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Foliations of Three Dimensional Manifolds∗
Foliations of Three Dimensional Manifolds∗ M. H. Vartanian December 17, 2007 Abstract The theory of foliations began with a question by H. Hopf in the 1930's: \Does there exist on S3 a completely integrable vector field?” By Frobenius' Theorem, this is equivalent to: \Does there exist on S3 a codimension one foliation?" A decade later, G. Reeb answered affirmatively by exhibiting a C1 foliation of S3 consisting of a single compact leaf homeomorphic to the 2-torus with the other leaves homeomorphic to R2 accumulating asymptotically on the compact leaf. Reeb's work raised the basic question: \Does every C1 codimension one foliation of S3 have a compact leaf?" This was answered by S. Novikov with a much stronger statement, one of the deepest results of foliation theory: Every C2 codimension one foliation of a compact 3-dimensional manifold with finite fundamental group has a compact leaf. The basic ideas leading to Novikov's Theorem are surveyed here.1 1 Introduction Intuitively, a foliation is a partition of a manifold M into submanifolds A of the same dimension that stack up locally like the pages of a book. Perhaps the simplest 3 nontrivial example is the foliation of R nfOg by concentric spheres induced by the 3 submersion R nfOg 3 x 7! jjxjj 2 R. Abstracting the essential aspects of this example motivates the following general ∗Survey paper written for S.N. Simic's Math 213, \Advanced Differential Geometry", San Jos´eState University, Fall 2007. 1We follow here the general outline presented in [Ca]. For a more recent and much broader survey of foliation theory, see [To]. -
General Topology
General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry). -
1.1.3 Reminder of Some Simple Topological Concepts Definition 1.1.17
1. Preliminaries The Hausdorffcriterion could be paraphrased by saying that smaller neigh- borhoods make larger topologies. This is a very intuitive theorem, because the smaller the neighbourhoods are the easier it is for a set to contain neigh- bourhoods of all its points and so the more open sets there will be. Proof. Suppose τ τ . Fixed any point x X,letU (x). Then, since U ⇒ ⊆ ∈ ∈B is a neighbourhood of x in (X,τ), there exists O τ s.t. x O U.But ∈ ∈ ⊆ O τ implies by our assumption that O τ ,soU is also a neighbourhood ∈ ∈ of x in (X,τ ). Hence, by Definition 1.1.10 for (x), there exists V (x) B ∈B s.t. V U. ⊆ Conversely, let W τ. Then for each x W ,since (x) is a base of ⇐ ∈ ∈ B neighbourhoods w.r.t. τ,thereexistsU (x) such that x U W . Hence, ∈B ∈ ⊆ by assumption, there exists V (x)s.t.x V U W .ThenW τ . ∈B ∈ ⊆ ⊆ ∈ 1.1.3 Reminder of some simple topological concepts Definition 1.1.17. Given a topological space (X,τ) and a subset S of X,the subset or induced topology on S is defined by τ := S U U τ . That is, S { ∩ | ∈ } a subset of S is open in the subset topology if and only if it is the intersection of S with an open set in (X,τ). Alternatively, we can define the subspace topology for a subset S of X as the coarsest topology for which the inclusion map ι : S X is continuous. -
Topology Via Higher-Order Intuitionistic Logic
Topology via higher-order intuitionistic logic Working version of 18th March 2004 These evolving notes will eventually be used to write a paper Mart´ınEscard´o Abstract When excluded middle fails, one can define a non-trivial topology on the one-point set, provided one doesn’t require all unions of open sets to be open. Technically, one obtains a subset of the subobject classifier, known as a dom- inance, which is to be thought of as a “Sierpinski set” that behaves as the Sierpinski space in classical topology. This induces topologies on all sets, rendering all functions continuous. Because virtually all theorems of classical topology require excluded middle (and even choice), it would be useless to reduce other topological notions to the notion of open set in the usual way. So, for example, our synthetic definition of compactness for a set X says that, for any Sierpinski-valued predicate p on X, the truth value of the statement “for all x ∈ X, p(x)” lives in the Sierpinski set. Similarly, other topological notions are defined by logical statements. We show that the proposed synthetic notions interact in the expected way. Moreover, we show that they coincide with the usual notions in cer- tain classical topological models, and we look at their interpretation in some computational models. 1 Introduction For suitable topologies, computable functions are continuous, and semidecidable properties of their inputs/outputs are open, but the converses of these two state- ments fail. Moreover, although semidecidable sets are closed under the formation of finite intersections and recursively enumerable unions, they fail to form the open sets of a topology in a literal sense. -
Math 4853 Homework 51. Let X Be a Set with the Cofinite Topology. Prove
Math 4853 homework 51. Let X be a set with the cofinite topology. Prove that every subspace of X has the cofinite topology (i. e. the subspace topology on each subset A equals the cofinite topology on A). Notice that this says that every subset of X is compact. So not every compact subset of X is closed (unless X is finite, in which case X and all of its subsets are finite sets with the discrete topology). Let U be a nonempty open subset of A for the subspace topology. Then U = V \ A for some open subset V of X. Now V = X − F for some finite subset F . So V \ A = (X − F ) \ A = (A \ X) − (A \ F ) = A − (A \ F ). Since A \ F is finite, this is open in the cofinite topology on A. Conversely, suppose that U is an open subset for the cofinite topology of A. Then U = A − F1 for some finite subset F1 of A. Then, X − F1 is open in X, and (X − F1) \ A = A − F1, so A − F1 is open in the subspace topology. 52. Let X be a Hausdorff space. Prove that every compact subset A of X is closed. Hint: Let x2 = A. For each a 2 A, choose disjoint open subsets Ua and Va of X such that x 2 Ua and a 2 Va. The collection fVa \ Ag is an open cover of A, so has a n n finite subcover fVai \ Agi=1. Now, let U = \i=1Uai , a neighborhood of x. Prove that U ⊂ X − A (draw a picture!), which proves that X − A is open. -
Basis (Topology), Basis (Vector Space), Ck, , Change of Variables (Integration), Closed, Closed Form, Closure, Coboundary, Cobou
Index basis (topology), Ôç injective, ç basis (vector space), ç interior, Ôò isomorphic, ¥ ∞ C , Ôý, ÔÔ isomorphism, ç Ck, Ôý, ÔÔ change of variables (integration), ÔÔ kernel, ç closed, Ôò closed form, Þ linear, ç closure, Ôò linear combination, ç coboundary, Þ linearly independent, ç coboundary map, Þ nullspace, ç cochain complex, Þ cochain homotopic, open cover, Ôò cochain homotopy, open set, Ôý cochain map, Þ cocycle, Þ partial derivative, Ôý cohomology, Þ compact, Ôò quotient space, â component function, À range, ç continuous, Ôç rank-nullity theorem, coordinate function, Ôý relative topology, Ôò dierential, Þ resolution, ¥ dimension, ç resolution of the identity, À direct sum, second-countable, Ôç directional derivative, Ôý short exact sequence, discrete topology, Ôò smooth homotopy, dual basis, À span, ç dual space, À standard topology on Rn, Ôò exact form, Þ subspace, ç exact sequence, ¥ subspace topology, Ôò support, Ô¥ Hausdor, Ôç surjective, ç homotopic, symmetry of partial derivatives, ÔÔ homotopy, topological space, Ôò image, ç topology, Ôò induced topology, Ôò total derivative, ÔÔ Ô trivial topology, Ôò vector space, ç well-dened, â ò Glossary Linear Algebra Denition. A real vector space V is a set equipped with an addition operation , and a scalar (R) multiplication operation satisfying the usual identities. + Denition. A subset W ⋅ V is a subspace if W is a vector space under the same operations as V. Denition. A linear combination⊂ of a subset S V is a sum of the form a v, where each a , and only nitely many of the a are nonzero. v v R ⊂ v v∈S v∈S ⋅ ∈ { } Denition.Qe span of a subset S V is the subspace span S of linear combinations of S. -
Section 11.3. Countability and Separability
11.3. Countability and Separability 1 Section 11.3. Countability and Separability Note. In this brief section, we introduce sequences and their limits. This is applied to topological spaces with certain types of bases. Definition. A sequence {xn} in a topological space X converges to a point x ∈ X provided for each neighborhood U of x, there is an index N such that n ≥ N, then xn belongs to U. The point x is a limit of the sequence. Note. We can now elaborate on some of the results you see early in a senior level real analysis class. You prove, for example, that the limit of a sequence of real numbers (under the usual topology) is unique. In a topological space, this may not be the case. For example, under the trivial topology every sequence converges to every point (at the other extreme, under the discrete topology, the only convergent sequences are those which are eventually constant). For an ad- ditional example, see Bonus 1 in my notes for Analysis 1 (MATH 4217/5217) at: http://faculty.etsu.edu/gardnerr/4217/notes/3-1.pdf. In a Hausdorff space (as in a metric space), points can be separated and limits of sequences are unique. Definition. A topological space (X, T ) is first countable provided there is a count- able base at each point. The space (X, T ) is second countable provided there is a countable base for the topology. 11.3. Countability and Separability 2 Example. Every metric space (X, ρ) is first countable since for all x ∈ X, the ∞ countable collection of open balls {B(x, 1/n)}n=1 is a base at x for the topology induced by the metric. -
Reviewworksheetsoln.Pdf
Math 351 - Elementary Topology Monday, October 8 Exam 1 Review Problems ⇤⇤ 1. Say U R is open if either it is finite or U = R. Why is this not a topology? ✓ 2. Let X = a, b . { } (a) If X is equipped with the trivial topology, which functions f : X R are continu- ! ous? What about functions g : R X? ! (b) If X is equipped with the topology ∆, a , X , which functions f : X R are { { } } ! continuous? What about functions g : R X? ! (c) If X is equipped with the discrete topology, which functions f : X R are contin- ! uous? What about functions g : R X? ! 3. Given an example of a topology on R (one we have discussed) that is not Hausdorff. 4. Show that if A X, then ∂A = ∆ if and only if A is both open and closed in X. ⇢ 5. Give an example of a space X and an open subset A such that Int(A) = A. 6 6. Let A X be a subspace. Show that C A is closed if and only if C = D A for some ⇢ ⇢ \ closed subset D X. ⇢ 7. Show that the addition function f : R2 R, defined by f (x, y)=x + y, is continuous. ! 8. Give an example of a function f : R R which is continuous only at 0 (in the usual ! topology). Hint: Define f piecewise, using the rationals and irrationals as the two pieces. Solutions. 1. This fails the union axiom. Any singleton set would be open. The set N is a union of single- ton sets and so should also be open, but it is not. -
General Topology
General Topology Andrew Kobin Fall 2013 | Spring 2014 Contents Contents Contents 0 Introduction 1 1 Topological Spaces 3 1.1 Topology . .3 1.2 Basis . .5 1.3 The Continuum Hypothesis . .8 1.4 Closed Sets . 10 1.5 The Separation Axioms . 11 1.6 Interior and Closure . 12 1.7 Limit Points . 15 1.8 Sequences . 16 1.9 Boundaries . 18 1.10 Applications to GIS . 20 2 Creating New Topological Spaces 22 2.1 The Subspace Topology . 22 2.2 The Product Topology . 25 2.3 The Quotient Topology . 27 2.4 Configuration Spaces . 31 3 Topological Equivalence 34 3.1 Continuity . 34 3.2 Homeomorphisms . 38 4 Metric Spaces 43 4.1 Metric Spaces . 43 4.2 Error-Checking Codes . 45 4.3 Properties of Metric Spaces . 47 4.4 Metrizability . 50 5 Connectedness 51 5.1 Connected Sets . 51 5.2 Applications of Connectedness . 56 5.3 Path Connectedness . 59 6 Compactness 63 6.1 Compact Sets . 63 6.2 Results in Analysis . 66 7 Manifolds 69 7.1 Topological Manifolds . 69 7.2 Classification of Surfaces . 70 7.3 Euler Characteristic and Proof of the Classification Theorem . 74 i Contents Contents 8 Homotopy Theory 80 8.1 Homotopy . 80 8.2 The Fundamental Group . 81 8.3 The Fundamental Group of the Circle . 88 8.4 The Seifert-van Kampen Theorem . 92 8.5 The Fundamental Group and Knots . 95 8.6 Covering Spaces . 97 9 Surfaces Revisited 101 9.1 Surfaces With Boundary . 101 9.2 Euler Characteristic Revisited . 104 9.3 Constructing Surfaces and Manifolds of Higher Dimension . -
Chapter 3. Topology of the Real Numbers. 3.1
3.1. Topology of the Real Numbers 1 Chapter 3. Topology of the Real Numbers. 3.1. Topology of the Real Numbers. Note. In this section we “topological” properties of sets of real numbers such as open, closed, and compact. In particular, we will classify open sets of real numbers in terms of open intervals. Definition. A set U of real numbers is said to be open if for all x ∈ U there exists δ(x) > 0 such that (x − δ(x), x + δ(x)) ⊂ U. Note. It is trivial that R is open. It is vacuously true that ∅ is open. Theorem 3-1. The intervals (a,b), (a, ∞), and (−∞,a) are open sets. (Notice that the choice of δ depends on the value of x in showing that these are open.) Definition. A set A is closed if Ac is open. Note. The sets R and ∅ are both closed. Some sets are neither open nor closed. Corollary 3-1. The intervals (−∞,a], [a,b], and [b, ∞) are closed sets. 3.1. Topology of the Real Numbers 2 Theorem 3-2. The open sets satisfy: n (a) If {U1,U2,...,Un} is a finite collection of open sets, then ∩k=1Uk is an open set. (b) If {Uα} is any collection (finite, infinite, countable, or uncountable) of open sets, then ∪αUα is an open set. Note. An infinite intersection of open sets can be closed. Consider, for example, ∞ ∩i=1(−1/i, 1 + 1/i) = [0, 1]. Theorem 3-3. The closed sets satisfy: (a) ∅ and R are closed. (b) If {Aα} is any collection of closed sets, then ∩αAα is closed. -
Let X Be a Topological Space and Let Y Be a Subset of X
Homework 2 Solutions I.19: Let X be a topological space and let Y be a subset of X . Check that the so-called subspace topology is indeed a topology on Y . In the subspace topology, a subset B of Y is open precisely if it can be written as B = A∩Y for some set A open in X . To check that this yields a topology on Y , we check that it satisfies the axioms. (i) ∅ an Y are open in Y : To verify this we write ∅ = ∅∩Y and Y = X ∩ Y ; the claim then follows because ∅ and X are open in X . (ii) Arbitrary unions of sets open in Y are open in Y : For this, suppose we have an arbitrary collection Bλ of open sets in Y . Then for each Bλ we may write Bλ = Aλ ∩ Y where Aλ is some open set in A. We know that A = S Aλ is open in X . Then (using deMorgan’s laws) we have [ Bλ = [(Aλ ∩ Y )=([ Aλ) ∩ Y = A ∩ Y. (iii) Finite intersections of sets open in Y are open in Y : Given a collection of open sets B1,...,Bn in Y , we write Bj = Aj ∩ Y for Aj open in X . We know that A = T Aj is open in X . Then \ Bj = \(Aj ∩ Y )=(\ Aj) ∩ Y = A ∩ Y. II.1: Verify each of the following for arbitrary subsets A and B of a space X : (a) A ∪ B = A ∪ B . To prove equality, we show containment in both directions. To show that A ∪ B ⊆ A ∪ B , we choose a point x ∈ A ∪ B . -
Math 131: Introduction to Topology 1
Math 131: Introduction to Topology 1 Professor Denis Auroux Fall, 2019 Contents 9/4/2019 - Introduction, Metric Spaces, Basic Notions3 9/9/2019 - Topological Spaces, Bases9 9/11/2019 - Subspaces, Products, Continuity 15 9/16/2019 - Continuity, Homeomorphisms, Limit Points 21 9/18/2019 - Sequences, Limits, Products 26 9/23/2019 - More Product Topologies, Connectedness 32 9/25/2019 - Connectedness, Path Connectedness 37 9/30/2019 - Compactness 42 10/2/2019 - Compactness, Uncountability, Metric Spaces 45 10/7/2019 - Compactness, Limit Points, Sequences 49 10/9/2019 - Compactifications and Local Compactness 53 10/16/2019 - Countability, Separability, and Normal Spaces 57 10/21/2019 - Urysohn's Lemma and the Metrization Theorem 61 1 Please email Beckham Myers at [email protected] with any corrections, questions, or comments. Any mistakes or errors are mine. 10/23/2019 - Category Theory, Paths, Homotopy 64 10/28/2019 - The Fundamental Group(oid) 70 10/30/2019 - Covering Spaces, Path Lifting 75 11/4/2019 - Fundamental Group of the Circle, Quotients and Gluing 80 11/6/2019 - The Brouwer Fixed Point Theorem 85 11/11/2019 - Antipodes and the Borsuk-Ulam Theorem 88 11/13/2019 - Deformation Retracts and Homotopy Equivalence 91 11/18/2019 - Computing the Fundamental Group 95 11/20/2019 - Equivalence of Covering Spaces and the Universal Cover 99 11/25/2019 - Universal Covering Spaces, Free Groups 104 12/2/2019 - Seifert-Van Kampen Theorem, Final Examples 109 2 9/4/2019 - Introduction, Metric Spaces, Basic Notions The instructor for this course is Professor Denis Auroux. His email is [email protected] and his office is SC539.