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 graduatelevel 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: SpringerVerlag, 1997. ISBN: 9780387974309.
Rotman, Joseph J. An Introduction to Algebraic Topology. New York, NY: SpringerVerlag, 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 inclass. 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
Course calendar.
WEEK # 
TOPICS 
1 
CWcomplexes, deltacomplexes, simplicial homology, exact sequences, diagram chasing 
2 
Singular homology, homotopies and chain homotopies, categories and functors, EilenbergSteenrod axioms 
3 
Excision, computations for spheres, equivalence of simplicial and singular homology 
4 
Cellular homology, MayerVietoris sequences, the MayerVietoris argument, homology with coefficients 
5 
Tensor products, Tor, universal coefficient theorem for homology, products of simplices 
6 
The EilenbergZilber shuffle "product" map, diagonal approximations, the AlexanderWhitney 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
