MA 332: Algebraic Topology

January 2024

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


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 will be posted roughly once in 2-3 weeks. Instead of submitting assignments, students will be asked to solve the problems in lecture, typically the lecture after the due date.

  1. $H_0$ and reduced homology due by Monday, Feb 5, 2024.

Course Details

Midterm Examination

The syllabus for the midterm examination is the material covered in lectures up to Thursday, February 15, 2024.

Microsoft Teams details

A Team for this course (within IISc) named Algebraic Topology 2024 has been created. Anyone from IISc can also join using the Team code znr3hi8.

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.