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研究生： 陳柏洹
Bo-Huan Chen
論文名稱： The isotopy classification of contact structures on S3
指導教授： 姚美琳
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2018
畢業學年度： 106
語文別： 英文
論文頁數： 48
中文關鍵詞： 微分幾何 、切觸幾何
外文關鍵詞： Differential geometry, Contact geometry
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    • 由Lutz、Martinet及Eliashberg的工作，我們可得知：若以同痕方式進行分類，在三維球面
    上的切觸結構已經被分類完成。
    本文將會藉由half及full Lutz twist方法，來為每一個同痕類找出更為具體且可算的代表元
    素。

    By the works of Lutz, Martinet and Eliashberg, we have known that the isotopy classes of
    contact structures on S^3 have been completely classified.
    In this thesis, we will find a representative for each class in a more explicit and computable
    form via the half and full Lutz twist.

    摘要............................................................................................................................................ i Abstract...................................................................................................................................... iii 目錄............................................................................................................................................ v 一、Introduction .......................................................................................................... 1 二、Contact Topology.................................................................................................. 3 三、Standard Contact Structure on S3........................................................................ 7 四、Lutz Twist............................................................................................................. 9 五、Basic Obstruction Theory ..................................................................................... 13 5.1 The Obstruction Cocycle of Maps . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.2 The Obstruction Cocycle of Homotopies . . . . . . . . . . . . . . . . . . . . . 13 5.3 Eilenberg-MacLane Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.4 On a Cell Decomposable 3-Manifold . . . . . . . . . . . . . . . . . . . . . . . 16 六、Cobordisms of Framed Links ................................................................................ 19 6.1 Definition of Framing and Cobordism . . . . . . . . . . . . . . . . . . . . . . . 19 6.2 Constructing Framed Cobordism from Oriented Cobordism . . . . . . . . . . 20 6.3 Relation Between Obstruction and Cobordism . . . . . . . . . . . . . . . . . . 20 七、Construction ......................................................................................................... 23 7.1 One Generator in H3(S3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 7.2 The Positive Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 7.3 The Other Generator and the Negative Classes . . . . . . . . . . . . . . . . . 28 Bibliography.................................................................................................................................... 31

    [1] Hansjörg Geiges, An Introduction to Contact Topology, Cambridge University Press, (2008).
    [2] Yakov Eliashberg, Classification of overtwisted contact structures on 3-manifolds, Invent.
    Math. 98, 623-637 (1989).
    [3] Robert Lutz, Sur quelques propriétés des formes différentielles en dimension trois, Université
    de Strasbourg, (1971).
    [4] James F. Davis and Paul Kirk, Lecture Notes in Algebraic Topology, American Mathematical
    Society, Graduate studies in mathematics ; 35, (2001).
    [5] Loring W. Tu, An Introduction to Manifolds, Springer, Second Edition, (2011).
    [6] Allen Hatcher, Algebraic Topology, Cambridge University Press, (2002).

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