Differential Topology, Spring 2014

Course Information

• Syllabus
• Information
• Ceiba
• Lectures: Monday, 10:20 ~ 11:10 and Thursday 15:30 ~ 17:20 at Astro-Math 102
• Office hours: Friday 16:00 ~ 17:00 or by appointment, at Astro-Math 458
• Grading scheme:
  • Homework 30%
    You have three jokers: the lowest three grades will be discarded.
  • Midterm 30%
  • Final report 30%
    The written report is due 6PM, Monday, June 23.
  • Course participation 10%
• References:
  • [GP] Victor Guillemin and Alan Pollack, Differential topology. MR
  • [H] Morris Hirsch, Differential topology. MR
  • [M1] John Milnor, Topology from the differentiable viewpoint. MR
  • [M2] John Milnor, Differential topology: the 1958 Princeton lecture notes by James Munkres. MR
  • [M3] John Milnor, Morse Theory. MR
  • [MS] John Milnor and James Stasheff, Characteristic classes. MR
  • [BT] Raoul Bott and Loring Tu, Differential forms in algebraic topology. MR

Lecture summaries and references

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

Homework

• Homework 01: due February 24.
• Homework 02: due March 3.
• Homework 03: due March 10.
• Homework 04: due March 17.
• Homework 05: due March 24.
• Homework 06: due March 31.
• Homework 07: due April 7.
• Homework 08: due April 14.
• Homework 09: due April 21.
• Homework 10: due May 5.
• Homework 11: due May 12.
• Homework 12: due May 19.
• Homework 13: due May 26.