課程資訊

Differential Topology

102-2

MATH5324

221 U3230

Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1022dt

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.

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

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).

Office Hours

[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.

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

(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.