DOCSLIB.ORG

From Braids to Mapping Class Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
From Braids to Mapping Class Groups From Braids to Mapping Class Groups Pratyush Kumar Mishra MP14008 A dissertation submitted for the partial fulfilment of MS degree in Mathematical Science Indian Institute of Science Education and Research Mohali April 2017 Certificate of Examination This is to certify that the dissertation titled \From Braids to Map- ping Class Groups" submitted by Mr. Pratyush Kumar Mishra (Reg. No. MP14008) for the partial fulfilment of MS degree programme in Mathematical Sciences of the Institute, has been examined by the the- sis committee duly appointed by the Institute. The committee finds the work done by the candidate satisfactory and recommends that the report be accepted. Prof.Kapil Hari Paranjape Dr.Pranab Sardar Dr.Mahender Singh (Supervisor) Dated: April 21, 2017 Declaration The work presented in this dissertation has been carried out by me un- der the guidance of Prof. Kapil Hari Paranjape at the Indian Institute of Science Education and Research Mohali. This work has not been submitted in part or in full for a degree, a diploma, or a fellowship to any other university or institute. Whenever contributions of others are involved, every effort is made to indicate this clearly, with due acknowledgement of collaborative research and discussions. This thesis is a bonafide record of original work done by me and all sources listed within have been detailed in the bibliography. Pratyush Kumar Mishra (Candidate) Dated: April 21, 2017 In my capacity as the supervisor of the candidate's project work, I cer- tify that the above statements by the candidate are true to the best of my knowledge. Prof. Kapil Hari Paranjape (Supervisor) Acknowledgements I would like to thank my supervisor Prof. Kapil Hari Paranjape, for his invaluable guidance, support and help in understanding the material presented in this thesis. I am very thankful to Mr. Soumya Dey, a senior research fellow at IISER Mohali for his continuous help in understanding many concepts throughout the thesis work. I would like to thank Dr. Pranab Sardar for clearing some of the major doubts during the thesis work and understanding some of the basic parts of the proofs. Also, I am thankful to Dr. Mahender Singh for allowing me to audit the course `Topics in Topology', which helped me in clearing some of the concepts in Higher Homotopy Theory and Dr. Chetan Balwe for helping me with understanding some of the concepts during my last phase of the project. I would like to express my gratitude to all my teachers at IISER Mohali Mathematics Department for being so supportive and helpful. IISER, Mohali Department of Mathematics provided an excellent li- brary and highly suggestive deadlines. I owe a debt of gratitude to the generous souls who proofread this work in its more and less spell checked stages. I would like to express my gratitude to my parents for always encour- aging me. Pratyush Kumar Mishra IISER Mohali Preface The central theme is Artin's braid group, and the many ways that the notion of a braid has proved to be important in low-dimensional topology. It will be assumed that the reader is familiar with the very basic ideas of homo- topy theory such as the ideas of homotopy equivalence, homomorphism, deformation retractions and the notions of fundamental groups(and its computation) etc. Chapter 1, as a preliminary develop the tools to be used in chapter 2 and 3 of the thesis. The materials here are based on my understanding from the texts: `Algebraic Topology' by Allan Hatcher; `Combinatorial Group Theory' by Magnus, Karrass and Solitar; `Homotopy Theory' by Sze-Tsen Hu. Chapter 2 starts with definition of braid group and deals with the concepts of a braid regarded as a group of motions of points in a manifold. Many algebraic and structural properties of the braid groups of two manifolds are studied, and defining relations are derived for the braid groups of E2 and S2. The materials presented in this section is based on my understanding of Chapter 1, from the text `Braids, Links and Mapping Class Groups' by J.S. Birman [1]. The proof of the theorem 13 is based on my understanding of the paper `Basic Results on Braids', 2004 by J. Gonzalez Meneses [22]. In Chapter 3, we will give some connections between braid groups and mapping class group of the surfaces. Also we compute the mapping class group of the n- punctured sphere. The contained of this chapter is based on my understanding of section 4.1 and 4.2 of chapter 4 from the text `Braids, Links and Mapping Class groups' by Birman [1]. The proof of Lemma 7, is based on my understanding from the text `A primer on Mapping class group' by 'Benson Farb and Dan Margalit'. Some of the figures are taken from Birman's book [1] and some other from the Internet. I tried my best to give detailed explanations for each theorems and results which were not that vivid in the original manuscript of Birman [1]. I mentioned the refer- ences whenever required in the `Bibliography'. Pratyush Kumar Mishra IISER Mohali Contents 1 Preliminaries 9 1.1 Covering Spaces . .9 1.1.1 Lifting Properties . 10 1.1.2 The Classification of Covering Spaces . 12 1.1.3 Deck Transformation and Group Actions . 13 1.2 Graphs and Free Groups . 16 1.3 Higher Homotopy Groups . 18 1.3.1 Whitehead's Theorem . 22 1.3.2 Cellular Approximation Theorem . 22 1.3.3 Freudenthal suspension theorem . 23 1.3.4 Fiber Bundles . 23 1.4 Homogeneous spaces . 25 1.5 Group Representation . 27 1.6 Schreier-Reidemeister Method . 28 2 Braid Groups 31 2.1 Definitions . 32 2.2 Configuration spaces . 38 2.3 Braid groups of manifolds . 41 2.4 The braid group of the plane . 43 2.4.1 Braids as Automorphisms of free groups or Mapping Class groups 56 2.5 The braid group of the sphere . 58 2.6 Survey of 2-manifold braid groups . 59 2.7 Some examples where braiding appears in mathematics, unexpectedly . 60 2.7.1 Algebraic geometry . 60 7 3 Mapping Class Groups 63 3.1 Definition and Examples . 63 3.2 The natural homomorphism from M(g,n) to M(g,0) . 64 3.3 The mapping class group of the n-punctured sphere . 71 Chapter 1 Preliminaries 1.1 Covering Spaces This section is based on the author's understanding of Chapters 0,1,2 from the Hatcher's book on Algebraic Topology [2]. We will just state the definitions and statements of the theorems(without proof), that we require for the rest of the section. Definition: A covering space of a space X is a space X~ together with a surjective ~ map p X X satisfying the following condition: There exists a open cover Uα of −1 ~ X such∶ that→ for each α, p Uα is a disjoint union of open sets in X, each of{ which} is mapped by p homeomorphically( ) onto Uα. Example: Take for example the map p R S1, defined by p t cos 2πt; sin 2πt . It's easy to check that it's a covering map.∶ → ( ) = ( ) Example: One can also think of the map p S1 S1, p z zn which is known as a n sheeted covering where we view z as the complex∶ → number( ) = with z 1 and n is any positive integer. S S = Simply-connected space: A path-connected space whose fundamental group is trivial. Proposition 1. A space X is simply-connected iff there is a unique homotopy class of paths connecting any two points in X. Example: It is easy to see that Rn is simply connected for all n. A simply-connected covering space of X S1 S1 is as in figure given below: 9 = ∨ Covering Spaces Section 1.3 59 A simply-connected covering space of X can be constructed in the following way. Start with the open intervals (−1, 1) in the coordinate R2 1 axes of . Next, for a fixed number λ,0<λ< /2, for = 1 example λ /3, adjoin four open segments of length 2λ, at distance λ from the ends of the previous seg- ments and perpendicular to them, the new shorter seg- ments being bisected by the older ones. For the third stage, add perpendicular open segments of length 2λ2 at distance λ2 from the endpoints of all the previous segments and bisected by them. The process is now − repeated indefinitely, at the nth stage adding open segments of length 2λn 1 at dis- n−1 tance λ from all theThis previous covering endpoints. space The is union called of all the theseuniversal open segments cover is of X because, it is a covering a graph, with vertices the intersection points of horizontal and vertical segments, and edges the subsegmentsspace between of every adjacent other vertices. connected We label covering all the horizontal space of X.edges a, oriented to the right, and all the vertical edges b , oriented upward. Universal cover: A covering space is a universal covering space if it is simply This covering space is called the universal cover of X because, as our general theory will show, itconnected. is a covering space of every other connected covering space of X . The covering spaces (1)–(14) in the table are all nonsimply-connected. Their fun- Remark: We will use covering space theory to deduce some of the interesting damental groups are free with bases represented by the loops specified by the listed e words in a and b ,properties starting at the in basepoint Infinite groupx0 indicated Theory by the like: heavilythe freeshaded group ver- on two generators can contain tex.
Load more

Recommended publications
  • (Pro-) Étale Cohomology 3. Exercise Sheet
    (Pro-) Étale Cohomology 3. Exercise Sheet Department of Mathematics Winter Semester 18/19 Prof. Dr. Torsten Wedhorn 2nd November 2018 Timo Henkel Homework Exercise H9 (Clopen subschemes) (12 points) Let X be a scheme. We define Clopen(X ) := Z X Z open and closed subscheme of X . f ⊆ j g Recall that Clopen(X ) is in bijection to the set of idempotent elements of X (X ). Let X S be a morphism of schemes. We consider the functor FX =S fromO the category of S-schemes to the category of sets, given! by FX =S(T S) = Clopen(X S T). ! × Now assume that X S is a finite locally free morphism of schemes. Show that FX =S is representable by an affine étale S-scheme which is of! finite presentation over S. Exercise H10 (Lifting criteria) (12 points) Let f : X S be a morphism of schemes which is locally of finite presentation. Consider the following diagram of S-schemes:! T0 / X (1) f T / S Let be a class of morphisms of S-schemes. We say that satisfies the 1-lifting property (resp. !-lifting property) ≤ withC respect to f , if for all morphisms T0 T in andC for all diagrams9 of the form (1) there exists9 at most (resp. exactly) one morphism of S-schemes T X!which makesC the diagram commutative. Let ! 1 := f : T0 T closed immersion of S-schemes f is given by a locally nilpotent ideal C f ! j g 2 := f : T0 T closed immersion of S-schemes T is affine and T0 is given by a nilpotent ideal C f ! j g 2 3 := f : T0 T closed immersion of S-schemes T is the spectrum of a local ring and T0 is given by an ideal I with I = 0 C f ! j g Show that the following assertions are equivalent: (i) 1 satisfies the 1-lifting property (resp.
    [Show full text]
  • [Math.GT] 9 Jul 2003
    LOW-DIMENSIONAL HOMOLOGY GROUPS OF MAPPING CLASS GROUPS: A SURVEY MUSTAFA KORKMAZ Abstract. In this survey paper, we give a complete list of known re- sults on the first and the second homology groups of surface mapping class groups. Some known results on higher (co)homology are also men- tioned. 1. Introduction n Let Σg,r be a connected orientable surface of genus g with r boundary n components and n punctures. The mapping class group of Σg,r may be defined in different ways. For our purpose, it is defined as the group of the n n isotopy classes of orientation-preserving diffeomorphisms Σg,r → Σg,r. The diffeomorphisms and the isotopies are assumed to fix each puncture and the n points on the boundary. We denote the mapping class group of Σg,r by n Γg,r. Here, we see the punctures on the surface as distinguished points. If r and/or n is zero, then we omit it from the notation. We write Σ for the n surface Σg,r when we do not want to emphasize g,r,n. The theory of mapping class groups plays a central role in low-dimensional n topology. When r = 0 and 2g + n ≥ 3, the mapping class group Γg acts properly discontinuously on the Teichm¨uller space which is homeomorphic to some Euclidean space and the stabilizer of each point is finite. The quotient of the Teichm¨uller space by the action of the mapping class group is the moduli space of complex curves. Recent developments in low-dimensional topology made the algebraic structure of the mapping class group more important.
    [Show full text]
  • On Modeling Homotopy Type Theory in Higher Toposes
    Review: model categories for type theory Left exact localizations Injective fibrations On modeling homotopy type theory in higher toposes Mike Shulman1 1(University of San Diego) Midwest homotopy type theory seminar Indiana University Bloomington March 9, 2019 Review: model categories for type theory Left exact localizations Injective fibrations Here we go Theorem Every Grothendieck (1; 1)-topos can be presented by a model category that interprets \Book" Homotopy Type Theory with: • Σ-types, a unit type, Π-types with function extensionality, and identity types. • Strict universes, closed under all the above type formers, and satisfying univalence and the propositional resizing axiom. Review: model categories for type theory Left exact localizations Injective fibrations Here we go Theorem Every Grothendieck (1; 1)-topos can be presented by a model category that interprets \Book" Homotopy Type Theory with: • Σ-types, a unit type, Π-types with function extensionality, and identity types. • Strict universes, closed under all the above type formers, and satisfying univalence and the propositional resizing axiom. Review: model categories for type theory Left exact localizations Injective fibrations Some caveats 1 Classical metatheory: ZFC with inaccessible cardinals. 2 Classical homotopy theory: simplicial sets. (It's not clear which cubical sets can even model the (1; 1)-topos of 1-groupoids.) 3 Will not mention \elementary (1; 1)-toposes" (though we can deduce partial results about them by Yoneda embedding). 4 Not the full \internal language hypothesis" that some \homotopy theory of type theories" is equivalent to the homotopy theory of some kind of (1; 1)-category. Only a unidirectional interpretation | in the useful direction! 5 We assume the initiality hypothesis: a \model of type theory" means a CwF.
    [Show full text]
  • NOTES on the TOPOLOGY of MAPPING CLASS GROUPS Before
    NOTES ON THE TOPOLOGY OF MAPPING CLASS GROUPS NICHOLAS G. VLAMIS Abstract. This is a short collection of notes on the topology of big mapping class groups stemming from discussions the author had at the AIM workshop \Surfaces of infinite type". We show that big mapping class groups are neither locally compact nor compactly generated. We also show that all big mapping class groups are homeomorphic to NN. Finally, we give an infinite family of big mapping class groups that are CB generated and hence have a well-defined quasi-isometry class. Before beginning, the author would like to make the disclaimer that the arguments contained in this note came out of discussions during a workshop at the American Institute of Mathematics and therefore the author does not claim sole credit, espe- cially in the case of Proposition 10 in which the author has completely borrowed the proof. For the entirety of the note, a surface is a connected, oriented, second countable, Hausdorff 2-manifold. The mapping class group of a surface S is denoted MCG(S). A mapping class group is big if the underlying surface is of infinite topological type, that is, if the fundamental group of the surface cannot be finitely generated. 1.1. Topology of mapping class groups. Let C(S) denote the set of isotopy classes of simple closed curves on S. Given a finite collection A of C(S), let UA = ff 2 MCG(S): f(a) = a for all a 2 Ag: We define the permutation topology on MCG(S) to be the topology with basis consisting of sets of the form f · UA, where A ⊂ C(S) is finite and f 2 MCG(S).
    [Show full text]
  • The Birman-Hilden Theory
    THE BIRMAN{HILDEN THEORY DAN MARGALIT AND REBECCA R. WINARSKI Abstract. In the 1970s Joan Birman and Hugh Hilden wrote several papers on the problem of relating the mapping class group of a surface to that of a cover. We survey their work, give an overview of the subsequent developments, and discuss open questions and new directions. 1. Introduction In the early 1970s Joan Birman and Hugh Hilden wrote a series of now- classic papers on the interplay between mapping class groups and covering spaces. The initial goal was to determine a presentation for the mapping class group of S2, the closed surface of genus two (it was not until the late 1970s that Hatcher and Thurston [33] developed an approach for finding explicit presentations for mapping class groups). The key innovation by Birman and Hilden is to relate the mapping class group Mod(S2) to the mapping class group of S0;6, a sphere with six marked points. Presentations for Mod(S0;6) were already known since that group is closely related to a braid group. The two surfaces S2 and S0;6 are related by a two-fold branched covering map S2 ! S0;6: arXiv:1703.03448v1 [math.GT] 9 Mar 2017 The six marked points in the base are branch points. The deck transforma- tion is called the hyperelliptic involution of S2, and we denote it by ι. Every element of Mod(S2) has a representative that commutes with ι, and so it follows that there is a map Θ : Mod(S2) ! Mod(S0;6): The kernel of Θ is the cyclic group of order two generated by (the homotopy class of) the involution ι.
    [Show full text]
  • A Godefroy-Kalton Principle for Free Banach Lattices 3
    A GODEFROY-KALTON PRINCIPLE FOR FREE BANACH LATTICES ANTONIO AVILES,´ GONZALO MART´INEZ-CERVANTES, JOSE´ RODR´IGUEZ, AND PEDRO TRADACETE Abstract. Motivated by the Lipschitz-lifting property of Banach spaces introduced by Godefroy and Kalton, we consider the lattice-lifting property, which is an analogous notion within the category of Ba- nach lattices and lattice homomorphisms. Namely, a Banach lattice X satisfies the lattice-lifting property if every lattice homomorphism to X having a bounded linear right-inverse must have a lattice homomor- phism right-inverse. In terms of free Banach lattices, this can be rephrased into the following question: which Banach lattices embed into the free Banach lattice which they generate as a lattice-complemented sublattice? We will provide necessary conditions for a Banach lattice to have the lattice-lifting property, and show that this property is shared by Banach spaces with a 1-unconditional basis as well as free Banach lattices. The case of C(K) spaces will also be analyzed. 1. Introduction In a fundamental paper concerning the Lipschitz structure of Banach spaces, G. Godefroy and N.J. Kalton introduced the Lipschitz-lifting property of a Banach space. In order to properly introduce this notion, and as a motivation for our work, let us recall the basic ingredients for this construction (see [9] for details). Given a Banach space E, let Lip0(E) denote the Banach space of all real-valued Lipschitz functions on E which vanish at 0, equipped with the norm |f(x) − f(y)| kfk = sup : x, y ∈ E, x 6= y . Lip0(E) kx − yk E The Lipschitz-free space over E, denoted by F(E), is the canonical predual of Lip0(E), that is the closed ∗ linear span of the evaluation functionals δ(x) ∈ Lip0(E) given by hδ(x),fi = f(x) for all x ∈ E and all f ∈ Lip0(E).
    [Show full text]
  • Lecture 11 Last Lecture We Defined Connections on Fiber Bundles. This Lecture We Will Attempt to Explain What a Connection Does
    Lecture 11 Last lecture we defined connections on fiber bundles. This lecture we will attempt to explain what a connection does for differential ge- ometers. We start with two basic concepts from algebraic topology. The first is that of homotopy lifting property. Let us recall: Definition 1. Let π : E → B be a continuous map and X a space. We say that (X, π) satisfies homotopy lifting property if for every H : X × I → B and map f : X → E with π ◦ f = H( , 0) there exists a homotopy F : X × I → E starting at f such that π ◦ F = H. If ({∗}, π) satisfies this property then we say that π has the homotopy lifting property for a point. Now, as is well known and easily proven, every fiber bundle satisfies the homotopy lifting property. A much stronger property is the notion of a covering space (which we will not recall with precision). Here we have the homotopy lifting property for a point plus uniqueness or the unique path lifting property. I.e. there is exactly one lift of a given path in the base starting at a specified point in the total space. The point of a connection is to take the ambiguity out of the homotopy lifting property for X = {∗} and force a fiber bundle to satisfy the unique path lifting property. We will see many conceptual advantages of this uniqueness, but let us start by making this a bit clearer via parallel transport. Definition 2. Let B be a topological space and PB the category whose objects are points of B and whose morphisms are defined via {γ : → B : ∃ c, d ∈ s.t.
    [Show full text]
  • Fibrations II - the Fundamental Lifting Property
    Fibrations II - The Fundamental Lifting Property Tyrone Cutler July 13, 2020 Contents 1 The Fundamental Lifting Property 1 2 Spaces Over B 3 2.1 Homotopy in T op=B ............................... 7 3 The Homotopy Theorem 8 3.1 Implications . 12 4 Transport 14 4.1 Implications . 16 5 Proof of the Fundamental Lifting Property Completed. 19 6 The Mutal Characterisation of Cofibrations and Fibrations 20 6.1 Implications . 22 1 The Fundamental Lifting Property This section is devoted to stating Theorem 1.1 and beginning its proof. We prove only the first of its two statements here. This part of the theorem will then be used in the sequel, and the proof of the second statement will follow from the results obtained in the next section. We will be careful to avoid circular reasoning. The utility of the theorem will soon become obvious as we repeatedly use its statement to produce maps having very specific properties. However, the true power of the theorem is not unveiled until x 5, where we show how it leads to a mutual characterisation of cofibrations and fibrations in terms of an orthogonality relation. Theorem 1.1 Let j : A,! X be a closed cofibration and p : E ! B a fibration. Assume given the solid part of the following strictly commutative diagram f A / E |> j h | p (1.1) | | X g / B: 1 Then the dotted filler can be completed so as to make the whole diagram commute if either of the following two conditions are met • j is a homotopy equivalence. • p is a homotopy equivalence.
    [Show full text]
  • Algebraic Topology (Math 592): Problem Set 6
    Algebraic topology (Math 592): Problem set 6 Bhargav Bhatt All spaces are assumed to be Hausdorff and locally path-connected, i.e., there exists a basis of path-connected open subsets of X. 1. Let f : X ! Y be a covering space. Assume that there exists s : Y ! X such that f ◦ s = id. Show that s(Y ) ⊂ X is clopen. Moreover, also check that (X − s(Y )) ! Y is a covering space, provided X 6= s(Y ) and Y is path-connected. 2. Let α : Z ! Y be a map of spaces. Fix a base point z 2 Z, and assume that Z and Y are path-connected, and that Y admits a universal cover. Show that α∗ : π1(Z; z) ! π1(Y; α(z)) is surjective if and only if for every connected covering space X ! Y , the pullback X ×Y Z ! Z is connected. All spaces are now further assumed to be path-connected unless otherwise specified. 3. Let G be a topological group, and let p : H ! G be a covering space. Fix a base point h 2 H living over the identity element e 2 G. Using the lifting property, show that H admits a unique topological group structure with identity element h such that p is a group homomorphism. Using a previous problem set, show that the kernel of p is abelian. 4. Let f : X ! Y be a covering space. Assume that there exists y 2 Y such that #f −1(y) = d is finite. Show that #f −1(y0) = d for all y0 2 Y .
    [Show full text]
  • Arxiv:1706.08798V1 [Math.GT] 27 Jun 2017
    WHAT'S WRONG WITH THE GROWTH OF SIMPLE CLOSED GEODESICS ON NONORIENTABLE HYPERBOLIC SURFACES Matthieu Gendulphe Dipartimento di Matematica Universit`adi Pisa Abstract. A celebrated result of Mirzakhani states that, if (S; m) is a finite area orientable hyperbolic surface, then the number of simple closed geodesics of length less than L on (S; m) is asymptotically equivalent to a positive constant times Ldim ML(S), where ML(S) denotes the space of measured laminations on S. We observed on some explicit examples that this result does not hold for nonorientable hyperbolic surfaces. The aim of this article is to explain this surprising phenomenon. Let (S; m) be a finite area nonorientable hyperbolic surface. We show that the set of measured laminations with a closed one{sided leaf has a peculiar structure. As a consequence, the action of the mapping class group on the projective space of measured laminations is not minimal. We determine a partial classification of its orbit closures, and we deduce that the number of simple closed geodesics of length less than L on (S; m) is negligible compared to Ldim ML(S). We extend this result to general multicurves. Then we focus on the geometry of the moduli space. We prove that its Teichm¨ullervolume is infinite, and that the Teichm¨ullerflow is not ergodic. We also consider a volume form introduced by Norbury. We show that it is the right generalization of the Weil{Petersson volume form. The volume of the moduli space with respect to this volume form is again infinite (as shown by Norbury), but the subset of hyperbolic surfaces whose one{sided geodesics have length at least " > 0 has finite volume.
    [Show full text]
  • The Quillen Model Category of Topological Spaces 3
    THE QUILLEN MODEL CATEGORY OF TOPOLOGICAL SPACES PHILIP S. HIRSCHHORN Abstract. We give a complete and careful proof of Quillen’s theorem on the existence of the standard model category structure on the category of topological spaces. We do not assume any familiarity with model categories. Contents 1. Introduction 2 2. Definitions and the main theorem 2 3. Lifting 4 3.1. Pushouts, pullbacks, and lifting 5 3.2. Coproducts, transfinite composition, and lifting 5 3.3. Retracts and lifting 6 4. Relative cell complexes 7 4.1. Compact subsets of relative cell complexes 9 4.2. Relative cell complexes and lifting 10 5. The small object argument 11 5.1. The first factorization 11 5.2. The second factorization 13 6. The factorization axiom 15 6.1. Cofibration and trivial fibration 15 6.2. Trivial cofibration and fibration 16 7. Homotopygroupsandmapsofdisks 16 7.1. Difference maps 17 7.2. Lifting maps of disks 18 arXiv:1508.01942v3 [math.AT] 22 Oct 2017 8. The lifting axiom 20 8.1. Cofibrations and trivial fibrations 20 8.2. Trivial cofibrations and fibrations 21 9. The proof of Theorem 2.5 21 References 22 Date: September 11, 2017. 2010 Mathematics Subject Classification. Primary 18G55, 55U35. Key words and phrases. model category, small object argument, relative cell complex, cell complex, difference map. c 2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/. 1 2 PHILIP S. HIRSCHHORN 1. Introduction Quillen defined model categories (see Definition 2.2) in [5, 6] to apply the tech- niques of homotopy theory to categories other than topological spaces or simplicial sets.
    [Show full text]
  • Actions of Mapping Class Groups Athanase Papadopoulos
    Actions of mapping class groups Athanase Papadopoulos To cite this version: Athanase Papadopoulos. Actions of mapping class groups. L. Ji, A. Papadopoulos and S.-T. Yau. Handbook of Group Actions, Vol. I, 31, Higher Education Press; International Press, p. 189-248., 2014, Advanced Lectures in Mathematics, 978-7-04-041363-2. hal-01027411 HAL Id: hal-01027411 https://hal.archives-ouvertes.fr/hal-01027411 Submitted on 21 Jul 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ACTIONS OF MAPPING CLASS GROUPS ATHANASE PAPADOPOULOS Abstract. This paper has three parts. The first part is a general introduction to rigidity and to rigid actions of mapping class group actions on various spaces. In the second part, we describe in detail four rigidity results that concern actions of mapping class groups on spaces of foliations and of laminations, namely, Thurston’s sphere of projective foliations equipped with its projective piecewise-linear structure, the space of unmeasured foliations equipped with the quotient topology, the reduced Bers boundary, and the space of geodesic laminations equipped with the Thurston topology. In the third part, we present some perspectives and open problems on other actions of mapping class groups.
    [Show full text]