Course Meeting Times
Lectures: 3 sessions / week, 1 hour / session
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.
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.
The textbook for this course is:
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:
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.
Your lowest assignment grade will be not be counted towards your final grade. The remaining assignments will be given equal weight.
|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|
Finish proof of Poincare duality
Intersection pairing and cup product
|14||Lefschetz fixed point theorem|
Finish proof of Lefschetz theorem
Assorted further topics