MA 332: Algebraic Topology

January 2022

This is the second course in the Algebraic Topology series, following MA 232: Introduction to Algebraic Topology which I taught online last academic year. A reasonable background in algebra is also a prerequisite, including at least the structure theorem for (finitely generated) abelian groups.

Syllabus

Homology : Singular homology, excision, Mayer-Vietoris theorem, acyclic models, CW-complexes, simplicial and cellular homology, homology with coefficients.

Cohomology : Comology groups, relative cohomology, cup products, cap product, orientation on manifolds, Poincaré duality.

Suggested books

We will be following Hacher’s book (available online).

  1. Hatcher, A., Algebraic Topology, Cambridge Univ. Press, 2002 (Indian edition is available).
  2. Rotman, J, An Introduction to Algebraic Topology, Graduate Texts in Mathematics, 119, Springer-Verlag, 1988.
  3. Munkres, I. R., Elements of Algebraic Topology, Addison-Wesiley, 1984.

Assignments

Assignments will be posted roughly once in 2-3 weeks.

  1. Categories and Chain Complexes due by Monday, Jan 31, 2022.

Course Details

Microsoft Teams details

A Team for this course (within IISc) named Algebraic Topology 2022 has been created. Anyone from IISc can also join using the Team code 8e7xdqs, and a link to join has been posted on the IISc intranet (under Department of Mathematics). Lectures will be initially on Teams at the scheduled times, and we will move offline when permitted by the IISc policy.

Assignments will be submitted through Teams. There are Teams apps available for Windows, Linux (Debian/Ubuntu and Redhat), Mac, Android and iOS.

I would encourage using the Teams chat to ask questions whenever needed.