Math 529: Differential Topology (Fall 2024)

Class Time: MWF 10-11am

Instructor: Sébastien Picard
Email: spicard@math
Office: MATH 236
Syllabus: Can be found here.

Topics: Differential forms and de Rham cohomology. The Poincare dual of a submanifold and intersection theory. Double complexes and spectral sequences.

Main Reference:
[BT] R. Bott and L.W. Tu, Differential Forms in Algebraic Topology

Additional References:
[GH] P. Griffiths and J. Harris, "Principles of Algebraic Geometry"
[Spivak] M. Spivak, "A Comprehensive Introduction to Differential Geometry - Volume One"
[Viaclovsky] J. Viaclovsky, Lecture Notes on Algebraic Topology
[Nicolaescu] L. Nicolaescu, Lectures on the Geometry of Manifolds

Problem Sets:
Homework 1 - Due Sept 20
Homework 2 - Due Oct 11
Homework 3 - Due Oct 25
Homework 4 - Due Nov 29

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Lecture Notes:
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(1) Review of differential forms
[BT] 1.1

(2) Integration over manifolds
[BT] 1.3

(3) The de Rham cohomology
[BT] 1.4

(4) Mayer-Vietoris sequence
[BT] 1.2

(5) Poincare duality
[BT] 1.5

(6) Vector bundles
[BT] 1.6, [GH] 0.5, 1.1

(7) Thom isomorphism
[BT] 1.6

(8) U(1) bundles
[BT] 1.6

(9) Euler class and Poincare duality
[BT] 2.12

(10) Intersection product
[GH] 1.1, 4.1

(11) Euler class of the tangent bundle
[BT] 2.11

(12) Cech cohomology
[BT] 2.10

(13) Spectral sequence of a double complex
[BT] 3.14

(14) Spectral sequence of a fiber bundle
[BT] 3.14, 2.13