MATH 6180 Lecture Notes

Lecture notes from MATH 6180: Algebraic Topology, Part I by Jonathan Block at the University of Pennsylvania for the Fall 2024 semester.

Although the class title has the phrase "Part I", the class assumes some pre-requisite in fundamental groups, covering spaces, homology and cohomology (ie. Chapter 1 to 3 of Hatcher) and some familiarity with category theory (categories, functors, limits and colimits).

The class is divided into four parts. The topics include but are not limited to:

1. Homotopy Theory

   • A. Homotopy of maps, homotopy equivalence, homotopy groups, relative homotopy groups, Whitehead's theorem, Cellular Approximation, CW-approximation.
   • B. Calculations, Blakers-Massey Theorem (excision in homotopy theory), Freudenthal Suspension Theorem, Hurewicz Theorem, Quillen's Plus Construction.

2. Fiber Bundles and Fibrations

   • A. Homotopy Lifting Property and fibrations, LES of homotopy groups from a fibration.
   • B. Fiber bundles are fibrations.
   • C. Structure fiber bundles (fiber bundle with the action of a topological group), principal G-bundles.
   • D. Homotopy classification of fiber bundles, classfying spaces.

3. Spectral Sequences

   • A. Double complexes, filtered complexes, exact couples, spectral sequences.
   • B. The Serre homological and cohomological spectral sequences, with applications.

4. Characteristic Classes

   • A. Chern and Stiefel-Whitney classes.
   • B. Applications of characteristics classes.