MA 232: Introduction to Algebraic Topology

October 2020

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 two weeks.

  1. Introduction due by Monday, Oct 12, 2020.
  2. Paths and Homotopies due by Monday, Oct 19, 2020.
  3. Path and Homotopy Lifting due by Monday, Oct 26, 2020.
  4. Functoriality and Applications due by Monday, Nov 2, 2020.
  5. Free Groups due by Monday, Nov 9, 2020.
  6. Map Lifting and Classification of Covers due by Monday, Nov 16, 2020.
  7. Galois Theory of Covering Spaces due by Monday, Nov 23, 2020.
  8. Free Products and the Seifert-Van Kampen theorem due by Monday, Nov 30, 2020.
  9. Graphs and Two-Complexes due by Monday, Dec 7, 2020.

Course Details

This course will be taught online, mostly through pre-recorded lectures. There will also be interactive sessions on Microsoft Teams.

Microsoft Teams details

A Team for this course (within IISc) named Introduction to Algebraic Topology 2020-2021 has been created. A link to join this Team is posted on the IISc intranet (under Department of Mathematics), and people from IISc can also join using the Team code d0b3gvb. Please join the Team if you register for the course or wish to attend interactive sessions. If you choose to leave the course after some lectures you can leave the Team.

Please download the appropriate Teams app before the first interactive session. There are Apps available for Windows, Linux (Debian/Ubuntu and Redhat), Mac, Android and iOS. Once you have joined the team, whenever I (or someone else) starts a meeting you will see in the chat that a meeting has started. Please click to join.

The initial timings for the interactive sessions are Monday, Wednesday, Friday, 8:00 am - 9:00 am. Since the bulk of the course will be through pre-recorded lectures, after October 15 we may reduce meeting times. The first meeting will be at 8:00 am on Friday, October 2, 2020.

I would encourage using the Teams chat when it is set up to ask questions whenever needed. We will also meet for interactive sessions.

Assignment submissions

Assignments will be posted on this web site and linked in Microsoft Teams. They have to be submitted through Microsoft Teams.

Alternative chat

I have also set up a gitter chat channel, which has the advantage of TeX support and not requiring an IISc account, but needs an account in one of Twitter, Github and GitLab to sign in.

Examinations

Due to the difficulty of online evaluation, there will be 3 term examinations as well as a final examination. These will all be written, followed by an oral section where the student must be able to explain anything in the written proof, well enough to establish to my satisfaction that the student wrote this based on understanding, not by copying. Suspicion of copying will be strongly penalized.