Foliations and Floer Theories
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Professor Smith Math 295 Lecture Notes
Professor Smith Math 295 Lecture Notes by John Holler Fall 2010 1 October 29: Compactness, Open Covers, and Subcovers. 1.1 Reexamining Wednesday's proof There seemed to be a bit of confusion about our proof of the generalized Intermediate Value Theorem which is: Theorem 1: If f : X ! Y is continuous, and S ⊆ X is a connected subset (con- nected under the subspace topology), then f(S) is connected. We'll approach this problem by first proving a special case of it: Theorem 1: If f : X ! Y is continuous and surjective and X is connected, then Y is connected. Proof of 2: Suppose not. Then Y is not connected. This by definition means 9 non-empty open sets A; B ⊂ Y such that A S B = Y and A T B = ;. Since f is continuous, both f −1(A) and f −1(B) are open. Since f is surjective, both f −1(A) and f −1(B) are non-empty because A and B are non-empty. But the surjectivity of f also means f −1(A) S f −1(B) = X, since A S B = Y . Additionally we have f −1(A) T f −1(B) = ;. For if not, then we would have that 9x 2 f −1(A) T f −1(B) which implies f(x) 2 A T B = ; which is a contradiction. But then these two non-empty open sets f −1(A) and f −1(B) disconnect X. QED Now we want to show that Theorem 2 implies Theorem 1. This will require the help from a few lemmas, whose proofs are on Homework Set 7: Lemma 1: If f : X ! Y is continuous, and S ⊆ X is any subset of X then the restriction fjS : S ! Y; x 7! f(x) is also continuous. -
Ricci Curvature and Minimal Submanifolds
Pacific Journal of Mathematics RICCI CURVATURE AND MINIMAL SUBMANIFOLDS Thomas Hasanis and Theodoros Vlachos Volume 197 No. 1 January 2001 PACIFIC JOURNAL OF MATHEMATICS Vol. 197, No. 1, 2001 RICCI CURVATURE AND MINIMAL SUBMANIFOLDS Thomas Hasanis and Theodoros Vlachos The aim of this paper is to find necessary conditions for a given complete Riemannian manifold to be realizable as a minimal submanifold of a unit sphere. 1. Introduction. The general question that served as the starting point for this paper was to find necessary conditions on those Riemannian metrics that arise as the induced metrics on minimal hypersurfaces or submanifolds of hyperspheres of a Euclidean space. There is an abundance of complete minimal hypersurfaces in the unit hy- persphere Sn+1. We recall some well known examples. Let Sm(r) = {x ∈ Rn+1, |x| = r},Sn−m(s) = {y ∈ Rn−m+1, |y| = s}, where r and s are posi- tive numbers with r2 + s2 = 1; then Sm(r) × Sn−m(s) = {(x, y) ∈ Rn+2, x ∈ Sm(r), y ∈ Sn−m(s)} is a hypersurface of the unit hypersphere in Rn+2. As is well known, this hypersurface has two distinct constant principal cur- vatures: One is s/r of multiplicity m, the other is −r/s of multiplicity n − m. This hypersurface is called a Clifford hypersurface. Moreover, it is minimal only in the case r = pm/n, s = p(n − m)/n and is called a Clifford minimal hypersurface. Otsuki [11] proved that if M n is a com- pact minimal hypersurface in Sn+1 with two distinct principal curvatures of multiplicity greater than 1, then M n is a Clifford minimal hypersur- face Sm(pm/n) × Sn−m(p(n − m)/n), 1 < m < n − 1. -
The Boundary Manifold of a Complex Line Arrangement
Geometry & Topology Monographs 13 (2008) 105–146 105 The boundary manifold of a complex line arrangement DANIEL CCOHEN ALEXANDER ISUCIU We study the topology of the boundary manifold of a line arrangement in CP2 , with emphasis on the fundamental group G and associated invariants. We determine the Alexander polynomial .G/, and more generally, the twisted Alexander polynomial associated to the abelianization of G and an arbitrary complex representation. We give an explicit description of the unit ball in the Alexander norm, and use it to analyze certain Bieri–Neumann–Strebel invariants of G . From the Alexander polynomial, we also obtain a complete description of the first characteristic variety of G . Comparing this with the corresponding resonance variety of the cohomology ring of G enables us to characterize those arrangements for which the boundary manifold is formal. 32S22; 57M27 For Fred Cohen on the occasion of his sixtieth birthday 1 Introduction 1.1 The boundary manifold Let A be an arrangement of hyperplanes in the complex projective space CPm , m > 1. Denote by V S H the corresponding hypersurface, and by X CPm V its D H A D n complement. Among2 the origins of the topological study of arrangements are seminal results of Arnol’d[2] and Cohen[8], who independently computed the cohomology of the configuration space of n ordered points in C, the complement of the braid arrangement. The cohomology ring of the complement of an arbitrary arrangement A is by now well known. It is isomorphic to the Orlik–Solomon algebra of A, see Orlik and Terao[34] as a general reference. -
TRANSVERSE STRUCTURE of LIE FOLIATIONS 0. Introduction This
TRANSVERSE STRUCTURE OF LIE FOLIATIONS BLAS HERRERA, MIQUEL LLABRES´ AND A. REVENTOS´ Abstract. We study if two non isomorphic Lie algebras can appear as trans- verse algebras to the same Lie foliation. In particular we prove a quite sur- prising result: every Lie G-flow of codimension 3 on a compact manifold, with basic dimension 1, is transversely modeled on one, two or countable many Lie algebras. We also solve an open question stated in [GR91] about the realiza- tion of the three dimensional Lie algebras as transverse algebras to Lie flows on a compact manifold. 0. Introduction This paper deals with the problem of the realization of a given Lie algebra as transverse algebra to a Lie foliation on a compact manifold. Lie foliations have been studied by several authors (cf. [KAH86], [KAN91], [Fed71], [Mas], [Ton88]). The importance of this study was increased by the fact that they arise naturally in Molino’s classification of Riemannian foliations (cf. [Mol82]). To each Lie foliation are associated two Lie algebras, the Lie algebra G of the Lie group on which the foliation is modeled and the structural Lie algebra H. The latter algebra is the Lie algebra of the Lie foliation F restricted to the clousure of any one of its leaves. In particular, it is a subalgebra of G. We remark that although H is canonically associated to F, G is not. Thus two interesting problems are naturally posed: the realization problem and the change problem. The realization problem is to know which pairs of Lie algebras (G, H), with H subalgebra of G, can arise as transverse and structural Lie algebras, respectively, of a Lie foliation F on a compact manifold M. -
Foliations Transverse to Fibers of a Bundle
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 42, Number 2, February 1974 FOLIATIONS TRANSVERSE TO FIBERS OF A BUNDLE J. F. PLANTE Abstract. Consider a fiber bundle where the base space and total space are compact, connected, oriented smooth manifolds and the projection map is smooth. It is shown that if the fiber is null-homologous in the total space, then the existence of a foliation of the total space which is transverse to each fiber and such that each leaf has the same dimension as the base implies that the fundamental group of the base space has exponential growth. Introduction. Let (E, p, B) be a locally trivial fiber bundle where /»:£-*-/? is the projection, E and B are compact, connected, oriented smooth manifolds and p is a smooth map. (By smooth we mean C for some r^l and, henceforth, all maps are assumed smooth.) Let b denote the dimension of B and k denote the dimension of the fiber (thus, dim E= b+k). By a section of the fibration we mean a smooth map a:B-+E such that p o a=idB. It is well known that if a section exists then the fiber over any point in B represents a nontrivial element in Hk(E; Z) since the image of a section is a compact orientable manifold which has intersection number one with the fiber over any point in B. The notion of a section may be generalized as follows. Definition. A polysection of (E,p,B) is a covering projection tr:B-*B together with a map Ç:B-*E such that the following diagram commutes. -
Base-Paracompact Spaces
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 128 (2003) 145–156 www.elsevier.com/locate/topol Base-paracompact spaces John E. Porter Department of Mathematics and Statistics, Faculty Hall Room 6C, Murray State University, Murray, KY 42071-3341, USA Received 14 June 2001; received in revised form 21 March 2002 Abstract A topological space is said to be totally paracompact if every open base of it has a locally finite subcover. It turns out this property is very restrictive. In fact, the irrationals are not totally paracompact. Here we give a generalization, base-paracompact, to the notion of totally paracompactness, and study which paracompact spaces satisfy this generalization. 2002 Elsevier Science B.V. All rights reserved. Keywords: Paracompact; Base-paracompact; Totally paracompact 1. Introduction A topological space X is paracompact if every open cover of X has a locally finite open refinement. A topological space is said to be totally paracompact if every open base of it has a locally finite subcover. Corson, McNinn, Michael, and Nagata announced [3] that the irrationals have a basis which does not have a locally finite subcover, and hence are not totally paracompact. Ford [6] was the first to give the definition of totally paracompact spaces. He used this property to show the equivalence of the small and large inductive dimension in metric spaces. Ford also showed paracompact, locally compact spaces are totally paracompact. Lelek [9] studied totally paracompact and totally metacompact separable metric spaces. He showed that these two properties are equivalent in separable metric spaces. -
Graph Manifolds and Taut Foliations Mark Brittenham1, Ramin Naimi, and Rachel Roberts2 Introduction
Graph manifolds and taut foliations 1 2 Mark Brittenham , Ramin Naimi, and Rachel Roberts UniversityofTexas at Austin Abstract. We examine the existence of foliations without Reeb comp onents, taut 1 1 foliations, and foliations with no S S -leaves, among graph manifolds. We show that each condition is strictly stronger than its predecessors, in the strongest p os- sible sense; there are manifolds admitting foliations of eachtyp e which do not admit foliations of the succeeding typ es. x0 Introduction Taut foliations have b een increasingly useful in understanding the top ology of 3-manifolds, thanks largely to the work of David Gabai [Ga1]. Many 3-manifolds admit taut foliations [Ro1],[De],[Na1], although some do not [Br1],[Cl]. To date, however, there are no adequate necessary or sucient conditions for a manifold to admit a taut foliation. This pap er seeks to add to this confusion. In this pap er we study the existence of taut foliations and various re nements, among graph manifolds. What we show is that there are many graph manifolds which admit foliations that are as re ned as wecho ose, but which do not admit foliations admitting any further re nements. For example, we nd manifolds which admit foliations without Reeb comp onents, but no taut foliations. We also nd 0 manifolds admitting C foliations with no compact leaves, but which do not admit 2 any C such foliations. These results p oint to the subtle nature b ehind b oth top ological and analytical assumptions when dealing with foliations. A principal motivation for this work came from a particularly interesting exam- ple; the manifold M obtained by 37/2 Dehn surgery on the 2,3,7 pretzel knot K . -
Isoparametric Hypersurfaces with Four Principal Curvatures Revisited
ISOPARAMETRIC HYPERSURFACES WITH FOUR PRINCIPAL CURVATURES REVISITED QUO-SHIN CHI Abstract. The classification of isoparametric hypersurfaces with four principal curvatures in spheres in [2] hinges on a crucial charac- terization, in terms of four sets of equations of the 2nd fundamental form tensors of a focal submanifold, of an isoparametric hypersur- face of the type constructed by Ferus, Karcher and Munzner.¨ The proof of the characterization in [2] is an extremely long calculation by exterior derivatives with remarkable cancellations, which is mo- tivated by the idea that an isoparametric hypersurface is defined by an over-determined system of partial differential equations. There- fore, exterior differentiating sufficiently many times should gather us enough information for the conclusion. In spite of its elemen- tary nature, the magnitude of the calculation and the surprisingly pleasant cancellations make it desirable to understand the under- lying geometric principles. In this paper, we give a conceptual, and considerably shorter, proof of the characterization based on Ozeki and Takeuchi's expan- sion formula for the Cartan-Munzner¨ polynomial. Along the way the geometric meaning of these four sets of equations also becomes clear. 1. Introduction In [2], isoparametric hypersurfaces with four principal curvatures and multiplicities (m1; m2); m2 ≥ 2m1 − 1; in spheres were classified to be exactly the isoparametric hypersurfaces of F KM-type constructed by Ferus Karcher and Munzner¨ [4]. The classification goes as follows. Let M+ be a focal submanifold of codimension m1 of an isoparametric hypersurface in a sphere, and let N be the normal bundle of M+ in the sphere. Suppose on the unit normal bundle UN of N there hold true 1991 Mathematics Subject Classification. -
MTH 304: General Topology Semester 2, 2017-2018
MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds.......................... -
FOLIATIONS Introduction. the Study of Foliations on Manifolds Has a Long
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 80, Number 3, May 1974 FOLIATIONS BY H. BLAINE LAWSON, JR.1 TABLE OF CONTENTS 1. Definitions and general examples. 2. Foliations of dimension-one. 3. Higher dimensional foliations; integrability criteria. 4. Foliations of codimension-one; existence theorems. 5. Notions of equivalence; foliated cobordism groups. 6. The general theory; classifying spaces and characteristic classes for foliations. 7. Results on open manifolds; the classification theory of Gromov-Haefliger-Phillips. 8. Results on closed manifolds; questions of compact leaves and stability. Introduction. The study of foliations on manifolds has a long history in mathematics, even though it did not emerge as a distinct field until the appearance in the 1940's of the work of Ehresmann and Reeb. Since that time, the subject has enjoyed a rapid development, and, at the moment, it is the focus of a great deal of research activity. The purpose of this article is to provide an introduction to the subject and present a picture of the field as it is currently evolving. The treatment will by no means be exhaustive. My original objective was merely to summarize some recent developments in the specialized study of codimension-one foliations on compact manifolds. However, somewhere in the writing I succumbed to the temptation to continue on to interesting, related topics. The end product is essentially a general survey of new results in the field with, of course, the customary bias for areas of personal interest to the author. Since such articles are not written for the specialist, I have spent some time in introducing and motivating the subject. -
A TEXTBOOK of TOPOLOGY Lltld
SEIFERT AND THRELFALL: A TEXTBOOK OF TOPOLOGY lltld SEI FER T: 7'0PO 1.OG 1' 0 I.' 3- Dl M E N SI 0 N A I. FIRERED SPACES This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: SAMUELEILENBERG AND HYMANBASS A list of recent titles in this series appears at the end of this volunie. SEIFERT AND THRELFALL: A TEXTBOOK OF TOPOLOGY H. SEIFERT and W. THRELFALL Translated by Michael A. Goldman und S E I FE R T: TOPOLOGY OF 3-DIMENSIONAL FIBERED SPACES H. SEIFERT Translated by Wolfgang Heil Edited by Joan S. Birman and Julian Eisner @ 1980 ACADEMIC PRESS A Subsidiary of Harcourr Brace Jovanovich, Publishers NEW YORK LONDON TORONTO SYDNEY SAN FRANCISCO COPYRIGHT@ 1980, BY ACADEMICPRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. 11 1 Fifth Avenue, New York. New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NWI 7DX Mit Genehmigung des Verlager B. G. Teubner, Stuttgart, veranstaltete, akin autorisierte englische Ubersetzung, der deutschen Originalausgdbe. Library of Congress Cataloging in Publication Data Seifert, Herbert, 1897- Seifert and Threlfall: A textbook of topology. Seifert: Topology of 3-dimensional fibered spaces. (Pure and applied mathematics, a series of mono- graphs and textbooks ; ) Translation of Lehrbuch der Topologic. Bibliography: p. Includes index. 1. -
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).