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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.
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