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課程名稱 |
流形分類導論
Introduction to classifications of manifolds |
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開課學期 |
112-2 |
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授課對象 |
理學院 數學系 |
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授課教師 |
王金龍 |
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課號 |
MATH5300 |
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課程識別碼 |
221 U9420 |
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班次 |
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學分 |
4.0 |
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全/半年 |
半年 |
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必/選修 |
選修 |
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上課時間 |
星期一3,4(10:20~12:10)星期三3,4(10:20~12:10) |
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上課地點 |
天數101天數101 |
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備註 |
總人數上限：30人 |
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課程簡介影片 |
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核心能力關聯 |
本課程尚未建立核心能力關連 |
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課程大綱
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為確保您我的權利,請尊重智慧財產權及不得非法影印
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課程概述 |
The following topics will be covered:
(a) Proof of h-cobordism theorem and generalized Poincare conjecture for dim 5 or higher.
(b) Discussions on dim 4, including an introduction to Donaldson/Freedman’s results.
(c) Cobordism theory, characteristic classes and exotic differentiable structures.
(d) C.T.C Wall’s theory on surgery in dimension 6 and its role in classification of Calabi—Yau 3-folds. |
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課程目標 |
The course intends to provide a basic training in differentiable topology and an introduction to classification theory of higher dimensional manifolds. After going through some details of famous historic advances in general higher dimensions theories, notably characteristic classes, h-cobordism and discoveries of exotic differentiable structures, I will give an introduction to the 4-dimensional story via Donaldson’s theory.
In the second half of the course, I will focus on C.T.C. Wall’s theory on surgery and its application to the classification of 6 dimensional manifolds with almost complex structures. The main goal is to understand the possible obstructions to relate two higher dimensional manifolds of the same dimension and with vanishing first Chern class. The missing details in the class will be assigned as final report topics. |
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課程要求 |
Students are required to have basic knowledge in manifolds and topology as in
[1] Differential geometry on manifolds in the level of Warner (GTM 94), Do Carmo (Riemannian geometry), or chapter 1 to 5 in my 2020 graduate course (lectures notes available in my course page).
[2] Algebraic topology in the level of Vick: Homology Theory, or Hatcher: Algebraic Topology.
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預期每週課後學習時數 |
12 hours |
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Office Hours |
每週四 14:00~15:00
每週五 09:00~10:00 |
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指定閱讀 |
[1] J. Milnor: Lectures on h-cobordism theorem, Princeton 1965.
[2] J.D. Moore: Lectures Notes on Seiberg-Witten Invariants (revised second edition), 2010.
[3] D. Salamon: Spin Geometry and Seiberg-Witten Invariants, 1994.
[4] C.T.C. Wall: Differential Topology (2016).
[5] C.T.C. Wall: Classification Problems in Differential Topology. V, on certain 6-manifolds, Invent 1, 355-374 (1966). |
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參考書目 |
[1] M. Hirsch: Differential Topology, GTM 33.
[2] Dubrovin—Fomenko—Novikov: Modern Geometry Part II, III, GTM 104, 124, Springer 1990.
[3] C.T.C. Wall: Surgery on Compact Manifolds, second edition, AMS 1999.
[4] W. L\"uck: A Basic Introduction to Surgery Theory, Lecture notes 2001.
[5] Gompf and Stipsicz: 4-Manifolds and Kirby Calculus, AMS 1999. |
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評量方式
(僅供參考) |
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針對學生困難提供學生調整方式 |
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上課形式 |
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作業繳交方式 |
延長作業繳交期限, 個人報告取代團體報告 |
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考試形式 |
書面(口頭)報告取代考試 |
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其他 |
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週次 |
日期 |
單元主題 |
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Week 1 |
2/19,2/21 |
Introduction lecture. h-cobordism theorem I: Morse theory |
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Week 2 |
2/26,2/28 |
h-cobordism theorem II: First cancellation theorem (Morse) |
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Week 3 |
3/04,3/06 |
h-cobordism theorem III: Second cancellation theorem (Whitney) |
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Week 4 |
3/11,3/13 |
h-cobordism theorem IV: Basis Theorem. Final step on index 0 and 1 |
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Week 5 |
3/18,3/20 |
Seiberg--Witten theory I: Spin geometry and SW equation |
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Week 6 |
3/25,3/27 |
Seiberg--Witten theory II: SW moduli spaces and invariants |
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Week 7 |
4/01,4/03 |
Seiberg--Witten theory III: Donaldson's theorems etc. |
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Week 8 |
4/08,4/10 |
Seiberg--Witten theory IV: Complexity of 4D h-cobordism |
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Week 9 |
4/15,4/17 |
Whitehead torsion and s-cobordism theorem |
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Week 10 |
4/22,4/24 |
Thom's cobordism theory and weak J structures |
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Week 11 |
4/29,5/01 |
Cohomology of classification spaces and cobordism rings |
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Week 12 |
5/06,5/08 |
Wall's surgery theory I |
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Week 13 |
5/13,5/15 |
Wall's surgery theory II |
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Week 14 |
5/20,5/22 |
Wall's surgery theory III |
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