Commentary on Thurston's Work on Foliations
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Realizing Cohomology Classes As Euler Classes 1
Ó DOI: 10.2478/s12175-012-0057-2 Math. Slovaca 62 (2012), No. 5, 949–966 REALIZING COHOMOLOGY CLASSES AS EULER CLASSES Aniruddha C Naolekar (Communicated by J´ulius Korbaˇs ) s ◦ ABSTRACT. For a space X,letEk(X), Ek(X)andEk (X) denote respectively the set of Euler classes of oriented k-plane bundles over X, the set of Euler classes of stably trivial k-plane bundles over X and the spherical classes in Hk(X; Z). s ◦ We prove some general facts about the sets Ek(X), Ek(X)andEk (X). We also compute these sets in the cases where X is a projective space, the Dold manifold P (m, 1) and obtain partial computations in the case that X is a product of spheres. c 2012 Mathematical Institute Slovak Academy of Sciences 1. Introduction Given a topological space X, we address the question of which classes x ∈ Hk(X; Z) can be realized as the Euler class e(ξ)ofanorientedk-plane bundle ξ over X. We also look at the question of which integral cohomology classes are spherical. It is convenient to make the following definition. s º ≥ ÒØÓÒ 1.1 For a space X and an integer k 1, the sets Ek(X), Ek(X) ◦ and Ek (X) are defined to be k Ek(X)= e(ξ) ∈ H (X; Z) | ξ is an oriented k-plane bundle over X s k Ek(X)= e(ξ) ∈ H (X; Z) | ξ is a stably trivial k-plane bundle over X ◦ k k ∗ Ek (X)= x ∈ H (X; Z) | there exists f : S −→ X with f (x) =0 . -
Notes on the Atiyah-Singer Index Theorem Liviu I. Nicolaescu
Notes on the Atiyah-Singer Index Theorem Liviu I. Nicolaescu Notes for a topics in topology course, University of Notre Dame, Spring 2004, Spring 2013. Last revision: November 15, 2013 i The Atiyah-Singer Index Theorem This is arguably one of the deepest and most beautiful results in modern geometry, and in my view is a must know for any geometer/topologist. It has to do with elliptic partial differential opera- tors on a compact manifold, namely those operators P with the property that dim ker P; dim coker P < 1. In general these integers are very difficult to compute without some very precise information about P . Remarkably, their difference, called the index of P , is a “soft” quantity in the sense that its determination can be carried out relying only on topological tools. You should compare this with the following elementary situation. m n Suppose we are given a linear operator A : C ! C . From this information alone we cannot compute the dimension of its kernel or of its cokernel. We can however compute their difference which, according to the rank-nullity theorem for n×m matrices must be dim ker A−dim coker A = m − n. Michael Atiyah and Isadore Singer have shown in the 1960s that the index of an elliptic operator is determined by certain cohomology classes on the background manifold. These cohomology classes are in turn topological invariants of the vector bundles on which the differential operator acts and the homotopy class of the principal symbol of the operator. Moreover, they proved that in order to understand the index problem for an arbitrary elliptic operator it suffices to understand the index problem for a very special class of first order elliptic operators, namely the Dirac type elliptic operators. -
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. -
K-Theoryand Characteristic Classes
MATH 6530: K-THEORY AND CHARACTERISTIC CLASSES Taught by Inna Zakharevich Notes by David Mehrle [email protected] Cornell University Fall 2017 Last updated November 8, 2018. The latest version is online here. Contents 1 Vector bundles................................ 3 1.1 Grassmannians ............................ 8 1.2 Classification of Vector bundles.................... 11 2 Cohomology and Characteristic Classes................. 15 2.1 Cohomology of Grassmannians................... 18 2.2 Characteristic Classes......................... 22 2.3 Axioms for Stiefel-Whitney classes................. 26 2.4 Some computations.......................... 29 3 Cobordism.................................. 35 3.1 Stiefel-Whitney Numbers ...................... 35 3.2 Cobordism Groups.......................... 37 3.3 Geometry of Thom Spaces...................... 38 3.4 L-equivalence and Transversality.................. 42 3.5 Characteristic Numbers and Boundaries.............. 47 4 K-Theory................................... 49 4.1 Bott Periodicity ............................ 49 4.2 The K-theory spectrum........................ 56 4.3 Some properties of K-theory..................... 58 4.4 An example: K-theory of S2 ..................... 59 4.5 Power Operations............................ 61 4.6 When is the Hopf Invariant one?.................. 64 4.7 The Splitting Principle........................ 66 5 Where do we go from here?........................ 68 5.1 The J-homomorphism ........................ 70 5.2 The Chern Character and e invariant............... -
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. -
Characteristic Classes and K-Theory Oscar Randal-Williams
Characteristic classes and K-theory Oscar Randal-Williams https://www.dpmms.cam.ac.uk/∼or257/teaching/notes/Kthy.pdf 1 Vector bundles 1 1.1 Vector bundles . 1 1.2 Inner products . 5 1.3 Embedding into trivial bundles . 6 1.4 Classification and concordance . 7 1.5 Clutching . 8 2 Characteristic classes 10 2.1 Recollections on Thom and Euler classes . 10 2.2 The projective bundle formula . 12 2.3 Chern classes . 14 2.4 Stiefel–Whitney classes . 16 2.5 Pontrjagin classes . 17 2.6 The splitting principle . 17 2.7 The Euler class revisited . 18 2.8 Examples . 18 2.9 Some tangent bundles . 20 2.10 Nonimmersions . 21 3 K-theory 23 3.1 The functor K ................................. 23 3.2 The fundamental product theorem . 26 3.3 Bott periodicity and the cohomological structure of K-theory . 28 3.4 The Mayer–Vietoris sequence . 36 3.5 The Fundamental Product Theorem for K−1 . 36 3.6 K-theory and degree . 38 4 Further structure of K-theory 39 4.1 The yoga of symmetric polynomials . 39 4.2 The Chern character . 41 n 4.3 K-theory of CP and the projective bundle formula . 44 4.4 K-theory Chern classes and exterior powers . 46 4.5 The K-theory Thom isomorphism, Euler class, and Gysin sequence . 47 n 4.6 K-theory of RP ................................ 49 4.7 Adams operations . 51 4.8 The Hopf invariant . 53 4.9 Correction classes . 55 4.10 Gysin maps and topological Grothendieck–Riemann–Roch . 58 Last updated May 22, 2018. -
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. -
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. -
On the Kinematic Formula in the Lives of the Saints Danny Calegari
SHORT STORIES On the Kinematic Formula in the Lives of the Saints Danny Calegari Saint Sebastian was an early Christian martyr. He served as a captain in the Praetorian Guard under Diocletian un- til his religious faith was discovered, at which point he was taken to a field, bound to a stake, and shot by archers “till he was as full of arrows as an urchin1 is full of pricks2.” Rather miraculously, he made a full recovery, but was later executed anyway for insulting the emperor. The trans- pierced saint became a popular subject for Renaissance painters, e.g., Figure 1. The arrows in Mantegna’s painting have apparently ar- rived from all directions, though they are conspicuously grouped around the legs and groin, almost completely missing the thorax. Intuitively, we should expect more ar- rows in the parts of the body that present a bigger cross section. This intuition is formalized by the claim that a subset of the surface of Saint Sebastian has an area pro- portional to its expected number of intersections with a random line (i.e., arrow). Since both area and expectation are additive, we may reduce the claim (by polygonal ap- proximation and limit) to the case of a flat triangular Saint Sebastian, in which case it is obvious. Danny Calegari is a professor of mathematics at the University of Chicago. His Figure 1. Andrea Mantegna’s painting of Saint Sebastian. email address is [email protected]. 1 hedgehog This is a 3-dimensional version of the classical Crofton 2quills formula, which says that the length of a plane curve is pro- For permission to reprint this article, please contact: portional to its expected number of intersections with a [email protected]. -
Groups of PL Homeomorphisms of Cubes
GROUPS OF PL HOMEOMORPHISMS OF CUBES DANNY CALEGARI AND DALE ROLFSEN D´edi´e`aMichel Boileau sur son soixanti`eme anniversaire. Sant´e! Abstract. We study algebraic properties of groups of PL or smooth homeo- morphisms of unit cubes in any dimension, fixed pointwise on the boundary, and more generally PL or smooth groups acting on manifolds and fixing point- wise a submanifold of codimension 1 (resp. codimension 2), and show that such groups are locally indicable (resp. circularly orderable). We also give many examples of interesting groups that can act, and discuss some other algebraic constraints that such groups must satisfy, including the fact that a group of PL homeomorphisms of the n-cube (fixed pointwise on the boundary) contains no elements that are more than exponentially distorted. 1. Introduction We are concerned in this paper with algebraic properties of the group of PL homeomorphisms of a PL manifold, fixed on some PL submanifold (usually of codimension 1 or 2, for instance the boundary) and some of its subgroups (usually those preserving some structure). The most important case is the group of PL homeomorphisms of In fixed pointwise on ∂In; hence these are “groups of PL homeomorphisms of the (n-)cube”. The algebraic study of transformation groups (often in low dimension, or preserv- ing some extra structure such as a symplectic or complex structure) has recently seen a lot of activity; however, much of this activity has been confined to the smooth category. It is striking that many of these results can be transplanted to the PL category. This interest is further strengthened by the possibility of working in the PL category over (real) algebraic rings or fields (this possibility has already been exploited in dimension 1, in the groups F and T of Richard Thompson). -
Classification of 3-Manifolds with Certain Spines*1 )
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 205, 1975 CLASSIFICATIONOF 3-MANIFOLDSWITH CERTAIN SPINES*1 ) BY RICHARD S. STEVENS ABSTRACT. Given the group presentation <p= (a, b\ambn, apbq) With m, n, p, q & 0, we construct the corresponding 2-complex Ky. We prove the following theorems. THEOREM 7. Jf isa spine of a closed orientable 3-manifold if and only if (i) \m\ = \p\ = 1 or \n\ = \q\ = 1, or (") (m, p) = (n, q) = 1. Further, if (ii) holds but (i) does not, then the manifold is unique. THEOREM 10. If M is a closed orientable 3-manifold having K^ as a spine and \ = \mq — np\ then M is a lens space L\ % where (\,fc) = 1 except when X = 0 in which case M = S2 X S1. It is well known that every connected compact orientable 3-manifold with or without boundary has a spine that is a 2-complex with a single vertex. Such 2-complexes correspond very naturally to group presentations, the 1-cells corre- sponding to generators and the 2-cells corresponding to relators. In the case of a closed orientable 3-manifold, there are equally many 1-cells and 2-cells in the spine, i.e., equally many generators and relators in the corresponding presentation. Given a group presentation one is motivated to ask the following questions: (1) Is the corresponding 2-complex a spine of a compact orientable 3-manifold? (2) If there are equally many generators and relators, is the 2-complex a spine of a closed orientable 3-manifold? (3) Can we say what manifold(s) have the 2-complex as a spine? L. -
Manifolds: Where Do We Come From? What Are We? Where Are We Going
Manifolds: Where Do We Come From? What Are We? Where Are We Going Misha Gromov September 13, 2010 Contents 1 Ideas and Definitions. 2 2 Homotopies and Obstructions. 4 3 Generic Pullbacks. 9 4 Duality and the Signature. 12 5 The Signature and Bordisms. 25 6 Exotic Spheres. 36 7 Isotopies and Intersections. 39 8 Handles and h-Cobordisms. 46 9 Manifolds under Surgery. 49 1 10 Elliptic Wings and Parabolic Flows. 53 11 Crystals, Liposomes and Drosophila. 58 12 Acknowledgments. 63 13 Bibliography. 63 Abstract Descendants of algebraic kingdoms of high dimensions, enchanted by the magic of Thurston and Donaldson, lost in the whirlpools of the Ricci flow, topologists dream of an ideal land of manifolds { perfect crystals of mathematical structure which would capture our vague mental images of geometric spaces. We browse through the ideas inherited from the past hoping to penetrate through the fog which conceals the future. 1 Ideas and Definitions. We are fascinated by knots and links. Where does this feeling of beauty and mystery come from? To get a glimpse at the answer let us move by 25 million years in time. 25 106 is, roughly, what separates us from orangutans: 12 million years to our common ancestor on the phylogenetic tree and then 12 million years back by another× branch of the tree to the present day orangutans. But are there topologists among orangutans? Yes, there definitely are: many orangutans are good at "proving" the triv- iality of elaborate knots, e.g. they fast master the art of untying boats from their mooring when they fancy taking rides downstream in a river, much to the annoyance of people making these knots with a different purpose in mind.