DOCSLIB.ORG

Chapter 1. Preface in 1963, John Milnor Put Forward a List of Problems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter 1. Preface in 1963, John Milnor Put Forward a List of Problems Chapter 1. Preface In 1963, John Milnor put forward a list of problems in geometric topology. 3 3 1. Let M be a homology 3-sphere with π1 =0. Is the double suspension of M homeomorphic to S5? 2. Is simple homotopy type a topological invariant? 3. Can rational Pontrjagin classes be defined as topological invariants? 4. (Hauptvermutung) If two PL manifolds are homeomorphic, does it follow that they are PL homeomorphic? 5. Can topological manifolds be triangulated? 6. The Poincar´e hypothesis in dimensions 3, 4. 7. (The annulus conjecture) Is the region bounded by two locally flat n-spheres in (n + 1)-space necessarily homeomorphic to Sn × [0, 1]? These were presented at the 1963 conference on differential and algebraic topology in Seattle, Washington. A much larger problem set from the conference is published in Ann. Math.81 (1965) pp. 565–591. In the last 30 years, much progress has been made on these problems. Problems 1, 2, 3, and 7 were solved affirmatively by Edwards-Cannon, Chapman, Novikov, and Kirby in the late 1960’s and early 1970’s, while problems 4 and 5 were solved negatively by Kirby- Siebenmann in 1967. Freedman solved the 4-dimensional (TOP) Poincar´e Conjecture in 1980. The 3-dimensional Poincar´e Conjecture and the 4-dimensional PL/DIFF Poincar´e Conjecture remain open as of this writing. This book introduces high-dimensional PL and TOP topology by providing solutions – or at least useful information pointing towards solutions – of problems 1, 2, 3, 4, 5, and 7. This sort of geometric topology has recently been applied to Gromov-style differential geometry, index theory, and algebraic geometry. One of the author’s goals in writing this book is to help workers in other areas to understand what the machinery of geometric topology can do. In a sense, this is intended as a book for people who haven’t decided, yet, whether they want to make the (substantial) investment of learning geometric topology! Accordingly, the focus here is on examples and techniques of proof, rather than on developing any one theory or technique to its logical limits. 1 Chapter 2. Some TOP topology We begin with a study of embeddings of Sn−1 into Sn. We know from algebraic topology that if i : Sn−1 → Sn is an embedding, then i(Sn−1) separates Sn into two parts. We remind the student of the statement of Alexander Duality: n n ∼ n−−1 If K is a polyhedron and i : K → S is an embedding, then H¯(S −K) = H (K). n−1 n n−1 ∼ n−1 n−1 If K = S and =0, this says that H¯0(S − S ) = H (S )=Z.Thus, Sn − Sn−1 has two path components. Further applications of Alexander duality show that each of these complementary domains has the homology of a point. The naive conjecture is that each complementary domain is homeomorphic to Dn and that all embeddings Sn−1 → Sn which preserve orientation should be topologically equivalent – that if i : Sn−1 → Sn is an orientation-preserving embedding, then there should be a homeomorphism h : Sn → Sn so that h ◦ i is the standard embedding of Sn−1 onto the equator of Sn. A classical example, the Alexander Horned sphere, shows that this conjecture is false for n =3. In Example 2.22, we provide a slightly different example of an embedded S2 in S3 which fails to bound a disk in S3. To achieve positive results in the face of such counterexamples, we must impose hy- potheses on the embedding i. The classical condition is that i should be either locally or globally collared in Sn. Accordingly, we begin the section with a proof of Morton Brown’s collaring theorem. In the case of Sn−1 ⊂ Sn, Brown’s theorem says that if either complementary domain of Sn−1 in Sn is a manifold, then boundary of this com- plementary domain has a neighborhood (in the complementary domain) homeomorphic to Sn−1 × [0, 1). Here is the general definition of a local collaring: Definition 2.1. Let X be a topological space and let B be a subset of X.ThenB is collared in X if there is an open embedding h : B × [0, 1) → X with h|B ×{0} = id.If B can be covered by a collection of open subsets, each of which is collared in X,thenB is said to be locally collared in X. Theorem 2.2 (M. Brown [B3], [Con]). If X and B are compact metric and B is locally collared in X,thenB is collared in X. Proof: The proof is by induction on the number of elements in the cover of B.This reduces immediately to the case in which B = U ∪ V with both U and V collared in X. 2 2. Some TOP topology 3 Form a space X+ = X ∪ B × [0, ∞). Using the collars on U and V , we can find open subsets U + and V + of X+ homeomorphic to U × [−∞, ∞]andV × [−∞, ∞]sothat U + ∪V + is a neighborhood of B. Choose functions σ, τ : B → [0, 1] so that σ+τ =1, and so that σ and τ are supported on closed subsets (in B) of U and V , respectively. Define + + + + homeomorphisms hσ : U → U and hτ : V → V so that hσ(u, t)=(u, t + σ(u)) and hτ (v, t)=(v, t + τ(v)). If we choose the inner collars on U and V carefully, perhaps by reparameterizing U × [−1, 0] and V × [−1, 0] to be the new U × [−∞, 0] and V × [−∞, 0], + we can assume that hσ and hτ extend continuously to X using the identity outside of + + U and V . The composition hσ ◦ hτ throws X onto X ∪ B × [0, 1], exhibiting a collar on B. X UV + X Corollary 2.3. The boundary of a topological manifold is collared. It is amazing that this was a new result as late as 1962. Our main goal for this section is to prove the Generalized Schoenfliess Theorem, which says that if both complementary domains of Sn−1 in Sn are topological manifolds, then they are homeomorphic to balls. Definition 2.4. By an n-ball B in a manifold M n, we mean the homeomorphic image ◦ of a standard n-ball in Rn.Ann-ball is collared if ∂B is collared in M n − B.The homeomorphic image of a k-ballisalsoreferredtoasak-cell. Many useful homeomorphisms in the topological category are constructed as compo- sitions of pushes. The next lemma constructs one such push. Lemma 2.5. Let B be a collared n-ball in Rn with collar C and let D be a compact subset of Rn. Then there is a homeomorphism h : Rn → Rn with compact support such 4 2. Some TOP topology that h|(B)=id and h(B ∪ C) ⊃ B ∪ D. ◦ ◦ n Proof: Choose a point b ∈ B and a standard ball B1 ⊂ B centered at b.Letρ : R → n R be a homeomorphism with compact support in B ∪ C shrinking B into B1.This homeomorphism is easy to construct after parametrizing B ∪ C as the cone from b on ∂(B ∪ C). n n Let σ : R → R be a homeomorphism with compact support so that σ|B1 = id and σ(B ∪ C) ⊃ B ∪ C ∪ ρ(D). Such a homeomorphism is obtained by radially stretching a −1 small collar on B1. Setting h = ρ ◦ σ ◦ ρ completes the proof, since we have h|B = id and h(B ∪ C)=ρ−1 ◦ σ ◦ ρ(B ∪ C)=ρ−1 ◦ σ(B ∪ C) ⊃ ρ−1(ρ(D)) = D. We obtain a striking characterization of euclidean n-space which is valid for any n. Theorem 2.6 (M. Brown [B2]). Let M n be a TOP n-manifold such that every com- ∼ pact subset D of M iscontainedinsomeopensubsetP of M n with P = Rn.Then ∼ M n = Rn. Proof: ∪∞ ⊂ ⊂ Write M = i=1Di with D1 D2 ... and Di compact for all i.Choosea ◦ small standard ball B ⊂ M with collar C.WriteB ∪ C = ∪Bi with Bi a ball, Bi+1 = ◦ Bi ∪ Ci,Ci a collar on Bi,B1 ⊂ B2 ⊂ .... We define a sequence of homeomorphisms n n hi : M → M so that hi|Bi = hi+1|Bi and hi(Bi) ⊃ Di. This will suffice to prove the Theorem, since limi→∞ hi|B ∪ C is a homeomorphism from B ∪ C onto M. Such a sequence of homeomorphisms is easily obtained by repeated application of ∼ n Lemma 2.5. Given hi,chooseP = R so that P ⊃ hi(B ∪ C) ∪ Di+1. Applying Lemma 2.5, we obtain a homeomorphism h : P → P with compact support so that h |hi(Bi)=id ◦ n and h (hi(Bi+1)) ⊃ Di+1. Setting hi+1 = h ◦ hi and extending to M by the identity completes the construction of hi+1. Theorem 2.7. Let M n be a compact TOP n-manifold which is the union of two open sets U and V which are homeomorphic to Rn.ThenM n is homeomorphic to Sn. Proof: Replace V by a smaller copy of Rn and choose a point p ∈ U −V¯ . We will show that M −{p} is homeomorphic to Rn, which will complete the proof, since the one-point compactification of Rn is Sn. Let D be a compact subset of M −{p} and let B1 be a standard ball in U −D centered at p.Now,U − V is a compact subset of U, so there is a homeomorphism σ : U → U with compact support so that σ(B1) ⊃ U − V .Extendσ by the identity to all of M.We 2.
Load more

Recommended publications
  • Commentary on the Kervaire–Milnor Correspondence 1958–1961
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 52, Number 4, October 2015, Pages 603–609 http://dx.doi.org/10.1090/bull/1508 Article electronically published on July 1, 2015 COMMENTARY ON THE KERVAIRE–MILNOR CORRESPONDENCE 1958–1961 ANDREW RANICKI AND CLAUDE WEBER Abstract. The extant letters exchanged between Kervaire and Milnor during their collaboration from 1958–1961 concerned their work on the classification of exotic spheres, culminating in their 1963 Annals of Mathematics paper. Michel Kervaire died in 2007; for an account of his life, see the obituary by Shalom Eliahou, Pierre de la Harpe, Jean-Claude Hausmann, and Claude We- ber in the September 2008 issue of the Notices of the American Mathematical Society. The letters were made public at the 2009 Kervaire Memorial Confer- ence in Geneva. Their publication in this issue of the Bulletin of the American Mathematical Society is preceded by our commentary on these letters, provid- ing some historical background. Letter 1. From Milnor, 22 August 1958 Kervaire and Milnor both attended the International Congress of Mathemati- cians held in Edinburgh, 14–21 August 1958. Milnor gave an invited half-hour talk on Bernoulli numbers, homotopy groups, and a theorem of Rohlin,andKer- vaire gave a talk in the short communications section on Non-parallelizability of the n-sphere for n>7 (see [2]). In this letter written immediately after the Congress, Milnor invites Kervaire to join him in writing up the lecture he gave at the Con- gress. The joint paper appeared in the Proceedings of the ICM as [10]. Milnor’s name is listed first (contrary to the tradition in mathematics) since it was he who was invited to deliver a talk.
    [Show full text]
  • George W. Whitehead Jr
    George W. Whitehead Jr. 1918–2004 A Biographical Memoir by Haynes R. Miller ©2015 National Academy of Sciences. Any opinions expressed in this memoir are those of the author and do not necessarily reflect the views of the National Academy of Sciences. GEORGE WILLIAM WHITEHEAD JR. August 2, 1918–April 12 , 2004 Elected to the NAS, 1972 Life George William Whitehead, Jr., was born in Bloomington, Ill., on August 2, 1918. Little is known about his family or early life. Whitehead received a BA from the University of Chicago in 1937, and continued at Chicago as a graduate student. The Chicago Mathematics Department was somewhat ingrown at that time, dominated by L. E. Dickson and Gilbert Bliss and exhibiting “a certain narrowness of focus: the calculus of variations, projective differential geometry, algebra and number theory were the main topics of interest.”1 It is possible that Whitehead’s interest in topology was stimulated by Saunders Mac Lane, who By Haynes R. Miller spent the 1937–38 academic year at the University of Chicago and was then in the early stages of his shift of interest from logic and algebra to topology. Of greater importance for Whitehead was the appearance of Norman Steenrod in Chicago. Steenrod had been attracted to topology by Raymond Wilder at the University of Michigan, received a PhD under Solomon Lefschetz in 1936, and remained at Princeton as an Instructor for another three years. He then served as an Assistant Professor at the University of Chicago between 1939 and 1942 (at which point he moved to the University of Michigan).
    [Show full text]
  • Millennium Prize for the Poincaré
    FOR IMMEDIATE RELEASE • March 18, 2010 Press contact: James Carlson: [email protected]; 617-852-7490 See also the Clay Mathematics Institute website: • The Poincaré conjecture and Dr. Perelmanʼs work: http://www.claymath.org/poincare • The Millennium Prizes: http://www.claymath.org/millennium/ • Full text: http://www.claymath.org/poincare/millenniumprize.pdf First Clay Mathematics Institute Millennium Prize Announced Today Prize for Resolution of the Poincaré Conjecture a Awarded to Dr. Grigoriy Perelman The Clay Mathematics Institute (CMI) announces today that Dr. Grigoriy Perelman of St. Petersburg, Russia, is the recipient of the Millennium Prize for resolution of the Poincaré conjecture. The citation for the award reads: The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman. The Poincaré conjecture is one of the seven Millennium Prize Problems established by CMI in 2000. The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude. The award of the Millennium Prize to Dr. Perelman was made in accord with their governing rules: recommendation first by a Special Advisory Committee (Simon Donaldson, David Gabai, Mikhail Gromov, Terence Tao, and Andrew Wiles), then by the CMI Scientific Advisory Board (James Carlson, Simon Donaldson, Gregory Margulis, Richard Melrose, Yum-Tong Siu, and Andrew Wiles), with final decision by the Board of Directors (Landon T.
    [Show full text]
  • PIECEWISE LINEAR TOPOLOGY Contents 1. Introduction 2 2. Basic
    PIECEWISE LINEAR TOPOLOGY J. L. BRYANT Contents 1. Introduction 2 2. Basic Definitions and Terminology. 2 3. Regular Neighborhoods 9 4. General Position 16 5. Embeddings, Engulfing 19 6. Handle Theory 24 7. Isotopies, Unknotting 30 8. Approximations, Controlled Isotopies 31 9. Triangulations of Manifolds 33 References 35 1 2 J. L. BRYANT 1. Introduction The piecewise linear category offers a rich structural setting in which to study many of the problems that arise in geometric topology. The first systematic ac- counts of the subject may be found in [2] and [63]. Whitehead’s important paper [63] contains the foundation of the geometric and algebraic theory of simplicial com- plexes that we use today. More recent sources, such as [30], [50], and [66], together with [17] and [37], provide a fairly complete development of PL theory up through the early 1970’s. This chapter will present an overview of the subject, drawing heavily upon these sources as well as others with the goal of unifying various topics found there as well as in other parts of the literature. We shall try to give enough in the way of proofs to provide the reader with a flavor of some of the techniques of the subject, while deferring the more intricate details to the literature. Our discussion will generally avoid problems associated with embedding and isotopy in codimension 2. The reader is referred to [12] for a survey of results in this very important area. 2. Basic Definitions and Terminology. Simplexes. A simplex of dimension p (a p-simplex) σ is the convex closure of a n set of (p+1) geometrically independent points {v0, .
    [Show full text]
  • The Arf-Kervaire Invariant Problem in Algebraic Topology: Introduction
    THE ARF-KERVAIRE INVARIANT PROBLEM IN ALGEBRAIC TOPOLOGY: INTRODUCTION MICHAEL A. HILL, MICHAEL J. HOPKINS, AND DOUGLAS C. RAVENEL ABSTRACT. This paper gives the history and background of one of the oldest problems in algebraic topology, along with a short summary of our solution to it and a description of some of the tools we use. More details of the proof are provided in our second paper in this volume, The Arf-Kervaire invariant problem in algebraic topology: Sketch of the proof. A rigorous account can be found in our preprint The non-existence of elements of Kervaire invariant one on the arXiv and on the third author’s home page. The latter also has numerous links to related papers and talks we have given on the subject since announcing our result in April, 2009. CONTENTS 1. Background and history 3 1.1. Pontryagin’s early work on homotopy groups of spheres 3 1.2. Our main result 8 1.3. The manifold formulation 8 1.4. The unstable formulation 12 1.5. Questions raised by our theorem 14 2. Our strategy 14 2.1. Ingredients of the proof 14 2.2. The spectrum Ω 15 2.3. How we construct Ω 15 3. Some classical algebraic topology. 15 3.1. Fibrations 15 3.2. Cofibrations 18 3.3. Eilenberg-Mac Lane spaces and cohomology operations 18 3.4. The Steenrod algebra. 19 3.5. Milnor’s formulation 20 3.6. Serre’s method of computing homotopy groups 21 3.7. The Adams spectral sequence 21 4. Spectra and equivariant spectra 23 4.1.
    [Show full text]
  • Pierre Deligne
    www.abelprize.no Pierre Deligne Pierre Deligne was born on 3 October 1944 as a hobby for his own personal enjoyment. in Etterbeek, Brussels, Belgium. He is Profes- There, as a student of Jacques Tits, Deligne sor Emeritus in the School of Mathematics at was pleased to discover that, as he says, the Institute for Advanced Study in Princeton, “one could earn one’s living by playing, i.e. by New Jersey, USA. Deligne came to Prince- doing research in mathematics.” ton in 1984 from Institut des Hautes Études After a year at École Normal Supériure in Scientifiques (IHÉS) at Bures-sur-Yvette near Paris as auditeur libre, Deligne was concur- Paris, France, where he was appointed its rently a junior scientist at the Belgian National youngest ever permanent member in 1970. Fund for Scientific Research and a guest at When Deligne was around 12 years of the Institut des Hautes Études Scientifiques age, he started to read his brother’s university (IHÉS). Deligne was a visiting member at math books and to demand explanations. IHÉS from 1968-70, at which time he was His interest prompted a high-school math appointed a permanent member. teacher, J. Nijs, to lend him several volumes Concurrently, he was a Member (1972– of “Elements of Mathematics” by Nicolas 73, 1977) and Visitor (1981) in the School of Bourbaki, the pseudonymous grey eminence Mathematics at the Institute for Advanced that called for a renovation of French mathe- Study. He was appointed to a faculty position matics. This was not the kind of reading mat- there in 1984.
    [Show full text]
  • Triangulations of 3-Manifolds with Essential Edges
    TRIANGULATIONS OF 3–MANIFOLDS WITH ESSENTIAL EDGES CRAIG D. HODGSON, J. HYAM RUBINSTEIN, HENRY SEGERMAN AND STEPHAN TILLMANN Dedicated to Michel Boileau for his many contributions and leadership in low dimensional topology. ABSTRACT. We define essential and strongly essential triangulations of 3–manifolds, and give four constructions using different tools (Heegaard splittings, hierarchies of Haken 3–manifolds, Epstein-Penner decompositions, and cut loci of Riemannian manifolds) to obtain triangulations with these properties under various hypotheses on the topology or geometry of the manifold. We also show that a semi-angle structure is a sufficient condition for a triangulation of a 3–manifold to be essential, and a strict angle structure is a sufficient condition for a triangulation to be strongly essential. Moreover, algorithms to test whether a triangulation of a 3–manifold is essential or strongly essential are given. 1. INTRODUCTION Finding combinatorial and topological properties of geometric triangulations of hyper- bolic 3–manifolds gives information which is useful for the reverse process—for exam- ple, solving Thurston’s gluing equations to construct an explicit hyperbolic metric from an ideal triangulation. Moreover, suitable properties of a triangulation can be used to de- duce facts about the variety of representations of the fundamental group of the underlying 3-manifold into PSL(2;C). See for example [32]. In this paper, we focus on one-vertex triangulations of closed manifolds and ideal trian- gulations of the interiors of compact manifolds with boundary. Two important properties of these triangulations are introduced. For one-vertex triangulations in the closed case, the first property is that no edge loop is null-homotopic and the second is that no two edge loops are homotopic, keeping the vertex fixed.
    [Show full text]
  • Math, Physics, and Calabi–Yau Manifolds
    Math, Physics, and Calabi–Yau Manifolds Shing-Tung Yau Harvard University October 2011 Introduction I’d like to talk about how mathematics and physics can come together to the benefit of both fields, particularly in the case of Calabi-Yau spaces and string theory. This happens to be the subject of the new book I coauthored, THE SHAPE OF INNER SPACE It also tells some of my own story and a bit of the history of geometry as well. 2 In that spirit, I’m going to back up and talk about my personal introduction to geometry and how I ended up spending much of my career working at the interface between math and physics. Along the way, I hope to give people a sense of how mathematicians think and approach the world. I also want people to realize that mathematics does not have to be a wholly abstract discipline, disconnected from everyday phenomena, but is instead crucial to our understanding of the physical world. 3 There are several major contributions of mathematicians to fundamental physics in 20th century: 1. Poincar´eand Minkowski contribution to special relativity. (The book of Pais on the biography of Einstein explained this clearly.) 2. Contributions of Grossmann and Hilbert to general relativity: Marcel Grossmann (1878-1936) was a classmate with Einstein from 1898 to 1900. he was professor of geometry at ETH, Switzerland at 1907. In 1912, Einstein came to ETH to be professor where they started to work together. Grossmann suggested tensor calculus, as was proposed by Elwin Bruno Christoffel in 1868 (Crelle journal) and developed by Gregorio Ricci-Curbastro and Tullio Levi-Civita (1901).
    [Show full text]
  • Experimental Statistics of Veering Triangulations
    EXPERIMENTAL STATISTICS OF VEERING TRIANGULATIONS WILLIAM WORDEN Abstract. Certain fibered hyperbolic 3-manifolds admit a layered veering triangulation, which can be constructed algorithmically given the stable lamination of the monodromy. These triangulations were introduced by Agol in 2011, and have been further studied by several others in the years since. We obtain experimental results which shed light on the combinatorial structure of veering triangulations, and its relation to certain topological invariants of the underlying manifold. 1. Introduction In 2011, Agol [Ago11] introduced the notion of a layered veering triangulation for certain fibered hyperbolic manifolds. In particular, given a surface Σ (possibly with punctures), and a pseudo-Anosov homeomorphism ' ∶ Σ → Σ, Agol's construction gives a triangulation ○ ○ ○ ○ T for the mapping torus M' with fiber Σ and monodromy ' = 'SΣ○ , where Σ is the surface resulting from puncturing Σ at the singularities of its '-invariant foliations. For Σ a once-punctured torus or 4-punctured sphere, the resulting triangulation is well understood: it is the monodromy (or Floyd{Hatcher) triangulation considered in [FH82] and [Jør03]. In this case T is a geometric triangulation, meaning that tetrahedra in T can be realized as ideal hyperbolic tetrahedra isometrically embedded in M'○ . In fact, these triangulations are geometrically canonical, i.e., dual to the Ford{Voronoi domain of the mapping torus (see [Aki99], [Lac04], and [Gu´e06]). Agol's definition of T was conceived as a generalization of these monodromy triangulations, and so a natural question is whether these generalized monodromy triangulations are also geometric. Hodgson{Issa{Segerman [HIS16] answered this question in the negative, by producing the first examples of non- geometric layered veering triangulations.
    [Show full text]
  • Equivalences to the Triangulation Conjecture 1 Introduction
    ISSN online printed Algebraic Geometric Topology Volume ATG Published Decemb er Equivalences to the triangulation conjecture Duane Randall Abstract We utilize the obstruction theory of GalewskiMatumotoStern to derive equivalent formulations of the Triangulation Conjecture For ex n ample every closed top ological manifold M with n can b e simplicially triangulated if and only if the two distinct combinatorial triangulations of 5 RP are simplicially concordant AMS Classication N S Q Keywords Triangulation KirbySieb enmann class Bo ckstein op erator top ological manifold Intro duction The Triangulation Conjecture TC arms that every closed top ological man n ifold M of dimension n admits a simplicial triangulation The vanishing of the KirbySieb enmann class KS M in H M Z is b oth necessary and n sucient for the existence of a combinatorial triangulation of M for n n by A combinatorial triangulation of a closed manifold M is a simplicial triangulation for which the link of every i simplex is a combinatorial sphere of dimension n i Galewski and Stern Theorem and Matumoto n indep endently proved that a closed connected top ological manifold M with n is simplicially triangulable if and only if KS M in H M ker where denotes the Bo ckstein op erator asso ciated to the exact sequence ker Z of ab elian groups Moreover the Triangulation Conjecture is true if and only if this exact sequence splits by or page The Ro chlin invariant morphism is dened on the homology b ordism group of oriented
    [Show full text]
  • Some Decomposition Lemmas of Diffeomorphisms Compositio Mathematica, Tome 84, No 1 (1992), P
    COMPOSITIO MATHEMATICA VICENTE CERVERA FRANCISCA MASCARO Some decomposition lemmas of diffeomorphisms Compositio Mathematica, tome 84, no 1 (1992), p. 101-113 <http://www.numdam.org/item?id=CM_1992__84_1_101_0> © Foundation Compositio Mathematica, 1992, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Compositio Mathematica 84: 101-113,101 1992. cg 1992 Kluwer Academic Publishers. Printed in the Netherlands. Some decomposition lemmas of diffeomorphisms VICENTE CERVERA and FRANCISCA MASCARO Departamento de Geometria, y Topologia Facultad de Matemàticas, Universidad de Valencia, 46100-Burjassot (Valencia) Spain Received 9 April 1991; accepted 14 February 1992 Abstract. Let Q be a volume element on an open manifold, M, which is the interior of a compact manifold M. We will give conditions for a non-compact supported Q-preserving diffeomorphism to decompose as a finite product of S2-preserving diffeomorphisms with supports in locally finite families of disjoint cells. A widely used and very powerful technique in the study of some subgroups of the group of diffeomorphisms of a differentiable manifold, Diff(M), is the decomposition of its elements as a finite product of diffeomorphisms with support in cells (See for example [1], [5], [7]). It was in a paper of Palis and Smale [7] that first appeared one of those decompositions for the case of a compact manifold, M.
    [Show full text]
  • Knots, 3-Manifolds and the Lickorish Wallace Theorem
    KNOTS, 3-MANIFOLDS AND THE LICKORISH WALLACE THEOREM A REPORT submitted in partial fulfillment of the requirements for the award of the dual degree of Bachelor of Science-Master of Science in MATHEMATICS by M V AJAY KUMAR NAIR (13076) DEPARTMENT OF MATHEMATICS INDIAN INSTITUTE OF SCIENCE EDUCATION AND RESEARCH BHOPAL BHOPAL - 462066 April 2018 i CERTIFICATE This is to certify that M V Ajay Kumar Nair, BS-MS (Dual Degree) student in Department of Mathematics has completed bonafide work on the dissertation entitled `Knots, 3-manifolds and the Lickorish Wallace theorem' under my supervision and guidance. April 2018 Dr. Kashyap Rajeevsarathy IISER Bhopal Committee Member Signature Date ii ACADEMIC INTEGRITY AND COPYRIGHT DISCLAIMER I hereby declare that this project is my own work and, to the best of my knowledge, it contains no materials previously published or written by another person, or substantial proportions of material which have been ac- cepted for the award of any other degree or diploma at IISER Bhopal or any other educational institution, except where due acknowledgement is made in the document. I certify that all copyrighted material incorporated into this document is in compliance with the Indian Copyright (Amendment) Act (2012) and that I have received written permission from the copyright owners for my use of their work, which is beyond the scope of the law. I agree to indemnify and saveguard IISER Bhopal from any claims that may be arise from any copyright violation. April 2018 M V Ajay Kumar Nair IISER Bhopal iii ACKNOWLEDGEMENT I would like to extend my heartfelt gratitude to my thesis supervisor Dr Kashyap Rajeevsarathy for the persistent inspiration and guidance.
    [Show full text]