The isotopy classification of contact structures on S3

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陳柏洹(Bo-Huan Chen) 查詢紙本館藏

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論文名稱

(The isotopy classification of contact structures on S3)

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摘要(中)

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。

摘要(英)

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.

關鍵字(中)

★ 微分幾何
★ 切觸幾何

關鍵字(英)

★ Differential geometry
★ Contact geometry

論文目次

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

參考文獻

[1] Hansjorg 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 proprietes des formes differentielles en dimension trois, Universite
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).

指導教授

姚美琳

審核日期

2018-7-24

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