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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. -
Part III 3-Manifolds Lecture Notes C Sarah Rasmussen, 2019
Part III 3-manifolds Lecture Notes c Sarah Rasmussen, 2019 Contents Lecture 0 (not lectured): Preliminaries2 Lecture 1: Why not ≥ 5?9 Lecture 2: Why 3-manifolds? + Intro to Knots and Embeddings 13 Lecture 3: Link Diagrams and Alexander Polynomial Skein Relations 17 Lecture 4: Handle Decompositions from Morse critical points 20 Lecture 5: Handles as Cells; Handle-bodies and Heegard splittings 24 Lecture 6: Handle-bodies and Heegaard Diagrams 28 Lecture 7: Fundamental group presentations from handles and Heegaard Diagrams 36 Lecture 8: Alexander Polynomials from Fundamental Groups 39 Lecture 9: Fox Calculus 43 Lecture 10: Dehn presentations and Kauffman states 48 Lecture 11: Mapping tori and Mapping Class Groups 54 Lecture 12: Nielsen-Thurston Classification for Mapping class groups 58 Lecture 13: Dehn filling 61 Lecture 14: Dehn Surgery 64 Lecture 15: 3-manifolds from Dehn Surgery 68 Lecture 16: Seifert fibered spaces 69 Lecture 17: Hyperbolic 3-manifolds and Mostow Rigidity 70 Lecture 18: Dehn's Lemma and Essential/Incompressible Surfaces 71 Lecture 19: Sphere Decompositions and Knot Connected Sum 74 Lecture 20: JSJ Decomposition, Geometrization, Splice Maps, and Satellites 78 Lecture 21: Turaev torsion and Alexander polynomial of unions 81 Lecture 22: Foliations 84 Lecture 23: The Thurston Norm 88 Lecture 24: Taut foliations on Seifert fibered spaces 89 References 92 1 2 Lecture 0 (not lectured): Preliminaries 0. Notation and conventions. Notation. @X { (the manifold given by) the boundary of X, for X a manifold with boundary. th @iX { the i connected component of @X. ν(X) { a tubular (or collared) neighborhood of X in Y , for an embedding X ⊂ Y . -
Ordering Thurston's Geometries by Maps of Non-Zero Degree
ORDERING THURSTON’S GEOMETRIES BY MAPS OF NON-ZERO DEGREE CHRISTOFOROS NEOFYTIDIS ABSTRACT. We obtain an ordering of closed aspherical 4-manifolds that carry a non-hyperbolic Thurston geometry. As application, we derive that the Kodaira dimension of geometric 4-manifolds is monotone with respect to the existence of maps of non-zero degree. 1. INTRODUCTION The existence of a map of non-zero degree defines a transitive relation, called domination rela- tion, on the homotopy types of closed oriented manifolds of the same dimension. Whenever there is a map of non-zero degree M −! N we say that M dominates N and write M ≥ N. In general, the domain of a map of non-zero degree is a more complicated manifold than the target. Gromov suggested studying the domination relation as defining an ordering of compact oriented manifolds of a given dimension; see [3, pg. 1]. In dimension two, this relation is a total order given by the genus. Namely, a surface of genus g dominates another surface of genus h if and only if g ≥ h. However, the domination relation is not generally an order in higher dimensions, e.g. 3 S3 and RP dominate each other but are not homotopy equivalent. Nevertheless, it can be shown that the domination relation is a partial order in certain cases. For instance, 1-domination defines a partial order on the set of closed Hopfian aspherical manifolds of a given dimension (see [18] for 3- manifolds). Other special cases have been studied by several authors; see for example [3, 1, 2, 25]. -
Computing Triangulations of Mapping Tori of Surface Homeomorphisms
Computing Triangulations of Mapping Tori of Surface Homeomorphisms Peter Brinkmann∗ and Saul Schleimer Abstract We present the mathematical background of a software package that computes triangulations of mapping tori of surface homeomor- phisms, suitable for Jeff Weeks’s program SnapPea. The package is an extension of the software described in [?]. It consists of two programs. jmt computes triangulations and prints them in a human-readable format. jsnap converts this format into SnapPea’s triangulation file format and may be of independent interest because it allows for quick and easy generation of input for SnapPea. As an application, we ob- tain a new solution to the restricted conjugacy problem in the mapping class group. 1 Introduction In [?], the first author described a software package that provides an en- vironment for computer experiments with automorphisms of surfaces with one puncture. The purpose of this paper is to present the mathematical background of an extension of this package that computes triangulations of mapping tori of such homeomorphisms, suitable for further analysis with Jeff Weeks’s program SnapPea [?].1 ∗This research was partially conducted by the first author for the Clay Mathematics Institute. 2000 Mathematics Subject Classification. 57M27, 37E30. Key words and phrases. Mapping tori of surface automorphisms, pseudo-Anosov au- tomorphisms, mapping class group, conjugacy problem. 1Software available at http://thames.northnet.org/weeks/index/SnapPea.html 1 Pseudo-Anosov homeomorphisms are of particular interest because their mapping tori are hyperbolic 3-manifolds of finite volume [?]. The software described in [?] recognizes pseudo-Anosov homeomorphisms. Combining this with the programs discussed here, we obtain a powerful tool for generating and analyzing large numbers of hyperbolic 3-manifolds. -
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. -
Singular Hyperbolic Structures on Pseudo-Anosov Mapping Tori
SINGULAR HYPERBOLIC STRUCTURES ON PSEUDO-ANOSOV MAPPING TORI A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MATHEMATICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Kenji Kozai June 2013 iv Abstract We study three-manifolds that are constructed as mapping tori of surfaces with pseudo-Anosov monodromy. Such three-manifolds are endowed with natural sin- gular Sol structures coming from the stable and unstable foliations of the pseudo- Anosov homeomorphism. We use Danciger’s half-pipe geometry [6] to extend results of Heusener, Porti, and Suarez [13] and Hodgson [15] to construct singular hyperbolic structures when the monodromy has orientable invariant foliations and its induced action on cohomology does not have 1 as an eigenvalue. We also discuss a combina- torial method for deforming the Sol structure to a singular hyperbolic structure using the veering triangulation construction of Agol [1] when the surface is a punctured torus. v Preface As a result of Perelman’s Geometrization Theorem, every 3-manifold decomposes into pieces so that each piece admits one of eight model geometries. Most 3-manifolds ad- mit hyperbolic structures, and outstanding questions in the field mostly center around these manifolds. Having an effective method for describing the hyperbolic structure would help with understanding topology and geometry in dimension three. The recent resolution of the Virtual Fibering Conjecture also means that every closed, irreducible, atoroidal 3-manifold can be constructed, up to a finite cover, from a homeomorphism of a surface by taking the mapping torus for the homeomorphism [2]. -
Inventiones Mathematicae 9 Springer-Verlag 1994
Invent. math. 118, 255 283 (1994) Inventiones mathematicae 9 Springer-Verlag 1994 Bundles and finite foliations D. Cooper 1'*, D.D. Long 1'**, A.W. Reid 2 .... 1 Department of Mathematics, University of California, Santa Barbara CA 93106, USA z Department of Pure Mathematics, University of Cambridge, Cambridge CB2 1SB, UK Oblatum 15-1X-1993 & 30-X11-1993 1 Introduction By a hyperbolic 3-manifold, we shall always mean a complete orientable hyperbolic 3-manifold of finite volume. We recall that if F is a Kleinian group then it is said to be geometrically finite if there is a finite-sided convex fundamental domain for the action of F on hyperbolic space. Otherwise, s is geometrically infinite. If F happens to be a surface group, then we say it is quasi-Fuchsian if the limit set for the group action is a Jordan curve C and F preserves the components of S~\C. The starting point for this work is the following theorem, which is a combination of theorems due to Marden [10], Thurston [14] and Bonahon [1]. Theorem 1.1 Suppose that M is a closed orientable hyperbolic 3-man!fold. If g: Sq-~ M is a lrl-injective map of a closed surface into M then exactly one of the two alternatives happens: 9 The 9eometrically infinite case: there is a finite cover lVl of M to which g l([ts and can be homotoped to be a homeomorphism onto a fiber of some fibration of over the circle. 9 The 9eometricallyfinite case: g,~zl (S) is a quasi-Fuchsian group. -
Endomorphisms of Mapping Tori
ENDOMORPHISMS OF MAPPING TORI CHRISTOFOROS NEOFYTIDIS Abstract. We classify in terms of Hopf-type properties mapping tori of residually finite Poincar´eDuality groups with non-zero Euler characteristic. This generalises and gives a new proof of the analogous classification for fibered 3-manifolds. Various applications are given. In particular, we deduce that rigidity results for Gromov hyperbolic groups hold for the above mapping tori with trivial center. 1. Introduction We classify in terms of Hopf-type properties mapping tori of residually finite Poincar´eDu- ality (PD) groups K with non-zero Euler characteristic, which satisfy the following finiteness condition: if there exists an integer d > 1 and ξ 2 Aut(K) such that θd = ξθξ−1 mod Inn(K); (∗) then θq 2 Inn(K) for some q > 1: One sample application is that every endomorphism onto a finite index subgroup of the 1 fundamental group of a mapping torus F oh S , where F is a closed aspherical manifold with 1 π1(F ) = K as above, induces a homotopy equivalence (equivalently, π1(F oh S ) is cofinitely 1 1 Hopfian) if and only if the center of π1(F oh S ) is trivial; equivalently, π1(F oh S ) is co- Hopfian. If, in addition, F is topologically rigid, then homotopy equivalence can be replaced by a map homotopic to a homeomorphism, by Waldhausen's rigidity [Wa] in dimension three and Bartels-L¨uck's rigidity [BL] in dimensions greater than four. Condition (∗) is known to be true for every aspherical surface by a theorem of Farb, Lubotzky and Minsky [FLM] on translation lengths. -
On Open Book Embedding of Contact Manifolds in the Standard Contact Sphere
ON OPEN BOOK EMBEDDING OF CONTACT MANIFOLDS IN THE STANDARD CONTACT SPHERE KULDEEP SAHA Abstract. We prove some open book embedding results in the contact category with a constructive ap- proach. As a consequence, we give an alternative proof of a Theorem of Etnyre and Lekili that produces a large class of contact 3-manifolds admitting contact open book embeddings in the standard contact 5-sphere. We also show that all the Ustilovsky (4m + 1)-spheres contact open book embed in the standard contact (4m + 3)-sphere. 1. Introduction An open book decomposition of a manifold M m is a pair (V m−1; φ), such that M m is diffeomorphic to m−1 m−1 2 m−1 MT (V ; φ)[id @V ×D . Here, V , the page, is a manifold with boundary, and φ, the monodromy, is a diffeomorphism of V m−1 that restricts to identity in a neighborhood of the boundary @V . MT (V m−1; φ) denotes the mapping torus of φ. We denote an open book, with page V m−1 and monodromy φ, by Aob(V; φ). The existence of open book decompositions, for a fairly large class of manifolds, is now known due the works of Alexander [Al], Winkelnkemper [Wi], Lawson [La], Quinn [Qu] and Tamura [Ta]. In particular, every closed, orientable, odd dimensional manifold admits an open book decomposition. Thurston and Winkelnkemper [TW] have shown that starting from an exact symplectic manifold (Σ2m;!) 2m as page and a boundary preserving symplectomorphism φs of (Σ ;!) as monodromy, one can produce a 2m+1 2m contact 1-form α on N = Aob(Σ ; φs). -
Topology and Geometry of 2 and 3 Dimensional Manifolds
Topology and Geometry of 2 and 3 dimensional manifolds Chris John May 3, 2016 Supervised by Dr. Tejas Kalelkar 1 Introduction In this project I started with studying the classification of Surface and then I started studying some preliminary topics in 3 dimensional manifolds. 2 Surfaces Definition 2.1. A simple closed curve c in a connected surface S is separating if S n c has two components. It is non-separating otherwise. If c is separating and a component of S n c is an annulus or a disk, then it is called inessential. A curve is essential otherwise. An observation that we can make here is that a curve is non-separating if and only if its complement is connected. For manifolds, connectedness implies path connectedness and hence c is non-separating if and only if there is some other simple closed curve intersecting c exactly once. 2.1 Decomposition of Surfaces Lemma 2.2. A closed surface S is prime if and only if S contains no essential separating simple closed curve. Proof. Consider a closed surface S. Let us assume that S = S1#S2 is a non trivial con- nected sum. The curve along which S1 n(disc) and S2 n(disc) where identified is an essential separating curve in S. Hence, if S contains no essential separating curve, then S is prime. Now assume that there exists a essential separating curve c in S. This means neither of 2 2 the components of S n c are disks i.e. S1 6= S and S2 6= S . -
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.