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:
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.
Your lowest assignment grade will be not be counted towards your final grade. The remaining assignments will be given equal weight.
||CW-complexes, delta-complexes, simplicial homology, exact sequences, diagram chasing
||Singular homology, homotopies and chain homotopies, categories and functors, Eilenberg-Steenrod axioms
||Excision, computations for spheres, equivalence of simplicial and singular homology
||Cellular homology, Mayer-Vietoris sequences, the Mayer-Vietoris argument, homology with coefficients
||Tensor products, Tor, universal coefficient theorem for homology, products of simplices
||The Eilenberg-Zilber shuffle "product" map, diagonal approximations, the Alexander-Whitney map, method of acyclic models, Kunneth formula
||Duality, cohomology, Ext, universal coefficients for cohomology
||Projective spaces and Grassmannians, cup products, relative cup products
||Dual Kunneth formula, field coefficients, cup products in cohomology of projective spaces
||Manifolds, local orientations, global orientations
||Cap products and choices of appropriate sign conventions, statement of Poincare duality, limits
||Compactly supported cohomology, proof of Poincare duality
Finish proof of Poincare duality
Intersection pairing and cup product
||Lefschetz fixed point theorem
Finish proof of Lefschetz theorem
Assorted further topics