Daniel A. Ramras



Homework and Lecture Notes for Math 643: Spectral Sequences in Algebraic Topology

Homework 1

Homework 2

Homework 3

Homework 4

Lecture notes:

Lecture 1: Higher Homotopy Groups

Lecture 2: Fibrations

Lecture 3: Examples of Fibrations

Lecture 4: Towers of Fibrations and Spectral Sequences

Lecture 5: The Spectral Sequence of a 3-term Filtration

Lectures 6 and 7: The Spectral Sequence of a Filtered Complex and the Serre Spectral Sequence

Lecture 8: Examples: The Cellular Chain Complex and the Unitary Groups.

Lecture 9: The E^2 term of the Serre Spectral Sequence.

Lectures 10-11: Further examples and applications of the Serre Spectral Sequence.

Lecture 12: Cohomology.

Lectures 13-14: The Serre Spectral Sequence for cohomology and applications.

Lecture 15: Vector bundles and K-theory.

Lectures 16-17: Cohomology theories and the Atiyah-Hirzebruch Spectral Sequence.

Notes on principal bundles: These are notes (from a previous course) covering the classification of principal bundles, which will be needed for the proof of Bott Periodicity.

Lecture 18: The classification of principal U(n)-bundles. (A condensed version of the above notes.)

Lecture 19: The cohomology of BU.

Lectures 20-21: The Bott map and the homology of SU(n).

Lectures 22-23: Conclusion to the proof of Bott Periodicity.