Course Meeting Times
Lectures: 3 sessions / week, 1 hour / session
Prerequisite
The prerequisites for the course are basic point set topology, such as Introduction to Topology (18.901 or equivalent) and algebra, such as Algebra I (18.701) or Modern Algebra (18.703) or equivalent.
Course Overview
This course is intended as a graduate-level introduction to the machinery of algebraic topology. Specifically, we will focus on singular homology and the dual theory of singular cohomology.
Text
The textbook for this course is:
Hatcher, Allen. Algebraic Topology. Cambridge, UK: Cambridge University Press, 2002. ISBN: 9780521795401.
The text is freely available online, but paperback copies are also available. We will be focusing on chapters 2 and 3.
Other texts you might find interesting or useful include the following:
Massey, William S. A Basic Course in Algebraic Topology. New York, NY: Springer-Verlag, 1997. ISBN: 9780387974309.
Rotman, Joseph J. An Introduction to Algebraic Topology. New York, NY: Springer-Verlag, 1998. ISBN: 9780387966786.
Munkres, James R. Elements of Algebraic Topology. Boulder, CO: Westview Press, 1993. ISBN: 9780201627282.
Exams and Assignments
There are no exams for this course. Your grade for this course will be based on weekly assignments due each Wednesday in-class. There are 12 assignments in all.
Grading
Your lowest assignment grade will be not be counted towards your final grade. The remaining assignments will be given equal weight.
Calendar
| WEEK # | TOPICS |
|---|---|
| 1 | CW-complexes, delta-complexes, simplicial homology, exact sequences, diagram chasing |
| 2 | Singular homology, homotopies and chain homotopies, categories and functors, Eilenberg-Steenrod axioms |
| 3 | Excision, computations for spheres, equivalence of simplicial and singular homology |
| 4 | Cellular homology, Mayer-Vietoris sequences, the Mayer-Vietoris argument, homology with coefficients |
| 5 | Tensor products, Tor, universal coefficient theorem for homology, products of simplices |
| 6 | The Eilenberg-Zilber shuffle "product" map, diagonal approximations, the Alexander-Whitney map, method of acyclic models, Kunneth formula |
| 7 | Duality, cohomology, Ext, universal coefficients for cohomology |
| 8 | Projective spaces and Grassmannians, cup products, relative cup products |
| 9 | Dual Kunneth formula, field coefficients, cup products in cohomology of projective spaces |
| 10 | Manifolds, local orientations, global orientations |
| 11 | Cap products and choices of appropriate sign conventions, statement of Poincare duality, limits |
| 12 | Compactly supported cohomology, proof of Poincare duality |
| 13 |
Finish proof of Poincare duality Intersection pairing and cup product |
| 14 | Lefschetz fixed point theorem |
| 15 |
Finish proof of Lefschetz theorem Assorted further topics |