課程大綱

課程資訊

課程名稱

微分拓樸
Differential Topology

開課學期

102-2

授課對象

理學院數學研究所

授課教師

蔡忠潤

課號

MATH5324

課程識別碼

221 U3230

班次

學分

全/半年

半年

必/選修

選修

上課時間

星期一3(10:20~11:10)星期四7,8(14:20~16:20)

上課地點

天數102天數102

備註

總人數上限：40人

Ceiba 課程網頁

http://ceiba.ntu.edu.tw/1022dt

課程簡介影片

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課程概述

The main course website is the following:
http://homepage.ntu.edu.tw/~cjtsai/teaching/14dt.html

1. Sard’s theorem and transversality argument.
2. Morse function.
3. Degree, Poincare--Hopf theorem, framed cobordism and Hopf theorem.
4. Vector bundles, Thom isomorphism and Euler class.
5. Chern classes and Pontrjagin classes.

課程目標

The main purpose of this course is to emphasize how to study the topology of smooth manifolds and smooth vector bundles using differentiable techniques.

課程要求

1. General Topology (topological spaces, product topology, quotient topology and quotient maps,
continuity, compactness, connectedness).
2. Differentiable manifolds (tangent spaces, differential maps, differential forms).
3. Basic algebraic topology (homology group and its Mayer--Vietoris sequence).

預期每週課後學習時數

Office Hours

每週五 16:00~17:00

指定閱讀

[GP] Victor Guillemin and Alan Pollack, Differential topology.

參考書目

[H] Morris Hirsch, Differential topology.
[M1] John Milnor, Topology from the differentiable viewpoint.
[M2] John Milnor, Differential topology. (1958 Princeton lecture notes by James
Munkres.)
[M3] John Milnor, Morse Theory.
[BT] Raoul Bott and Loring Tu, Differential forms in algebraic topology.

評量方式
(僅供參考)

No.	項目	百分比	說明
1.	Homework	30%
2.	Midterm	30%
3.	Final exam/report	30%
4.	Course participation	10%

課程進度

週次	日期	單元主題
第7週	3/31	oriented intersection theory (continued). Reference: [GP, §III.3].
第16週	6/05	cohomology and holomogy with Z/2 coefficient
第1-1週	2/17	introduction and overview
第1-2週	2/20	manifolds, immersion and embedding. Reference: [GP, §I.1~§I.3].
第2-1週	2/24	submersion. Reference: [GP, §I.4].
第2-2週	2/27	transversal and homotopy. Reference: [GP, §I.5 ~ §I.6].
第3-1週	3/03	Sard theorem. Reference: [M1, §3].
第3-2週	3/06	tangent bundle, partition of unity, (weak) Whitney embedding theorem. Reference: [GP, §I.8].
第4-1週	3/10	manifold with boundary. Reference: [GP, §II.1].
第4-2週	3/13	Brouwer fixed point theorem, parametric transversality theorem. Reference: [GP, §II.2 ~ §II.3].
第5-1週	3/17	mod 2 intersection number. Reference: [GP, §II.3 ~ §II.4].
第5-2週	3/20	mod 2 intersection number (continued), the Jordan--Brouwer separation theorem. Reference: [GP, §II.4 ~ §II.5].
第6-1週	3/24	the Borsuk--Ulam theorem. Reference: [GP, §II.6].
第6-2週	3/27	orientation, oriented intersection number, fundamental theorem of algebra. Reference: [GP, §III.2 ~ §III.3].
第8-1週	4/07	Lefschetz fixed-point theory. Reference: [GP, §III.4].
第8-2週	4/10	Lefschetz fixed-point theory (continued), the Poincare--Hopf theorem. Reference: [GP, §III.4 ~ §III.5].
第9-1週	4/14	the Poincare--Hopf theorem (continued). Reference: [GP, §III.5].
第9-2週	4/17	framed cobordism theory. Reference: [M1, §7].
第10-1週	4/21	framed cobordism theory (continued). Reference: [M1, §7].
第10-2週	4/24	the Hopf theorem, fiber bundle, linking number. Reference: [M1, §7].
第11-1週	4/28	linking number (continued). Reference: [M1, §Exercise].
第11-2週	5/01	Hopf invariant, vector bundles. Reference: [M1, §Exercise], [M2] and [MS, §2].
第12-1週	5/05	vector bundles (continued). Reference: [M2] and [MS, §2].
第12-2週	5/08	kernel and quotient bundles, Grassmannian manifolds. Reference: [M2].
第13-1週	5/12	universal bundle. Reference: [M2].
第13-2週	5/15	(unoriented) cobordism group, Thom's theorem, homotopy group. Reference: [M2].
第14-1週	5/19	approximation lemmata. Reference: [M2].
第14-2週	5/22	the map from the homotopy group of the Thom space to the cobordism group, the surjectivity of the map. Reference: [M2].
第15-1週	5/26	Whitney immersion and embedding revisted. Reference: [M2].
第15-2週	5/29	the injectivity of the map from the homotopy group to the cobordism group, the principal bundle of the universal bundle. Reference: [M2].
第17-1週	6/09	duality between the chain complexes, the cup product.
第17-2週	6/12	cohomology ring of real projective spaces, projectification of a vector bundle, Stiefel--Whitney classes.