Syllabus

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:

Amazon logo 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:

Amazon logo Massey, William S. A Basic Course in Algebraic Topology. New York, NY: Springer-Verlag, 1997. ISBN: 9780387974309.

Amazon logo Rotman, Joseph J. An Introduction to Algebraic Topology. New York, NY: Springer-Verlag, 1998. ISBN: 9780387966786.

Amazon logo 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