\( \def\bbN{{\mathbb N}} \def\bbQ{{\mathbb Q}} \def\bbR{{\mathbb R}} \def\bbZ{{\mathbb Z}} \def\calA{{\mathcal A}} \def\calD{{\mathcal D}} \def\calT{{\mathcal T}} \def\Lim{{\operatorname{Lim}}} \) © | Dror Bar-Natan: Classes: 2024-25:

MAT1301S - Algebraic Topology

Toronto, Spring 2025

Agenda. Learn about the surprising relation between the easily deformed (topology) and the most rigid (algebra).

Ambition. Get to the Wirtinger presentation of the fundamental group of knot complements and to the definition of the Alexander polynomial as the order ideal of the first homology of the universal Abelian cover of a knot complement. Both of these goals are just a bit too far, yet they can serve as perfect motivators for all that isn't too far.

Instructor. Dror Bar-Natan, drorbn@math.toronto.edu (for course administration matters only; math on email is slow and prone to misunderstandings, so I generally avoid it). Office: Bahen 6178.


Hadi Azizi

Teaching Assistant. Hadi Azizi, hadi.azizi@mail.utoronto.ca.

Classes. Mondays 1-2 and Tuesdays 2:30-4:30, at Bahen 6183.

Office Hours. Tuesdays at 9:30-10:30 at Bahen 6178 and online at https://drorbn.net/vchat.

Text. Mostly Alan Hacther's Algebraic Topology, but also several specialized sources for specialized topics.

URL. https://drorbn.net/25-1301.

Course Calendar

# Week of ... Things
1 January 6-10 Blackboards for Monday January 6.
Handout: About This Class.
Blackboards for Tuesday January 7.
Homework Assignment 1.
2 January 13-17 Blackboards for Monday January 13.
A category (groupoid) with 16 objects.
Blackboards for Tuesday January 14.
Homework Assignment 2.
3 January 20-24 Blackboards for Monday January 20: Borsuk-Ulam.
Tuesday: Class photo during break.
Blackboards for Tuesday January 21: Homotopy equivalences, van Kampen.
4 January 27-31 Blackboards for Monday January 27: van Kampen examples.
van Kampen for $T_{8/3}$.
Blackboards for Tuesday January 28: More van Kampen examples; proof of van Kampen.
Homework Assignment 3.
5 February 3-7 A mock Term Test.
Everything You Always Wanted to Know About Covering Spaces.
Blackboards for Monday February 3: examples of coverings.
Blackboards for Tuesday February 4: The fundamental theorems on covering spaces.
Thursday: our term test took place at 6-8pm at Bahen 1220.
6 February 10-14 Blackboards for Monday February 10: Proof of the fundamental theorems on covering spaces.
Blackboards for Tuesday February 11: Proof of the fundamental theorems on covering spaces.
Nobody Solves the Quintic.
Homework Assignment 4.
Friday is the last date for undergraduate students to drop this class without penalty.
R February 17-21 Monday is Family Day, and the week is Reading Week - no classes and no office hours.
7 February 24-28 Blackboards for Monday February 24: Homology basics.
Blackboards for Tuesday February 25: Functoriallity, invariance under homotopy.
Homework Assignment 5.
Wednesday was the last date for graduate students to drop this class without penalty.
8 March 3-7 Blackboards for Monday March 3: Prizms and exact sequences.
Blackboards for Tuesday March 4: short and long exact sequences, excision.
Homework Assignment 6.
9 March 10-14 Blackboards for Monday March 10: More on Excision.
Some pictures for the barycentric subdivision.
Proofs of the subdivision formulas.
Blackboards for Tuesday March 11: barycentric details, reduced homology, quotients vs. relative homology.
10 March 17-21 Blackboards for Monday March 17: Mayer-Vietoris and some Rn/Sn theorems.
Topological Pathologies in R3.
Blackboards for Tuesday March 18: The Rn/Sn theorems, invariance of domain, and Borsuk-Ulam.
Homework Assignment 7.
11 March 24-28 Tuesdsay or Wednesday: HW will likley be assigned.
12 March 31 - April 4  
F April 7-111 Wednesday: our final exam will take place at 10am-1pm at Bahen 6183.

Further resources: