1. Homotopy as Path of Paths (23/8/2023).
  2. Components & Inclusions (11/9/2023).
  3. Strange Covers (2/11/2023).
  4. Fundamental Groups of Graphs (20/11/2023).

MA 232: Introduction to Algebraic Topology

August 2023

Syllabus

The fundamental group: Homotopy of maps, multiplication of paths, the fundamental group, induced homomorphisms, the fundamental group of the circle, covering spaces, lifting theorems, the universal covering space, Seifert-van Kampen theorem, applications.

Simplicial Homology: Simplicial complexes, chain complexes, definitions of the simplicial homology groups, properties of homology groups, applications.

Suggested books:

  1. Hatcher, A., Algebraic Topology, Cambridge University Press, 2002.
  2. Armstrong, M.A., Basic Topology, Springer (India), 2004.

Assignments

Assignments will be posted roughly once in 2 weeks.

  1. Homotopy as Path of Paths due by Wednesday, Aug 23, 2023.
  2. Components & Inclusions due by Monday, Sep 11, 2023.
  3. Strange Covers due by Thursday, Nov 2, 2023.
  4. Fundamental Groups of Graphs due by Monday, Nov 20, 2023.

Course Details

Microsoft Teams details

A Team for this course (within IISc) named Introduction to Algebraic Topology 2023 has been created. Anyone from IISc can also join using the Team code u1hkivl. 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.

Assignment submissions

Assignments will be posted on this web site and linked in Microsoft Teams. They have to be submitted through Microsoft Teams. The ea siest way to do this is to write on paper, photograph or scan this and submit pdf or image files. It is recommended that you use a n app like Adobe scan or Microsoft Lens to generate a single pdf file for each assignment (instead of a collection of images).

If you wish you can instead use LaTeX and submit pdf files.

Additional Resources

This course was taught online in 2021. The lectures are online.

Note that students are responsible for all the material covered in the lectures this semester, which is likely to be more than that in the above resources. Experience also suggests that offline lectures are more effective. Thus, it is wise to use the above as supplements, not substitutes, for the lectures.

Midterm Examination

Date: Wednesday, September 27, 2023.

Time: 10:00 - 11:30 am.

Venue: LH-5, Department of Mathematics, IISc.

Weightage: 40% of the total marks.

Syllabus: Till the lecture on Monday, September 18, 2023.

Final Examination

Date: Friday, December 1, 2023.

Time: 9:00 am - 12:00 noon.

Venue: LH-5, Department of Mathematics, IISc.

Weightage: 50% of the total marks.

Syllabus: The entire course.