$ \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

$$ $Essentials $$

Course Calendar

# Week of ... Things

1 January 6-10 $$framed_page {name => 'BB-0106', title => 'Blackboards for Monday January 6', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250106-142329'}$$.
Handout: $$make_page {name => 'About', title => 'About This Class'}$$.
$$framed_page {name => 'BB-0107', title => 'Blackboards for Tuesday January 7', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250107-162817'}$$.
$$framed_page {name => 'HW01', title => 'Homework Assignment 1', external => 0, url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/HW01.pdf'}$$.

2 January 13-17 $$framed_page {name => 'BB-0113', title => 'Blackboards for Monday January 13', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250114-075239'}$$.
$$framed_page {name => '15Puzzle', title => 'A category (groupoid) with 16 objects', url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/15Puzzle.png'}$$.
$$framed_page {name => 'BB-0114', title => 'Blackboards for Tuesday January 14', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250115-112346'}$$.
$$framed_page {name => 'HW02', title => 'Homework Assignment 2', external => 0, url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/HW02.pdf'}$$.

3 January 20-24 $$framed_page {name => 'BB-0120', title => 'Blackboards for Monday January 20: Borsuk-Ulam', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250121-162901'}$$.
Tuesday: Class photo during break.
$$framed_page {name => 'BB-0121', title => 'Blackboards for Tuesday January 21: Homotopy equivalences, van Kampen', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250122-064610'}$$.

4 January 27-31 $$framed_page {name => 'BB-0127', title => 'Blackboards for Monday January 27: van Kampen examples', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250130-145613'}$$.
$$framed_page {name => 'T83', title => 'van Kampen for $T_{8/3}$', url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/nb/T83.pdf'}$$.
$$framed_page {name => 'BB-0128', title => 'Blackboards for Tuesday January 28: More van Kampen examples; proof of van Kampen', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250130-154049'}$$.
$$framed_page {name => 'HW03', title => 'Homework Assignment 3', external => 0, url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/HW03.pdf'}$$.

5 February 3-7 $$framed_page {name => 'MockTT', title => 'A mock Term Test', url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/25-1301-MockTT.pdf'}$$.
$$framed_page {name => 'CoveringSpaces', title => 'Everything You Always Wanted to Know About Covering Spaces', url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/CoveringSpaces.pdf'}$$.
$$framed_page {name => 'BB-0203', title => 'Blackboards for Monday February 3: examples of coverings', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250206-072721'}$$.
$$framed_page {name => 'BB-0204', title => 'Blackboards for Tuesday February 4: The fundamental theorems on covering spaces', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250206-073359'}$$.
Thursday: our $$framed_page {name => 'TT', title => 'Term Test', url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/25-1301-TT.pdf'}$$ took place at 6-8pm at Bahen 1220.

6 February 10-14 $$framed_page {name => 'BB-0210', title => 'Blackboards for Monday February 10: Proof of the fundamental theorems on covering spaces', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250213-084628'}$$.
$$framed_page {name => 'BB-0211', title => 'Blackboards for Tuesday February 11: Proof of the fundamental theorems on covering spaces', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250213-084759'}$$.
$$framed_page {name => 'Quintic', title => 'Nobody Solves the Quintic', url => 'https://drorbn.net/AcademicPensieve/Talks/CUMC-200820/Quintic-Handout.pdf'}$$.
$$framed_page {name => 'HW04', title => 'Homework Assignment 4', external => 0, url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/HW04.pdf'}$$.
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 $$framed_page {name => 'BB-0224', title => 'Blackboards for Monday February 24: Homology basics', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250224-165159'}$$.
$$framed_page {name => 'BB-0225', title => 'Blackboards for Tuesday February 25: Functoriallity, invariance under homotopy', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250303-143312'}$$.
$$framed_page {name => 'HW05', title => 'Homework Assignment 5', external => 0, url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/HW05.pdf'}$$.
Wednesday was the last date for graduate students to drop this class without penalty.

8 March 3-7 $$framed_page {name => 'BB-0303', title => 'Blackboards for Monday March 3: Prizms and exact sequences', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250303-143832'}$$.
$$framed_page {name => 'BB-0304', title => 'Blackboards for Tuesday March 4: short and long exact sequences, excision', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250305-172545'}$$.
$$framed_page {name => 'HW06', title => 'Homework Assignment 6', external => 0, url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/HW06.pdf'}$$.

9 March 10-14 $$framed_page {name => 'BB-0310', title => 'Blackboards for Monday March 10: More on Excision', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250311-111221'}$$.
$$framed_page {name => 'BarycentricSubdivision', title => 'Some pictures for the barycentric subdivision', url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/BarycentricSubdivision.png'}$$.
$$framed_page {name => 'SubdivisionProofs', title => 'Proofs of the subdivision formulas', url => 'https://drorbn.net/AcademicPensieve/Classes/17-SummerHomology/AccraNotes-5-3.jpg'}$$.
$$framed_page {name => 'BB-0311', title => 'Blackboards for Tuesday March 11: barycentric details, reduced homology, quotients vs. relative homology', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250314-155310'}$$.

10 March 17-21 $$framed_page {name => 'BB-0317', title => 'Blackboards for Monday March 17: Mayer-Vietoris and some Rn/Sn theorems', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250318-112344'}$$.
$$framed_page {name => 'Pathologies', title => 'Topological Pathologies in R3', url => 'https://www.math.toronto.edu/drorbn/classes/0102/AlgTop/Pathologies/'}$$.
$$framed_page {name => 'BB-0318', title => 'Blackboards for Tuesday March 18: The Rn/Sn theorems, invariance of domain, and Borsuk-Ulam', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250318-185347'}$$.
$$framed_page {name => 'HW07', title => 'Homework Assignment 7', external => 0, url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/HW07.pdf'}$$.

11 March 24-28 $$framed_page {name => 'BB-0324', title => 'Blackboards for Monday March 24: Proof of Borsuk-Ulam', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250325-110650'}$$.
$$framed_page {name => 'BB-0325', title => 'Blackboards for Tuesday March 25: Degrees, CW complexes', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250325-140557'}$$.
$$framed_page {name => 'HW08', title => 'Homework Assignment 8', external => 0, url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/HW08.pdf'}$$.

12 March 31 - April 4 $$framed_page {name => 'Axioms4HomologyFollowingBredon', title => 'Axioms For Homology Following Bredon', url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/Axioms4HomologyFollowingBredon.html'}$$.
$$framed_page {name => 'BB-0331', title => 'Blackboards for Monday March 31: Axiomatics', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250401-095204'}$$.
$$framed_page {name => 'BB-0401', title => 'Blackboards for Tuesday April 1: CW homology proof and examples', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250408-150320'}$$.

F April 7-111 Wednesday April 9: our final exam will take place at 10am-1pm at Bahen 6183.

#	Week of ...	Things
1	January 6-10	$$framed_page {name => 'BB-0106', title => 'Blackboards for Monday January 6', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250106-142329'}$$. Handout: $$make_page {name => 'About', title => 'About This Class'}$$. $$framed_page {name => 'BB-0107', title => 'Blackboards for Tuesday January 7', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250107-162817'}$$. $$framed_page {name => 'HW01', title => 'Homework Assignment 1', external => 0, url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/HW01.pdf'}$$.
2	January 13-17	$$framed_page {name => 'BB-0113', title => 'Blackboards for Monday January 13', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250114-075239'}$$. $$framed_page {name => '15Puzzle', title => 'A category (groupoid) with 16 objects', url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/15Puzzle.png'}$$. $$framed_page {name => 'BB-0114', title => 'Blackboards for Tuesday January 14', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250115-112346'}$$. $$framed_page {name => 'HW02', title => 'Homework Assignment 2', external => 0, url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/HW02.pdf'}$$.
3	January 20-24	$$framed_page {name => 'BB-0120', title => 'Blackboards for Monday January 20: Borsuk-Ulam', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250121-162901'}$$. Tuesday: Class photo during break. $$framed_page {name => 'BB-0121', title => 'Blackboards for Tuesday January 21: Homotopy equivalences, van Kampen', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250122-064610'}$$.
4	January 27-31	$$framed_page {name => 'BB-0127', title => 'Blackboards for Monday January 27: van Kampen examples', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250130-145613'}$$. $$framed_page {name => 'T83', title => 'van Kampen for $T_{8/3}$', url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/nb/T83.pdf'}$$. $$framed_page {name => 'BB-0128', title => 'Blackboards for Tuesday January 28: More van Kampen examples; proof of van Kampen', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250130-154049'}$$. $$framed_page {name => 'HW03', title => 'Homework Assignment 3', external => 0, url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/HW03.pdf'}$$.
5	February 3-7	$$framed_page {name => 'MockTT', title => 'A mock Term Test', url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/25-1301-MockTT.pdf'}$$. $$framed_page {name => 'CoveringSpaces', title => 'Everything You Always Wanted to Know About Covering Spaces', url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/CoveringSpaces.pdf'}$$. $$framed_page {name => 'BB-0203', title => 'Blackboards for Monday February 3: examples of coverings', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250206-072721'}$$. $$framed_page {name => 'BB-0204', title => 'Blackboards for Tuesday February 4: The fundamental theorems on covering spaces', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250206-073359'}$$. Thursday: our $$framed_page {name => 'TT', title => 'Term Test', url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/25-1301-TT.pdf'}$$ took place at 6-8pm at Bahen 1220.
6	February 10-14	$$framed_page {name => 'BB-0210', title => 'Blackboards for Monday February 10: Proof of the fundamental theorems on covering spaces', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250213-084628'}$$. $$framed_page {name => 'BB-0211', title => 'Blackboards for Tuesday February 11: Proof of the fundamental theorems on covering spaces', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250213-084759'}$$. $$framed_page {name => 'Quintic', title => 'Nobody Solves the Quintic', url => 'https://drorbn.net/AcademicPensieve/Talks/CUMC-200820/Quintic-Handout.pdf'}$$. $$framed_page {name => 'HW04', title => 'Homework Assignment 4', external => 0, url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/HW04.pdf'}$$. 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	$$framed_page {name => 'BB-0224', title => 'Blackboards for Monday February 24: Homology basics', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250224-165159'}$$. $$framed_page {name => 'BB-0225', title => 'Blackboards for Tuesday February 25: Functoriallity, invariance under homotopy', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250303-143312'}$$. $$framed_page {name => 'HW05', title => 'Homework Assignment 5', external => 0, url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/HW05.pdf'}$$. Wednesday was the last date for graduate students to drop this class without penalty.
8	March 3-7	$$framed_page {name => 'BB-0303', title => 'Blackboards for Monday March 3: Prizms and exact sequences', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250303-143832'}$$. $$framed_page {name => 'BB-0304', title => 'Blackboards for Tuesday March 4: short and long exact sequences, excision', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250305-172545'}$$. $$framed_page {name => 'HW06', title => 'Homework Assignment 6', external => 0, url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/HW06.pdf'}$$.
9	March 10-14	$$framed_page {name => 'BB-0310', title => 'Blackboards for Monday March 10: More on Excision', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250311-111221'}$$. $$framed_page {name => 'BarycentricSubdivision', title => 'Some pictures for the barycentric subdivision', url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/BarycentricSubdivision.png'}$$. $$framed_page {name => 'SubdivisionProofs', title => 'Proofs of the subdivision formulas', url => 'https://drorbn.net/AcademicPensieve/Classes/17-SummerHomology/AccraNotes-5-3.jpg'}$$. $$framed_page {name => 'BB-0311', title => 'Blackboards for Tuesday March 11: barycentric details, reduced homology, quotients vs. relative homology', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250314-155310'}$$.
10	March 17-21	$$framed_page {name => 'BB-0317', title => 'Blackboards for Monday March 17: Mayer-Vietoris and some Rn/Sn theorems', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250318-112344'}$$. $$framed_page {name => 'Pathologies', title => 'Topological Pathologies in R3', url => 'https://www.math.toronto.edu/drorbn/classes/0102/AlgTop/Pathologies/'}$$. $$framed_page {name => 'BB-0318', title => 'Blackboards for Tuesday March 18: The Rn/Sn theorems, invariance of domain, and Borsuk-Ulam', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250318-185347'}$$. $$framed_page {name => 'HW07', title => 'Homework Assignment 7', external => 0, url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/HW07.pdf'}$$.
11	March 24-28	$$framed_page {name => 'BB-0324', title => 'Blackboards for Monday March 24: Proof of Borsuk-Ulam', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250325-110650'}$$. $$framed_page {name => 'BB-0325', title => 'Blackboards for Tuesday March 25: Degrees, CW complexes', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250325-140557'}$$. $$framed_page {name => 'HW08', title => 'Homework Assignment 8', external => 0, url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/HW08.pdf'}$$.
12	March 31 - April 4	$$framed_page {name => 'Axioms4HomologyFollowingBredon', title => 'Axioms For Homology Following Bredon', url => 'https://drorbn.net/AcademicPensieve/Classes/25-1301-AlgebraicTopology/Axioms4HomologyFollowingBredon.html'}$$. $$framed_page {name => 'BB-0331', title => 'Blackboards for Monday March 31: Axiomatics', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250401-095204'}$$. $$framed_page {name => 'BB-0401', title => 'Blackboards for Tuesday April 1: CW homology proof and examples', url => 'https://drorbn.net/bbs/show?prefix=25-1301#250408-150320'}$$.
F	April 7-111	Wednesday April 9: our final exam will take place at 10am-1pm at Bahen 6183.

Further resources:

The University of Toronto School of Graduate Studies Calendar.
A similar class I gave in 2002 in Jeruasalem.
A similar class I gave in 2004-05 in Toronto, which included also point-set topology.
A similar classs I gave in 2007-08 in Toronto, which included also differential geometry.
A summer class on homology given in Ghana in 2010.
An informal summer class on homology given in Toronto in 2017.
My 25-1301 Pensieve Folder.
The source files used to create this page: sources.zip.