -
Immersion of Self-Intersecting Solids and Surfaces: Supplementary Material 1: Topology
Immersion of Self-Intersecting Solids and Surfaces: Supplementary Material 1: Topology Yijing Li and Jernej Barbiˇc,University of Southern California May 15, 2018 1 Topological definitions A mapping f ∶ X → Y is injective (also called \one-to-one") if for every x; y ∈ X such that x ≠ y; we have f(x) ≠ f(y): It is surjective (also called \onto") if for each y ∈ Y; there exists x ∈ X such that f(x) = y; i.e., f(X) = Y: A topological space is a set X equipped with a topology. A topology is a collection τ of subsets of X satisfying the following properties: (1) the empty set and X belong to τ; (2) any union of members of τ still belongs to τ; and (3) the intersection of any finite number of members of τ belongs to τ: A neighborhood of x ∈ X is any set that contains an open set that contains x: A homeomorphism between two topological spaces is a bijection that is continuous in both directions. A mapping between two topological spaces is continuous if the pre-image of each open set is continuous. Homeomorphic topological spaces are considered equivalent for topological purposes. An immersion between two topological spaces X and Y is a continuous mapping f that is locally injective, but not necessarily globally injective. I.e., for each x ∈ X; there exists a neighborhood N of x such that the restriction of f onto N is injective. Topological closure of a subset A of a topological set X is the set A of elements of x ∈ X with the property that every neighborhood of x contains a point of A: A topological space X is path-connected if, for each x; y ∈ X; there is a continuous mapping γ ∶ [0; 1] → X (a curve), such that γ(0) = x and γ(1) = y: A manifold of dimension d is a topological space with the property that every point x has a d d neighborhood that is homeomorphic to R (or, equivalently, to the open unit ball in R ), or a d d halfspace of R (or, equivalently, to the open unit ball in R intersected with the upper halfspace of d R ). -
Lecture 9: the Whitney Embedding Theorem
LECTURE 9: THE WHITNEY EMBEDDING THEOREM Historically, the word \manifold" (Mannigfaltigkeit in German) first appeared in Riemann's doctoral thesis in 1851. At the early times, manifolds are defined extrinsi- cally: they are the set of all possible values of some variables with certain constraints. Translated into modern language,\smooth manifolds" are objects that are (locally) de- fined by smooth equations and, according to last lecture, are embedded submanifolds in Euclidean spaces. In 1912 Weyl gave an intrinsic definition for smooth manifolds. A natural question is: what is the difference between the extrinsic definition and the intrinsic definition? Is there any \abstract" manifold that cannot be embedded into any Euclidian space? In 1930s, Whitney and others settled this foundational problem: the two ways of defining smooth manifolds are in fact the same. In fact, Whitney's result is much more stronger than this. He showed that not only one can embed any smooth manifold into some Euclidian space, but that the dimension of the Euclidian space can be chosen to be (as low as) twice the dimension of the manifold itself! Theorem 0.1 (The Whitney embedding theorem). Any smooth manifold M of di- mension m can be embedded into R2m+1. Remark. In 1944, by using completely different techniques (now known as the \Whitney trick"), Whitney was able to prove Theorem 0.2 (The Strong Whitney Embedding Theorem). Any smooth man- ifold M of dimension m ≥ 2 can be embedded into R2m (and can be immersed into R2m−1). We will not prove this stronger version in this course, but just mention that the Whitney trick was further developed in h-cobordism theory by Smale, using which he proved the Poincare conjecture in dimension ≥ 5 in 1961! Remark. -
Milnor, John W. Groups of Homotopy Spheres
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 52, Number 4, October 2015, Pages 699–710 http://dx.doi.org/10.1090/bull/1506 Article electronically published on June 12, 2015 SELECTED MATHEMATICAL REVIEWS related to the paper in the previous section by JOHN MILNOR MR0148075 (26 #5584) 57.10 Kervaire, Michel A.; Milnor, John W. Groups of homotopy spheres. I. Annals of Mathematics. Second Series 77 (1963), 504–537. The authors aim to study the set of h-cobordism classes of smooth homotopy n-spheres; they call this set Θn. They remark that for n =3 , 4thesetΘn can also be described as the set of diffeomorphism classes of differentiable structures on Sn; but this observation rests on the “higher-dimensional Poincar´e conjecture” plus work of Smale [Amer. J. Math. 84 (1962), 387–399], and it does not really form part of the logical structure of the paper. The authors show (Theorem 1.1) that Θn is an abelian group under the connected sum operation. (In § 2, the authors give a careful treatment of the connected sum and of the lemmas necessary to prove Theorem 1.1.) The main task of the present paper, Part I, is to set up methods for use in Part II, and to prove that for n = 3 the group Θn is finite (Theorem 1.2). (For n =3the authors’ methods break down; but the Poincar´e conjecture for n =3wouldimply that Θ3 = 0.) We are promised more detailed information about the groups Θn in Part II. The authors’ method depends on introducing a subgroup bPn+1 ⊂ Θn;asmooth homotopy n-sphere qualifies for bPn+1 if it is the boundary of a parallelizable man- ifold.