Dror Bar-Natan: Classes: 2004-05: | FEEDBACK |
Agenda: Learn about the surprising relation between the easily deformed (topology) and the most rigid (algebra).
Instructor: Dror Bar-Natan, drorbn@math.toronto.edu, Sidney Smith 5016G, 416-946-5438. Office hours: Thursdays 12:30-1:30.
Teaching Assistant: Toan Ho Minh, hmtoan@math.toronto.edu, Sidney Smith 623A, 416-978-2967.
Classes: Tuesdays 1-3 and Thursdays 2-3 at Sidney Smith 5017A.
# | Week of ... | |
Fall Semester: | ||
1 | September 6 |
Classes begin on Thursday September 9.
Handout: About This Class Class notes for Thursday September 9, 2004 (categories, functors and the brouwer fixed point theorem) |
2 | September 13 |
Class notes for Tuesday September 14, 2004 (topologies, bases, continuous functions, the product topology)
Questionnaire Class notes for Thursday September 16, 2004 (the product and subset topology, closed sets) |
3 | September 20 |
Class notes for Tuesday September 21, 2004 (closure and limit points, T2 spaces, equivalent conditions for continuity, the box and the product topologies)
Thursday's class was cancelled. Instead of Classnotes |
4 | September 27 |
Homework Assignment 1
Agenda for September 28, 2004 Class notes for Tuesday September 28, 2004 (topologies on products, metrics and metrizability, metrizability of countable products) We took our Class Photo this Tuesday! Class notes for Thursday September 30, 2004 (non-metrizability of products in the box topology, non-metrizability of uncountable products, sequences and sequencial closure, clopens and connectedness) |
5 | October 4 |
Class notes for Tuesday October 5, 2004 (connected sets, intervals are connected, finite and infinite products are connected, connectedness in the box topology, connectedness and path-connectedness)
Homework Assignment 2 Class notes for Thursday October 7, 2004 (compactness, the unit interval is compact, compactness and closedness, compactness of finite products) |
6 | October 11 |
Monday is Thanksgiving.
Class notes for Tuesday October 12, 2004 (The finite intersection property, Zorn's lemma, the axiom of choice and Tychonoff's theorem) Handouts: Max August Zorn, Ultrafilters, Compactness, and the Stone-Cech Compactification. Class notes for Thursday October 14, 2004 (Lims from Stone-Cech) Thursday: HW1 was due. |
7 | October 18 |
Class notes for Tuesday October 19, 2004 (the cardinality of Stone-Cech, the general definition of Stone-Cech, completely regular spaces and embeddings into cubes)
Homework Assignment 3 Class notes for Thursday October 21, 2004 (coordinate systems, existence of Stone-Cech, normal spaces, compacts and metrizable are normal, Urysohn's lemma (statement)) Thursday: HW2 was due. |
8 | October 25 |
Class notes for Tuesday October 26, 2004 (Urysohn and Tietze, compactness in metric spaces)
Thursday: Dror was away giving two talks. |
9 | November 1 |
Makeup class:
Class notes for Monday November 1, 2004 (compactness in metric spaces)
Class notes for Tuesday November 2, 2004 (completing a metric space, Baire spaces, nowhere-differentiable functions, metric is Baire) Homework Assignment 4 Class notes for Thursday November 4, 2004 (compact is Baire, local compactness and the one-point compactification, a word about the fundamental group) Thursday: HW3 was due. |
10 | November 8 |
Handout: Math 131 Midterm, March 1993.
Class notes for Tuesday November 9, 2004 (homotopy of paths and the fundamental group, the fundamental group of a circle and path lifting) Handout: The Double Cover of an Annulus Class notes for Thursday November 11, 2004 (from path lifting to the fundamental group of a circle) |
11 | November 15 |
Class notes for Tuesday November 16, 2004 (end of the proof of the path lifting property)
Term Exam 1 took place on Tuesday November 16 from 6PM until 8PM. Class notes for Thursday November 18, 2004 (homotopy theory lite, categories and functors) |
12 | November 22 |
Class notes for Tuesday November 23, 2004 (homotopies between maps, homotopy equivalence and retracts, Brouwer's theorem, the fundamental theorem of algebra, the fundamental group of speheres)
Homework Assignment 5 Handout: The Torus Knot T(8,3). Class notes for Thursday November 25, 2004 (even/odd degrees, topological groups and their fundamental groups, Borsuk-Ulam) Thursday: HW4 was due. |
13 | November 29 |
Handout: A Blown Torus Knot.
Class notes for Tuesday November 30, 2004 (Van-Kampen: statement and many examples) Class notes for Thursday December 2, 2004 (pushouts and Van-Kampen) |
14 | December 6 |
Class notes for Tuesday December 7, 2004 (proof of the Van-Kampen theorem)
Classes end December 8 (no Thursday class). |
Winter Semester: | ||
15 | January 3 |
Handout: Errata to Munkres' Book.
Handout: Covering Spaces in One Swoosh. Handout: Coverings of the Figure "8". Class notes for Tuesday January 4, 2005 (very quick reminders, many examples of coverings and maps between them, the category of coverings and the category of G-sets) Homework Assignment 6 Advertisement: Gödel's Incompleteness Theorem. Class notes for Thursday January 6, 2005 (the main theorem of covering spaces, classification of G-sets (and coverings)) Thursday: HW5 was due. |
16 | January 10 |
Class notes for Tuesday January 11, 2005 (predicitng the universal cover, π_{1}(S^{1}) and π_{1}(SO(3)), start of the proof of the main theorem, the idea of spelunking)
Class notes for Thursday January 13, 2005 (existence of the universal covering) |
17 | January 17 |
Class notes for Tuesday January 18, 2005 (hints for the conclusion of the proof of the main theorem, the idea of homology)
Homework Assignment 7 Class notes for Thursday January 20, 2005 (simplices and the formal definition of homology) Thursday: HW6 was due. |
18 | January 24 |
Class notes for Tuesday January 25, 2005 (the square of the boundary is 0, the homology of a point, the 0th homology, functoriality, homotopy invariance and the idea of the proof, homotopies on the level of complexes)
Class notes for Thursday January 27, 2005 (conclusion of homotopy invariance, short and long exact sequences of groups, the long exact sequence for X/A) |
19 | January 31 |
Class notes for Tuesday February 1, 2005 (quotient spaces and pathologies, the homology of spheres, the long exact sequence of homologies of a short exact sequence of complexes, relative homology)
Homework Assignment 8 Class notes for Thursday February 3, 2005 (foot play with relative homology) Thursday: HW7 was due. |
20 | February 7 |
Class notes for Tuesday February 8, 2005 (more foot play towards the proof of excision)
Special class on Khovanov Homology: Handout Side 1, Handout Side 2. Transparencies: The Khovanov Homology of T(9,5), My Paper, Frenkel's Conjecture. Class notes for Thursday February 10, 2005 (knots, Khovanov homology) Sunday February 13: last chance to drop this course. |
Reading | February 14 | |
21 | February 21 |
Homework Assignment 9
Transparency: The Barycentric Subdivision. Class notes for Tuesday February 22, 2005 (end of excision, Δ-complexes) Handout: Coffin Assembly and Geomag Panels. Class notes for Thursday February 24, 2005 (geometric realization, the Δ-homology of the torus, Δ-homology equals homology) Thursday: HW8 was due. |
22 | February 28 |
Class notes for Tuesday March 1, 2005 (Δ-homology equals homology, the generators of H^{n}(S^{n}) etc.)
Thursday: Dror's office hour is 1-2PM. A Sample Term Exam 2. Class notes for Thursday March 3, 2005 (the topology of Δ-complexes, degrees and their basic properties) |
23 | March 7 |
Stickers: Rayman Excision
Class notes for Tuesday March 8, 2005 (degrees and local degrees, the linking number) Tuesday: Term Exam 2 took place in the evening, 6-8PM, at Sidney Smith 2127. Class notes for Thursday March 10, 2005 (CW complexes, the genus g surface, CW homology) |
24 | March 14 |
Tuesday: HW9 was due.
Class notes for Tuesday March 15, 2005 (homology of RP^{n}, π_{1} and H_{1}, based homology) Homework Assignment 10 Handouts: Topological Theorems About R^{n}, Topological Pathologies In R^{n}. Class notes for Thursday March 17, 2005 (Mayer-Vietoris, pathologies in R^{n}) |
25 | March 21 |
Class notes for Tuesday March 22, 2005 (the topology of a disk in R^{n}, the topology of a sphere in R^{n}, invariance of domain, the salad bowl theorem)
Thursday: HW10 was due. Homework Assignment 11 Handout: A 5 Level Alexander's Horned Sphere by Gideon Weisz. Class notes for Thursday March 24, 2005 (Borsuk-Ulam) |
26 | March 28 |
Class notes for Tuesday March 29, 2005 (the transfer sequence and the proof of Borsuk-Ulam)
Handouts: Dreams on the (Co)Homology of Manifolds, Jules Henri Poincare. Class notes for Thursday March 31, 2005 (dreams on the (co)homology of manifolds, definition of cup) |
27 | April 4 |
Class notes for Tuesday April 5, 2005 (more on the significance of homology and intersection theory, the (co)homology of Σ_{2})
Thursday: HW11 due. Homework Assignment 12 Handouts: A Cup Product Example, A Genus 2 Surface by Mister Bailey. Class notes for Thursday April 7, 2005 (cup on Σ_{2}, homotopy invariance and excision for cohomology, cup on CP^{2} and the Hopf fibration) |
Study | April 11 | A Sample Final Exam. |
Study | April 18 | |
Final | April 25 |
Thursday 9-5 and Friday 9-12: I will be in the math lounge, or in
my office with my door open, available to answer questions. If a
significant group of students will hang out at the math lounge, so
will I. (Though allow me a lunch break on Thursday).
The Final Exam took place on Friday April 29, 2-5PM, at SS 1085. See some Exam Artwork by a bored student. |
Further resources: