1.2 Smooth Manifolds
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Introduction to Gauge Theory Arxiv:1910.10436V1 [Math.DG] 23
Introduction to Gauge Theory Andriy Haydys 23rd October 2019 This is lecture notes for a course given at the PCMI Summer School “Quantum Field The- ory and Manifold Invariants” (July 1 – July 5, 2019). I describe basics of gauge-theoretic approach to construction of invariants of manifolds. The main example considered here is the Seiberg–Witten gauge theory. However, I tried to present the material in a form, which is suitable for other gauge-theoretic invariants too. Contents 1 Introduction2 2 Bundles and connections4 2.1 Vector bundles . .4 2.1.1 Basic notions . .4 2.1.2 Operations on vector bundles . .5 2.1.3 Sections . .6 2.1.4 Covariant derivatives . .6 2.1.5 The curvature . .8 2.1.6 The gauge group . 10 2.2 Principal bundles . 11 2.2.1 The frame bundle and the structure group . 11 2.2.2 The associated vector bundle . 14 2.2.3 Connections on principal bundles . 16 2.2.4 The curvature of a connection on a principal bundle . 19 arXiv:1910.10436v1 [math.DG] 23 Oct 2019 2.2.5 The gauge group . 21 2.3 The Levi–Civita connection . 22 2.4 Classification of U(1) and SU(2) bundles . 23 2.4.1 Complex line bundles . 24 2.4.2 Quaternionic line bundles . 25 3 The Chern–Weil theory 26 3.1 The Chern–Weil theory . 26 3.1.1 The Chern classes . 28 3.2 The Chern–Simons functional . 30 3.3 The modui space of flat connections . 32 3.3.1 Parallel transport and holonomy . -
Math 600 Day 6: Abstract Smooth Manifolds
Math 600 Day 6: Abstract Smooth Manifolds Ryan Blair University of Pennsylvania Tuesday September 28, 2010 Ryan Blair (U Penn) Math 600 Day 6: Abstract Smooth Manifolds Tuesday September 28, 2010 1 / 21 Outline 1 Transition to abstract smooth manifolds Partitions of unity on differentiable manifolds. Ryan Blair (U Penn) Math 600 Day 6: Abstract Smooth Manifolds Tuesday September 28, 2010 2 / 21 A Word About Last Time Theorem (Sard’s Theorem) The set of critical values of a smooth map always has measure zero in the receiving space. Theorem Let A ⊂ Rn be open and let f : A → Rp be a smooth function whose derivative f ′(x) has maximal rank p whenever f (x) = 0. Then f −1(0) is a (n − p)-dimensional manifold in Rn. Ryan Blair (U Penn) Math 600 Day 6: Abstract Smooth Manifolds Tuesday September 28, 2010 3 / 21 Transition to abstract smooth manifolds Transition to abstract smooth manifolds Up to this point, we have viewed smooth manifolds as subsets of Euclidean spaces, and in that setting have defined: 1 coordinate systems 2 manifolds-with-boundary 3 tangent spaces 4 differentiable maps and stated the Chain Rule and Inverse Function Theorem. Ryan Blair (U Penn) Math 600 Day 6: Abstract Smooth Manifolds Tuesday September 28, 2010 4 / 21 Transition to abstract smooth manifolds Now we want to define differentiable (= smooth) manifolds in an abstract setting, without assuming they are subsets of some Euclidean space. Many differentiable manifolds come to our attention this way. Here are just a few examples: 1 tangent and cotangent bundles over smooth manifolds 2 Grassmann manifolds 3 manifolds built by ”surgery” from other manifolds Ryan Blair (U Penn) Math 600 Day 6: Abstract Smooth Manifolds Tuesday September 28, 2010 5 / 21 Transition to abstract smooth manifolds Our starting point is the definition of a topological manifold of dimension n as a topological space Mn in which each point has an open neighborhood homeomorphic to Rn (locally Euclidean property), and which, in addition, is Hausdorff and second countable. -
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. -
Derived Smooth Manifolds
DERIVED SMOOTH MANIFOLDS DAVID I. SPIVAK Abstract. We define a simplicial category called the category of derived man- ifolds. It contains the category of smooth manifolds as a full discrete subcat- egory, and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local C1-rings that is obtained by patching together homotopy zero-sets of smooth functions on Eu- clidean spaces. We show that derived manifolds come equipped with a stable normal bun- dle and can be imbedded into Euclidean space. We define a cohomology theory called derived cobordism, and use a Pontrjagin-Thom argument to show that the derived cobordism theory is isomorphic to the classical cobordism theory. This allows us to define fundamental classes in cobordism for all derived man- ifolds. In particular, the intersection A \ B of submanifolds A; B ⊂ X exists on the categorical level in our theory, and a cup product formula [A] ^ [B] = [A \ B] holds, even if the submanifolds are not transverse. One can thus consider the theory of derived manifolds as a categorification of intersection theory. Contents 1. Introduction 1 2. The axioms 8 3. Main results 14 4. Layout for the construction of dMan 21 5. C1-rings 23 6. Local C1-ringed spaces and derived manifolds 26 7. Cotangent Complexes 32 8. Proofs of technical results 39 9. Derived manifolds are good for doing intersection theory 50 10. Relationship to similar work 52 References 55 1. Introduction Let Ω denote the unoriented cobordism ring (though what we will say applies to other cobordism theories as well, e.g. -
MAT 531: Topology&Geometry, II Spring 2011
MAT 531: Topology&Geometry, II Spring 2011 Solutions to Problem Set 1 Problem 1: Chapter 1, #2 (10pts) Let F be the (standard) differentiable structure on R generated by the one-element collection of ′ charts F0 = {(R, id)}. Let F be the differentiable structure on R generated by the one-element collection of charts ′ 3 F0 ={(R,f)}, where f : R−→R, f(t)= t . Show that F 6=F ′, but the smooth manifolds (R, F) and (R, F ′) are diffeomorphic. ′ ′ (a) We begin by showing that F 6=F . Since id∈F0 ⊂F, it is sufficient to show that id6∈F , i.e. the overlap map id ◦ f −1 : f(R∩R)=R −→ id(R∩R)=R ′ from f ∈F0 to id is not smooth, in the usual (i.e. calculus) sense: id ◦ f −1 R R f id R ∩ R Since f(t)=t3, f −1(s)=s1/3, and id ◦ f −1 : R −→ R, id ◦ f −1(s)= s1/3. This is not a smooth map. (b) Let h : R −→ R be given by h(t)= t1/3. It is immediate that h is a homeomorphism. We will show that the map h:(R, F) −→ (R, F ′) is a diffeomorphism, i.e. the maps h:(R, F) −→ (R, F ′) and h−1 :(R, F ′) −→ (R, F) are smooth. To show that h is smooth, we need to show that it induces smooth maps between the ′ charts in F0 and F0. In this case, there is only one chart in each. So we need to show that the map f ◦ h ◦ id−1 : id h−1(R)∩R =R −→ R is smooth: −1 f ◦ h ◦ id R R id f h (R, F) (R, F ′) Since f ◦ h ◦ id−1(t)= f h(t) = f(t1/3)= t1/3 3 = t, − this map is indeed smooth, and so is h. -
Categorical Differential Geometry Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 35, No 4 (1994), P
CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES M. V. LOSIK Categorical differential geometry Cahiers de topologie et géométrie différentielle catégoriques, tome 35, no 4 (1994), p. 274-290 <http://www.numdam.org/item?id=CTGDC_1994__35_4_274_0> © Andrée C. Ehresmann et les auteurs, 1994, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression 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/ CAHIERS DE TOPOLOGIE ET VolumeXXXV-4 (1994) GEOMETRIE DIFFERENTIELLE CATEGORIQUES CATEGORICAL DIFFERENTIAL GEOMETRY by M.V. LOSIK R6sum6. Cet article d6veloppe une th6orie g6n6rale des structures g6om6triques sur des vari6t6s, bas6e sur la tlieorie des categories. De nombreuses generalisations connues des vari6t6s et des vari6t6s de Rie- mann r.entrent dans le cadre de cette th6orie g6n6rale. On donne aussi une construction des classes caractéristiques des objets ainsi obtenus, tant classiques que generalises. Diverses applications sont indiqu6es, en parti- culier aux feuilletages. Introduction This paper is the result of an attempt to give a precise meaning to some ideas of the "formal differential geometry" of Gel’fand. These ideas of Gel’fand have not been formalized in general but they may be explained by the following example. Let C*(Wn; IR) be the complex of continuous cochains of the Lie algebra Wn of formal vector fields on IRn with coefficients in the trivial Wn-module IR and let C*(Wn,GL(n, R)), C*(Wn, O(n)) be its subcomplexes of relative cochains of Wn relative to the groups GL(n, R), 0(n), respectively. -
On Colimits in Various Categories of Manifolds
ON COLIMITS IN VARIOUS CATEGORIES OF MANIFOLDS DAVID WHITE 1. Introduction Homotopy theorists have traditionally studied topological spaces and used methods which rely heavily on computation, construction, and induction. Proofs often go by constructing some horrendously complicated object (usually via a tower of increasingly complicated objects) and then proving inductively that we can understand what’s going on at each step and that in the limit these steps do what is required. For this reason, a homotopy theorist need the freedom to make any construction he wishes, e.g. products, subobjects, quotients, gluing towers of spaces together, dividing out by group actions, moving to fixed points of group actions, etc. Over the past 50 years tools have been developed which allow the study of many different categories. The key tool for this is the notion of a model category, which is the most general place one can do homotopy theory. Model categories were introduced by the Fields Medalist Dan Quillen, and the best known reference for model categories is the book Model Categories by Mark Hovey. Model categories are defined by a list of axioms which are all analogous to properties that the category Top of topological spaces satisfies. The most important property of a model category is that it has three distinguished classes of morphisms–called weak equivalences, cofibrations, and fibrations–which follow rules similar to the classes of maps with those names in the category Top (so the homotopy theory comes into play by inverting the weak equivalences to make them isomorphisms). The fact that homotopy theorists need to be able to do lots of constructions is contained in the axiom that M must be complete and cocomplete. -
Basics of Differential Geometry
Cambridge University Press 978-0-521-86891-4 - Lectures on Kahler Geometry Andrei Moroianu Excerpt More information Part 1 Basics of differential geometry © Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-86891-4 - Lectures on Kahler Geometry Andrei Moroianu Excerpt More information CHAPTER 1 Smooth manifolds 1.1. Introduction A topological manifold of dimension n is a Hausdorff topological space M which locally “looks like” the space Rn. More precisely, M has an open covering U such that for every U ∈Uthere exists a homeomorphism φU : U → U˜ ⊂ Rn, called a local chart. The collection of all charts is called an atlas of M. Since every open ball in Rn is homeomorphic to Rn itself, the definition above amounts to saying that every point of M has a neighbourhood homeomorphic to Rn. Examples. 1. The sphere Sn ⊂ Rn+1 is a topological manifold of di- mension n, with the atlas consisting of the two stereographic projections n n n n φN : S \{S}→R and φS : S \{N}→R where N and S are the North and South poles of Sn. 2. The union Ox∪Oy of the two coordinate lines in R2 is not a topological manifold. Indeed, for every neighbourhood U of the origin in Ox ∪ Oy,the set U \{0} has 4 connected components, so U can not be homeomorphic to R. We now investigate the possibility of defining smooth functions on a given topological manifold M.Iff : M → R is a continuous function, one is tempted to define f to be smooth if for every x ∈ M there exists U ∈U containing x such that the composition −1 ˜ n fU := f ◦ φU : U ⊂ R → R is smooth. -
DIFFERENTIABLE STRUCTURES 1. Classification of Manifolds 1.1
DIFFERENTIABLE STRUCTURES 1. Classification of Manifolds 1.1. Geometric topology. The goals of this topic are (1) Classify all spaces - ludicrously impossible. (2) Classify manifolds - only provably impossible (because any finitely presented group is the fundamental group of a 4-manifold, and finitely presented groups are not able to be classified). (3) Classify manifolds with a given homotopy type (sort of worked out in 1960's by Wall and Browder if π1 = 0. There is something called the Browder-Novikov-Sullivan-Wall surgery (structure) sequence: ! L (π1 (X)) !S0 (X) ! [X; GO] LHS=algebra, RHS=homotopy theory, S0 (X) is the set of smooth manifolds h.e. to X up to some equivalence relation. Classify means either (1) up to homeomorphism (2) up to PL homeomorphism (3) up to diffeomorphism A manifold is smooth iff the tangent bundle is smooth. These are classified by the orthog- onal group O. There are analogues of this for the other types (T OP, PL). Given a space X, then homotopy classes of maps [X; T OPPL] determine if X is PL-izable. It turns out T OPPL = a K (Z2; 3). It is a harder question to look at if X has a smooth structure. 1.2. Classifying homotopy spheres. X is a homotopy sphere if it is a topological manifold n that is h.e. to a sphere. It turns out that if π1 (X) = 0 and H∗ (X) = H∗ (S ), then it is a homotopy sphere. The Poincare Conjecture is that every homotopy n-sphere is Sn. We can ask this in several categories : Diff, PL, Top: In dimensions 0, 1, all is true. -
Suppose N Is a Topological Manifold with Smooth Structure U. Suppose M Is a Topological Manifold and F Is a Homeomorphism M → N
Suppose n is a topological manifold with smooth structure U. Suppose M is a topological manifold and F is a homeomorphism M → N. We can use F to “pull back” the smooth structure U from N to M as follows: for each chart (U, φ) in U we have a chart (F −1(U), φ◦F ) on M. These charts form a maximal smooth atlas F ∗(U) for M, that is a smooth structure on M. Almost by definition, the map F : M → N is a diffeomorphism from (M, F ∗(U)) to (N, U). Now consider the case M = N. If F is the identity map, then F ∗(U) = U. More generally, this is true any time F is a diffeormorphism with respect to the smooth structure U (that is, a diffeomorphism from the smooth manifold (M, U) to itself). But suppose F is a homeomorphism which is not a diffeomorphism. Then (N, F ∗(U)) = (M, F ∗(U)) is a distinct differentible structure on the same differentiable manifold M. For ∗ instance, a real-valued function g : M → R is smooth w.r.t. U iff g ◦ F is smooth w.r.t. F (U). But, as mentioned above, the map F is a diffeomorphism as a map (M, F ∗(U)) → (M, U). In particular, the two smooth manifolds are really the same (the are diffeomorphic to each other). Thus, this simple construction never produces “exotic” smooth structures (two different smooth manifolds with the same underlying topological manifold). Facts: (1) Up to diffeomorphism, there is a unique smooth structure on any topological manifold n n M in dimension n ≤ 3. -
Differential Topology Forty-Six Years Later
Differential Topology Forty-six Years Later John Milnor n the 1965 Hedrick Lectures,1 I described its uniqueness up to a PL-isomorphism that is the state of differential topology, a field that topologically isotopic to the identity. In particular, was then young but growing very rapidly. they proved the following. During the intervening years, many problems Theorem 1. If a topological manifold Mn without in differential and geometric topology that boundary satisfies Ihad seemed totally impossible have been solved, 3 n 4 n often using drastically new tools. The following is H (M ; Z/2) = H (M ; Z/2) = 0 with n ≥ 5 , a brief survey, describing some of the highlights then it possesses a PL-manifold structure that is of these many developments. unique up to PL-isomorphism. Major Developments (For manifolds with boundary one needs n > 5.) The corresponding theorem for all manifolds of The first big breakthrough, by Kirby and Sieben- dimension n ≤ 3 had been proved much earlier mann [1969, 1969a, 1977], was an obstruction by Moise [1952]. However, we will see that the theory for the problem of triangulating a given corresponding statement in dimension 4 is false. topological manifold as a PL (= piecewise-linear) An analogous obstruction theory for the prob- manifold. (This was a sharpening of earlier work lem of passing from a PL-structure to a smooth by Casson and Sullivan and by Lashof and Rothen- structure had previously been introduced by berg. See [Ranicki, 1996].) If B and B are the Top PL Munkres [1960, 1964a, 1964b] and Hirsch [1963]. -
Problems in Low-Dimensional Topology
Problems in Low-Dimensional Topology Edited by Rob Kirby Berkeley - 22 Dec 95 Contents 1 Knot Theory 7 2 Surfaces 85 3 3-Manifolds 97 4 4-Manifolds 179 5 Miscellany 259 Index of Conjectures 282 Index 284 Old Problem Lists 294 Bibliography 301 1 2 CONTENTS Introduction In April, 1977 when my first problem list [38,Kirby,1978] was finished, a good topologist could reasonably hope to understand the main topics in all of low dimensional topology. But at that time Bill Thurston was already starting to greatly influence the study of 2- and 3-manifolds through the introduction of geometry, especially hyperbolic. Four years later in September, 1981, Mike Freedman turned a subject, topological 4-manifolds, in which we expected no progress for years, into a subject in which it seemed we knew everything. A few months later in spring 1982, Simon Donaldson brought gauge theory to 4-manifolds with the first of a remarkable string of theorems showing that smooth 4-manifolds which might not exist or might not be diffeomorphic, in fact, didn’t and weren’t. Exotic R4’s, the strangest of smooth manifolds, followed. And then in late spring 1984, Vaughan Jones brought us the Jones polynomial and later Witten a host of other topological quantum field theories (TQFT’s). Physics has had for at least two decades a remarkable record for guiding mathematicians to remarkable mathematics (Seiberg–Witten gauge theory, new in October, 1994, is the latest example). Lest one think that progress was only made using non-topological techniques, note that Freedman’s work, and other results like knot complements determining knots (Gordon- Luecke) or the Seifert fibered space conjecture (Mess, Scott, Gabai, Casson & Jungreis) were all or mostly classical topology.