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Cup Products in Surface Bundles, Higher Johnson Invariants, and Mmm Classes
CUP PRODUCTS IN SURFACE BUNDLES, HIGHER JOHNSON INVARIANTS, AND MMM CLASSES NICK SALTER Abstract. In this paper we prove a family of results connecting the problem of computing cup products in surface bundles to various other objects that appear in the theory of the cohomology of the mapping class group Modg and the Torelli group Ig. We show that N. Kawazumi's twisted MMM class m0;k can be used to compute k-fold cup products in surface bundles, and that m0;k provides an extension of the higher Johnson invariant τk−2 k−2 k to H (Modg;∗; ^ H1). These results are used to show that the behavior of the restriction 4i 1 4i+2 of the even MMM classes e2i to H (Ig ) is completely determined by Im(τ4i) ≤ ^ H1, and to give a partial answer to a question of D. Johnson. We also use these ideas to show that all surface bundles with monodromy in the Johnson kernel Kg;∗ have cohomology rings isomorphic to that of a trivial bundle, implying the vanishing of all τi when restricted to Kg;∗. 1. Introduction The theme of this paper is the central role that the structure of the cup product in surface bundles plays in the understanding of the cohomology of the mapping class group and its subgroups. We use this perspective to gain a new understanding of the relationships between several well-known cohomology classes, and we also use these ideas to study the topology of surface bundles. 1 Denote by Modg (resp. Modg;∗; Modg) the mapping class group of a closed oriented surface of genus g (resp. -
Lecture II - Preliminaries from General Topology
Lecture II - Preliminaries from general topology: We discuss in this lecture a few notions of general topology that are covered in earlier courses but are of frequent use in algebraic topology. We shall prove the existence of Lebesgue number for a covering, introduce the notion of proper maps and discuss in some detail the stereographic projection and Alexandroff's one point compactification. We shall also discuss an important example based on the fact that the sphere Sn is the one point compactifiaction of Rn. Let us begin by recalling the basic definition of compactness and the statement of the Heine Borel theorem. Definition 2.1: A space X is said to be compact if every open cover of X has a finite sub-cover. If X is a metric space, this is equivalent to the statement that every sequence has a convergent subsequence. If X is a topological space and A is a subset of X we say that A is compact if it is so as a topological space with the subspace topology. This is the same as saying that every covering of A by open sets in X admits a finite subcovering. It is clear that a closed subset of a compact subset is necessarily compact. However a compact set need not be closed as can be seen by looking at X endowed the indiscrete topology, where every subset of X is compact. However, if X is a Hausdorff space then compact subsets are necessarily closed. We shall always work with Hausdorff spaces in this course. For subsets of Rn we have the following powerful result. -
Proper Maps and Universally Closed Maps
Proper Maps and Universally Closed Maps Reinhard Schultz Continuous functions defined on compact spaces generally have special properties. For exam- ple, if X is a compact space and Y is a Hausdorff space, then f is a closed mapping. There is a useful class of continuous mappings called proper, perfect, or compact maps that satisfy many properties of continuous maps with compact domains. In this note we shall study a few basic properties of these maps. Further information about proper maps can be found in Bourbaki [B2], Section I.10), Dugundji [D] (Sections XI.5{6, pages 235{240), and Kasriel [K] (Sections 95 and 105, pages 214{217 and 243{247). There is a slight difference between the notions of perfect and compact mappings in [B2], [D], and [K] and the notion of proper map considered here. Specifically, the definitions in [D] and [K] include a requirement that the maps in question be surjective. The definition in [B2] is entirely different and corresponds to the notion of univer- sally closed map discussed later in these notes; the equivalence of this definition with ours for a reasonable class of spaces is proved both in [B2] and in these notes (see the section Universally closed maps). PROPER MAPS If f : X ! Y is a continuous map of topological spaces, then f is said to be proper if for each compact subset K ⊂ Y the inverse image f −1[K] is also compact. EXAMPLE. Suppose that A is a finite subset of Rn and f : Rn − A ! Rm is a continuous function such that for all a 2 A we have lim jf(x)j = lim jf(x)j = 1: x!a x!1 Then f is proper. -
On Milnor Classes of Constructible Functions
Universidade Federal da Paraíba Universidade Federal de Campina Grande Programa Associado de Pós-Graduação em Matemática Doutorado em Matemática On Milnor classes of constructible functions por Mauri Pereira da Silva João Pessoa - PB Setembro/2019 On Milnor classes of constructible functions por Mauri Pereira da Silva y sob orientação do Prof. Dr. Roberto Callejas Bedregal Tese apresentada ao Corpo Docente do Programa Associado de Pós-Graduação em Matemática - UFPB/UFCG, como requisito parcial para obtenção do título de Doutor em Matemática. João Pessoa - PB Setembro/2019 yEste trabalho contou com apoio nanceiro da CAPES ii Catalogação na publicação Seção de Catalogação e Classificação S586o Silva, Mauri Pereira da. On Milnor classes of constructible functions / Mauri Pereira da Silva. - João Pessoa, 2019. 77 f. Orientação: Roberto Callejas-Bedregal. Tese (Doutorado) - UFPB/CCEN. 1. Número de Milnor. 2. Classe de Milnor. 3. Classe de Segre. 4. Classes características. 5. Variedades singulares. I. Callejas-Bedregal, Roberto. II. Título. UFPB/BC Resumo O principal objetivo deste trabalho é apresentar uma generalização do importante inva- riante da Teoria das Singularidades, chamado número de Milnor. Tal generalização é o que chamamos de número de Milnor logarítmico. Bem como explanar sobre denições um pouco mais gerais no contexto de funções construtíveis, apresentando observações, exemplos e propriedades. Dentre os conceitos que trabalhamos estão também a classe de Fulton-Johnson, a classe de Schwartz-MacPherson, a classe de Milnor e a classe de Segre. Palavras-chave: Número de Milnor; Classe de Milnor; Classe de Segre; Classes ca- racterísticas; variedades singulares. iv Abstract The main goal of this thesis is to present a generalization of the important invariant of the singularity theory, called the Milnor number. -
The Bogomolov-Prokhorov Invariant of Surfaces As Equivariant Cohomology
Bull. Korean Math. Soc. 54 (2017), No. 5, pp. 1725{1741 https://doi.org/10.4134/BKMS.b160689 pISSN: 1015-8634 / eISSN: 2234-3016 THE BOGOMOLOV-PROKHOROV INVARIANT OF SURFACES AS EQUIVARIANT COHOMOLOGY Evgeny Shinder Abstract. For a complex smooth projective surface M with an action of a finite cyclic group G we give a uniform proof of the isomorphism between 1 2 the invariant H (G; H (M; Z)) and the first cohomology of the divisors fixed by the action, using G-equivariant cohomology. This generalizes the main result of Bogomolov and Prokhorov [4]. 1. Introduction Let M be a smooth projective complex variety and G be a finite group acting faithfully on M. In this note we consider the cohomology group 1 2 (1.1) H (G; H (M; Z)) of the singular cohomology H2(M; Z) considered as a G-module. It is easy to see that this cohomology group is a stable birational invariant of the pair (G; M) (see Section 2). In a slightly different context, when M is defined over an arbitrary field K, the analogous group H1(Gal(Ksep=K); P ic(M)) has been considered by Manin who used it to study rationality of del Pezzo surfaces over K [9, 10]. Bogomolov and Prokhorov used the cohomology group (1.2) H1(G; P ic(M)) in [4] when M is a smooth rational surface and G is a finite group to study conjugacy classes in the Cremona groups. Note that for such a surface we have a canonical isomorphism H2(M; Z) ' P ic(M) so that in this case (1.1) and (1.2) are the same. -
Foliations and Global Inversion
Foliations and Global Inversion Eduardo Cabral Balreira Trinity University Mathematics Department San Antonio, TX 78212 [email protected] http://www.trinity.edu/ebalreir XVI Encontro Brasileiro de Topologia (Brazilian Topology Meeting) - p. 1/47 The Global Inversion Problem Program: Understand injectivity mechanism for a local diffeomorphism f : M → N to be invertible. (M and N are non-compact manifolds) Focus: Use Geometric and Topological methods to understand global invertibility of maps on Rn. Outline: I) Motivation II) Classical results III) Topological Results IV) Recent Progress and Holomorphic Results XVI Encontro Brasileiro de Topologia (Brazilian Topology Meeting) - p. 2/47 The Global Inversion Problem Motivation: 1. Algebraic Geometry. Jacobian Conjecture: Let F : Cn → Cn be a local polynomial biholomorphism, i.e., det(DF(z)) = 1, then F admits a polynomial inverse. • It suffices to show injectivity. • Pinchuck has examples of real polynomial maps with non-zero Jacobian determinant everywhere and not injective Understanding the structure of Aut(Cn), n > 1. XVI Encontro Brasileiro de Topologia (Brazilian Topology Meeting) - p. 3/47 The Global Inversion Problem 2. Dynamical Systems. Markus-Yamabe Conjecture: Let f : Rn → Rnbe a C1 map with Spec(Df ) ⊆ {Re < 0} and f (0)= 0, then 0 is a global attractor of x˙ = f (x). • Solved when n = 2 by Fessler, Glutsiuk, Gutierrez, 95. [Gutierrez] If f : R2 → R2 is a C1 map with Spec(Df ) ∩ [0,∞)= 0/, then f is injective. [Fernandes, Gutierrez, Rabanal, 04] Let f : R2 → R2 be a differentiable map, not necessarily C1. If there is ε > 0 such that Spec(Df ) ∩ [0,ε)= 0/, then f is injective. -
University of Warsaw the Gysin Homomorphism for Homogeneous
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repozytorium UW University of Warsaw Faculty of Mathematics, Informatics and Mechanics Magdalena Zielenkiewicz The Gysin homomorphism for homogeneous spaces via residues PhD dissertation Supervisor dr hab. Andrzej Weber Institute of Mathematics University of Warsaw 1 2 Author’s declaration: I hereby declare that this dissertation is my own work. December 12, 2016 ................................................... Magdalena Zielenkiewicz Supervisor’s declaration: The dissertation is ready to be reviewed. December 12, 2016 ................................................... dr hab. Andrzej Weber 3 Abstract The subject of this dissertation is the Gysin homomorphism in equivariant cohomology for spaces with torus action. We consider spaces which are quo- tients of classical semisimple complex linear algebraic groups by a parabolic subgroup with the natural action of a maximal torus, the so-called partial flag varieties. We derive formulas for the Gysin homomorphism for the projection to a point, of the form Z α = Resz=1 Z(z; t) · α(t); X for a certain residue operation and a map Z(z; t), associated to the root system. The mentioned description relies on two following generalizations of theorems in symplectic geometry. We adapt to the equivariant setting (for torus actions) the Jeffrey–Kirwan nonabelian localization and a theorem relating the coho- mology of symplectic reductions by a compact Lie group and by its maximal torus, following the approach by Martin. Applying the generalized theorems to certain contractible spaces acted upon by a product of unitary groups we derive the push-forward formula for partial flag varieties of types A, B, C and D. -
Proof by Proper Maps
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear factors. The original form of this theorem makes no mention of complex polynomials or even complex numbers: it says that in R[x], every nonconstant polynomial can be factored into a product of linear and quadratic factors. (For any polynomial f(x) 2 C[x], the product f(x)f(x) has real coefficients and this permits a passage between the real and complex formulations of the theorem.) That the theorem can be stated without complex numbers doesn't mean it can be proved without complex numbers, and indeed nearly all proofs of the Fundamental Theorem of Algebra make some use of complex numbers, either analytically (e.g., holomorphic functions) or algebraically (e.g., the only quadratic extension field of R is C) or topologically (e.g., GLn(C) is path connected). In the articles [3] and [4], Pukhlikov and Pushkar' give proofs of the Fundamental The- orem of Algebra that make absolutely no use of the concept of a complex number. Both articles are in Russian. The purpose of this note is to describe these two proofs in English so they may become more widely known. As motivation for the two proofs, let's consider what it means to say a polynomial can be factored, in terms of the coefficients. To say that every polynomial x4 +Ax3 +Bx2 +Cx+D in R[x] can be written as a product of two monic quadratic polynomials in R[x], so x4 + Ax3 + Bx2 + Cx + D = (x2 + ax + b)(x2 + cx + d) = x4 + (a + c)x3 + (b + d + ac)x2 + (ad + bc)x + bd; which amounts to saying that given any four real numbers A; B; C; D there is a real solution (a; b; c; d) to the system of equations A = a + c; B = b + d + ac; C = ad + bc; D = bd: This is a system of four (nonlinear) equations in four unknowns. -
A Note on Properly Discontinuous Actions
A NOTE ON PROPERLY DISCONTINUOUS ACTIONS MICHAEL KAPOVICH This note is meant to clarify the relation between different commonly used definitions of proper discontinuity without the local compactness assumption for the underlying topo- logical space. Much of the discussion applies to actions of nondiscrete topological groups, but, since my primary interest is geometric group theory, I will work only with discrete groups. All group actions are assumed to be continuous, in other words, these are homo- morphisms from abstract groups to groups of homeomorphisms of topological spaces. This combination of continuous and properly discontinuous, sadly, leads to the ugly terminology \a continuous properly discontinuous action." A better terminology might be that of a properly discrete action, since it refers to proper actions of discrete groups. Throughout this note, I will be working only with topological spaces which are 1st count- able, since spaces most common in metric geometry, geometric topology, algebraic topology and geometric group theory satisfy this property. One advantage of this assumption is that if (xn) is a sequence converging to a point x 2 X, then the subset fxg [ fxn : n 2 Ng is compact, which is not true if we work with nets instead of sequences. However, I will try to avoid the local compactness assumption whenever possible, since many spaces appear- ing in metric geometry and geometric group theory (e.g. asymptotic cones) and algebraic topology (e.g. CW complexes) are not locally compact. (Recall that topological space X is locally compact if every point has a basis of topology consisting of relatively compact subsets.) Lastly, all the material of this note is elementary and by no means original. -
THE GLEASON COVER of a TOPOS. II P.T. JOHNSTONE This Paper Is a Sequel to [17], in Which We Showed How to Construct for an Arbit
Journal of Pure and Applied Algebra 22 (1981) 229-217 229 North-Holland Publishing Company THE GLEASON COVER OF A TOPOS. II P.T. JOHNSTONE Department of Pure ,Mathematics, University of Cambridge, England Communicated by P.J. Freyd Received 18 July 1979 0. Introduction This paper is a sequel to [17], in which we showed how to construct for an arbitrary topos 6 a ‘best approximation’ to 6 by a topos y6 in which De Morgan’s law is satisfied. This construction generalizes the well-known Gleason cover of a topological space, in that if 6 is the topos of sheaves on a space X, then yK is (equivalent to) the topos of sheaves on the Gleason cover of X. Now the Gleason cover of a space was originally introduced as a by-product of Gleason’s investigation [5] of the connection between extremal disconnectedness and projectivity in categories of topological spaces, and it is natural to ask whether this projectivity theorem also has a topos-theoretic generalization. In this paper, we shall show that it does; we shall show that the projectivity of a topos I! (with respect to a suitable class of ‘proper maps’) in Zap is equivalent to the validity, in the internal logic of ci”,of the completeness theorem for coherent propositional logic, and that (under suitable assumptions on 6 ) this completeness theorem holds precisely when De Morgan’s law is valid in 6. In particular, this result enables us to recover all known cases of the projectivity theorem for topological spaces; it also extends it in a natural way to the category of locales [l 11. -
ORBIFOLDS AS DIFFEOLOGIES 1. Introduction an Orbifold Is a Space
ORBIFOLDS AS DIFFEOLOGIES PATRICK IGLESIAS, YAEL KARSHON, AND MOSHE ZADKA Abstract. We consider orbifolds as diffeological spaces. This gives rise to a natural notion of differentiable maps between orb- ifolds, making them into a subcategory of diffeology. We prove that the diffeological approach to orbifolds is equivalent to Satake’s no- tion of a V-manifold and to Haefliger’s notion of an orbifold. This follows from a lemma: a diffeomorphism (in the diffeological sense) of finite linear quotients lifts to an equivariant diffeomorphism. 1. Introduction An orbifold is a space that is locally modeled on quotients of Rn by finite linear group actions. The precise definition of its global struc- ture is more complicated. The situation in the literature is somewhat problematic: different authors give definitions that are a priori different from each other, and the relations between them are not made clear. We propose to approach orbifolds through the notion of a diffeology. A diffeology on a set X specifies, for each whole number m and each open subset U of Rm, which functions from U to X are differentiable. A diffeological space is a set equipped with a diffeology. A quotient of a diffeological space is again a diffeological space; a map to the quotient is declared to be differentiable if and only if it locally lifts. We define a diffeological orbifold to be a diffeological space that is locally diffeomorphic at each point to the quotient of Rn by a finite linear group action. For details, see Section 2. In Section 3 we show that a diffeomorphism (in the diffeological sense) between finite linear quotients of Rn lifts to an equivariant dif- feomorphism. -
Computing Topological Invariants Using Fixed Points
COMPUTING TOPOLOGICAL INVARIANTS USING FIXED POINTS LORING W. TU ABSTRACT. When a torus acts on a compact oriented manifold with isolated fixed points, the equivariant localization formula of Atiyah–Bott and Berline–Vergne converts the integral of an equivariantly closed form into a finite sum over the fixed points of the action, thus providing a powerful tool for computing integrals on a manifold. An integral can also be viewed as a pushforward map from a manifold to a point, and in this guise it is intimately related to the Gysin homomorphism. This article highlights two applications of the equivariant localization formula. We show how to use it to compute characteristic numbers of a homogeneous space and to derive a formula for the Gysin map of a fiber bundle. Many invariants in geometry and topology can be represented as integrals. For example, according to the Gauss–Bonnet theorem, the Euler characteristic of a compact oriented surface in R3 is 1/2π times the integral of its Gaussian curvature: 1 χ(M)= Kvol. 2π ZM The Euler characteristic can be generalized to other characteristic numbers. For example, if E is a complex vector bundle of rank r over a complex manifold M of complex dimension n, and c1,...,cr are the Chern classes of E, then the integrals r i1 ir c1 ···cr , where ∑ k · ik = n, ZM k=1 are the Chern numbers of E. Taking E to be the holomorphic tangent bundle TM of M, the Chern numbers of TM are called the Chern numbers of the complex manifold M.