Differential Topology
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I0 and Rank-Into-Rank Axioms
I0 and rank-into-rank axioms Vincenzo Dimonte∗ July 11, 2017 Abstract Just a survey on I0. Keywords: Large cardinals; Axiom I0; rank-into-rank axioms; elemen- tary embeddings; relative constructibility; embedding lifting; singular car- dinals combinatorics. 2010 Mathematics Subject Classifications: 03E55 (03E05 03E35 03E45) 1 Informal Introduction to the Introduction Ok, that’s it. People who know me also know that it is years that I am ranting about a book about I0, how it is important to write it and publish it, etc. I realized that this particular moment is still right for me to write it, for two reasons: time is scarce, and probably there are still not enough results (or anyway not enough different lines of research). This has the potential of being a very nasty vicious circle: there are not enough results about I0 to write a book, but if nobody writes a book the diffusion of I0 is too limited, and so the number of results does not increase much. To avoid this deadlock, I decided to divulge a first draft of such a hypothetical book, hoping to start a “conversation” with that. It is literally a first draft, so it is full of errors and in a liquid state. For example, I still haven’t decided whether to call it I0 or I0, both notations are used in literature. I feel like it still lacks a vision of the future, a map on where the research could and should going about I0. Many proofs are old but never published, and therefore reconstructed by me, so maybe they are wrong. -
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. -
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". -
Elementary Functions Part 1, Functions Lecture 1.6D, Function Inverses: One-To-One and Onto Functions
Elementary Functions Part 1, Functions Lecture 1.6d, Function Inverses: One-to-one and onto functions Dr. Ken W. Smith Sam Houston State University 2013 Smith (SHSU) Elementary Functions 2013 26 / 33 Function Inverses When does a function f have an inverse? It turns out that there are two critical properties necessary for a function f to be invertible. The function needs to be \one-to-one" and \onto". Smith (SHSU) Elementary Functions 2013 27 / 33 Function Inverses When does a function f have an inverse? It turns out that there are two critical properties necessary for a function f to be invertible. The function needs to be \one-to-one" and \onto". Smith (SHSU) Elementary Functions 2013 27 / 33 Function Inverses When does a function f have an inverse? It turns out that there are two critical properties necessary for a function f to be invertible. The function needs to be \one-to-one" and \onto". Smith (SHSU) Elementary Functions 2013 27 / 33 Function Inverses When does a function f have an inverse? It turns out that there are two critical properties necessary for a function f to be invertible. The function needs to be \one-to-one" and \onto". Smith (SHSU) Elementary Functions 2013 27 / 33 One-to-one functions* A function like f(x) = x2 maps two different inputs, x = −5 and x = 5, to the same output, y = 25. But with a one-to-one function no pair of inputs give the same output. Here is a function we saw earlier. This function is not one-to-one since both the inputs x = 2 and x = 3 give the output y = C: Smith (SHSU) Elementary Functions 2013 28 / 33 One-to-one functions* A function like f(x) = x2 maps two different inputs, x = −5 and x = 5, to the same output, y = 25. -
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. -
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. -
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. -
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. -
Monomorphism - Wikipedia, the Free Encyclopedia
Monomorphism - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Monomorphism Monomorphism From Wikipedia, the free encyclopedia In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation . In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism, that is, an arrow f : X → Y such that, for all morphisms g1, g2 : Z → X, Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below. The categorical dual of a monomorphism is an epimorphism, i.e. a monomorphism in a category C is an epimorphism in the dual category Cop. Every section is a monomorphism, and every retraction is an epimorphism. Contents 1 Relation to invertibility 2 Examples 3 Properties 4 Related concepts 5 Terminology 6 See also 7 References Relation to invertibility Left invertible morphisms are necessarily monic: if l is a left inverse for f (meaning l is a morphism and ), then f is monic, as A left invertible morphism is called a split mono. However, a monomorphism need not be left-invertible. For example, in the category Group of all groups and group morphisms among them, if H is a subgroup of G then the inclusion f : H → G is always a monomorphism; but f has a left inverse in the category if and only if H has a normal complement in G. -
Tangent Spaces
CHAPTER 4 Tangent spaces So far, the fact that we dealt with general manifolds M and not just embedded submanifolds of Euclidean spaces did not pose any serious problem: we could make sense of everything right away, by moving (via charts) to opens inside Euclidean spaces. However, when trying to make sense of "tangent vectors", the situation is quite different. Indeed, already intuitively, when trying to draw a "tangent space" to a manifold M, we are tempted to look at M as part of a bigger (Euclidean) space and draw planes that are "tangent" (in the intuitive sense) to M. The fact that the tangent spaces of M are intrinsic to M and not to the way it sits inside a bigger ambient space is remarkable and may seem counterintuitive at first. And it shows that the notion of tangent vector is indeed of a geometric, intrinsic nature. Given our preconception (due to our intuition), the actual general definition may look a bit abstract. For that reason, we describe several different (but equivalent) approaches to tangent m spaces. Each one of them starts with one intuitive perception of tangent vectors on R , and then proceeds by concentrating on the main properties (to be taken as axioms) of the intuitive perception. However, before proceeding, it is useful to clearly state what we expect: • tangent spaces: for M-manifold, p 2 M, have a vector space TpM. And, of course, m~ for embedded submanifolds M ⊂ R , these tangent spaces should be (canonically) isomorphic to the previously defined tangent spaces TpM. • differentials: for smooths map F : M ! N (between manifolds), p 2 M, have an induced linear map (dF )p : TpM ! TF (p)N; m called the differential of F at p. -
Differential Topology from the Point of View of Simple Homotopy Theory
PUBLICATIONS MATHÉMATIQUES DE L’I.H.É.S. BARRY MAZUR Differential topology from the point of view of simple homotopy theory Publications mathématiques de l’I.H.É.S., tome 15 (1963), p. 5-93 <http://www.numdam.org/item?id=PMIHES_1963__15__5_0> © Publications mathématiques de l’I.H.É.S., 1963, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou im- pression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CHAPTER 1 INTRODUCTION It is striking (but not uncharacteristic) that the « first » question asked about higher dimensional geometry is yet unsolved: Is every simply connected 3-manifold homeomorphic with S3 ? (Its original wording is slightly more general than this, and is false: H. Poincare, Analysis Situs (1895).) The difficulty of this problem (in fact of most three-dimensional problems) led mathematicians to veer away from higher dimensional geometric homeomorphism-classificational questions. Except for Whitney's foundational theory of differentiable manifolds and imbed- dings (1936) and Morse's theory of Calculus of Variations in the Large (1934) and, in particular, his analysis of the homology structure of a differentiable manifold by studying critical points of 0°° functions defined on the manifold, there were no classificational results about high dimensional manifolds until the era of Thorn's Cobordisme Theory (1954), the beginning of" modern 9? differential topology. -
Topics in Low Dimensional Computational Topology
THÈSE DE DOCTORAT présentée et soutenue publiquement le 7 juillet 2014 en vue de l’obtention du grade de Docteur de l’École normale supérieure Spécialité : Informatique par ARNAUD DE MESMAY Topics in Low-Dimensional Computational Topology Membres du jury : M. Frédéric CHAZAL (INRIA Saclay – Île de France ) rapporteur M. Éric COLIN DE VERDIÈRE (ENS Paris et CNRS) directeur de thèse M. Jeff ERICKSON (University of Illinois at Urbana-Champaign) rapporteur M. Cyril GAVOILLE (Université de Bordeaux) examinateur M. Pierre PANSU (Université Paris-Sud) examinateur M. Jorge RAMÍREZ-ALFONSÍN (Université Montpellier 2) examinateur Mme Monique TEILLAUD (INRIA Sophia-Antipolis – Méditerranée) examinatrice Autre rapporteur : M. Eric SEDGWICK (DePaul University) Unité mixte de recherche 8548 : Département d’Informatique de l’École normale supérieure École doctorale 386 : Sciences mathématiques de Paris Centre Numéro identifiant de la thèse : 70791 À Monsieur Lagarde, qui m’a donné l’envie d’apprendre. Résumé La topologie, c’est-à-dire l’étude qualitative des formes et des espaces, constitue un domaine classique des mathématiques depuis plus d’un siècle, mais il n’est apparu que récemment que pour de nombreuses applications, il est important de pouvoir calculer in- formatiquement les propriétés topologiques d’un objet. Ce point de vue est la base de la topologie algorithmique, un domaine très actif à l’interface des mathématiques et de l’in- formatique auquel ce travail se rattache. Les trois contributions de cette thèse concernent le développement et l’étude d’algorithmes topologiques pour calculer des décompositions et des déformations d’objets de basse dimension, comme des graphes, des surfaces ou des 3-variétés.