1. Categories and Chain Complexes (31/1/2022).
  2. Simplicial Complexes and Simplicial Homology (7/2/2022).
  3. CW-Complexes (14/2/2022).
  4. Homology : Overview (21/2/2022).
Hybrid format for lectures

As discussed earlier, some of the material that is naturally part of this course and has not been covered in versions of the first course in Algebraic Topology various students have taken is available in the form of prerecorded lectures that I made for MA 232: Introduction to Algebraic Topology which I taught in the first term of the academic year 202-21. So for the month of February, we will follow a format that mixes the present format of live Teams lectures with one of interactive sessions to discuss prerecorded lectures.

Some of the live lectures will depend on material in prerecorded lectures. I have made a schedule, given below, to account for this. Assignments will also be given based on this.

Midterm I

In the last week of February we will have Midterm I as an oral examination which is based on assignments. This means that I will set up individual Teams meetings with each of you and ask questions that are from the assignments, but seeking more details, or are closely related to the assignments. I will specify which assignments are included before the midterm. The exam will be for 20% of the composite score.

The oral examinations are tentatively on Tuesday, February 22 and Thursday, February 24. Ideally I would like to have these on a single day, if most of you are free from 10:00 - 11:30. Please let me know if you have a lecture in the time slot Tuesday-Thursday 10:00 - 11:30 as soon as possible, and certainly by Friday, February 4, 2021.

Topics in the prerecorded lectures

There are four topics that are covered in the prerecorded lectures (at http://math.iisc.ac.in/~gadgil/introduction-algebraic-topology-2020/all-lectures/) that will be part of the course.

  1. Simplicial complexes and Simplicial homology: Lectures 27 and 28 (these do not depend on the earlier lectures).

  2. CW complexes (and surfaces): Lectures 20 and 22.

  3. Higher homotopy groups: Lectures 23 and 24

  4. Whitehead theory (introduction): Lecture 25.

The first two are prerequisites in February as per the schedule below. The third will be a prerequisite for a topic that is later in the course. The fourth is part of the syllabus but will not be required for any live lecture.

Weekly schedule for February

Below is the tentative schedule for the lectures and interactive sessions. Note that some of the material of the lectures on Tuesday may spill over into the interactive sessions. The interactive sessions are to answer questions about the prerecorded lectures, which it is assumed that you have seen by the time of the session.

  1. Week 1: February 1 and 3: No prerequisites from recorded lectures.

    • * Tuesday, February 1 (lecture) : Degree of a map; application: vector fields on spheres; local homology and degree
    • * Thursday, February 3 (interactive session) : Simplicial complexes and simplicial homology

  2. Week 2: February 8 and 10: I will assume familiarity with Simplicial complexes and simplicial homology

    • * Tuesday, February 8 (lecture) : Simplicial approximation, Euler characteristing and the Lefschetz fixed point theorem.
    • * Thursday, February 10 (interactive session) : CW complexes (and surfaces)

  3. Week 3: February 15 and 17: I will also assume familiarity with CW complexes

    • * Tuesday, February 15 and Thursday, February 17: Cellular homology, examples and applications of homology.

  4. Week 4: February 22 and 24: Midterm I