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.
2. Simplicial Complexes and Simplicial Homology due by Monday, Feb 7, 2022.
3. CW-Complexes due by Monday, Feb 14, 2022.
4. Homology : Overview due by Monday, Feb 21, 2022.

Course Details

• Instructor: Siddhartha Gadgil
• E-mail: siddhartha.gadgil@gmail.com
• Office: N-15, Department of Mathematics, IISc.
• Timing: Tuesday, Thursday 8:30 am to 10:00 am
• Venue: Teams (see below) initially; LH-5, Department of Mathematics, IISc (if and when offline courses resume at IISc).
• First meeting: Tuesday, January 4, 2021.

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.

Prerecorded and live lectures

We will be using some prerecorded lectures as part of the course. The schedule for February and links are posted.

Midterm I

Midterm I will be a written, in class examination which is based on the first four assignments. This does not mean that the questions will be the same as those in the assignment or perturbations of these, but that most of the results you will need to use as well as most of the ideas needed to solve the problems are the same as in the assignments. The exam will be for 20% of the composite score. Details of the timings and venue are as below.

• Date: Tuesday, February 22, 2022.
• Time: 9:00 am to 10:00 am.
• Venue: LH-5, Department of Mathematics, IISc.

Midterm II

Midterm II will be a written, in class examination which is based on topics of the lectures up to Tuesday, March 22, 2022. Specifically all of homology theory that has been covered in the course is included as are cohomology groups. However the cup product is not in the syllabus. The exam will be for 20% of the composite score. Details of the timings and venue are as below.

• Date: Thursday, March 31, 2022.
• Time: 8:45 am to 9:45 am.
• Venue: LH-5, Department of Mathematics, IISc.

Final Examination

The final examination will be a written, in class examination which is based on all the material covered in the course. The exam will be for 50% of the composite score. Details of the timings and venue are as below.

• Date: Wednesday, April 20, 2022.
• Time: 9:00 am to 12 noon.
• Venue: LH-5, Department of Mathematics, IISc.