Elementary Algebraic Topology
Peter Koroteev
833 Evans, pkoroteev@math.berkeley.edu, /~pkoroteev
Lectures: Tues/Thurs 2:00 pm - 3:30 pm, 9 Evans Hall
Office Hours: Tuesday 11:10 am -- 12:40 pm, Thursday 4:45 pm -- 6:15 pm
Textbooks: main book: `Topology' by J. Munkres. Additional sources: `Basic Topology' by M. A. Armstrong (can be downloaded on a campus connection.), `Algebraic Topology' by A. Hatcher, `Quantum Field Theory and Topology' by A. Schwarz, as well as online MIT courses.
Useful Links: Academic honesty in mathematics courses: (courtesy of Michael Hutchings), How to get an A in this class (courtesy of Kathryn Mann).
Plan of the lectures:
(will be updated throughout the course)
| Week 1 (Jan 22nd and 24th) |
Topological spaces, continuous functions Munkres: Chapter 2 (Sec 12,17,18) |
| Week 2 (Jan 29th and 31st) |
Continuity, product topology, identification spaces, quotient topology, compactness Munkres: Sec 19, 22, Sec 3 Armstrong: Chapter 4 then Chapter 3 |
| Week 3 (Feb 5th and 7th) |
Compactness, Connectedness Munkres: Chapter 3 Armstrong: Sec 3.1, 3.3, 3.4 (compactness), 3.5, 3.6 (connectedness) |
| Week 4 (Feb 12th and 14th) |
Connectedness, Separation Axioms, Normal Spaces, Homotopy Munkres: Sec 31, 32 (separation axioms) Armstrong: 3.5, 3.6 (connectedness) 5.1 5.4 (homotopy) |
| Week 5 (Feb 19th and 21st) |
Homotopy, Fundamental Group Munkres: Chapter 9 (51-55) Armstrong: 5.1 -- 5.4 Hatcher: Chapters 0,1 |
| Week 6 (Feb 26th and 28th) |
Review of the material. Midterm 1. |
| Week 7 (March 5th and 7th) |
Fundamental groups of orbit spaces. Covering spaces. Fixed points. Munkres: Sec. 53, 58, 60 Armstrong: 4.4, 5.3, 5,4, 10.4 |
| Week 8 (March 12th and 14th) |
Manifolds. Classification of 1-manifolds Some useful links: Introduction to topological manifolds (available with campus connection) (see Chapter 2 from p.38 and Chapter 3 p. 73-77 for general background on manifolds). Note on classification of closed connected 1-manifolds (part of the proof in class). |
| Week 9 (March 19th and 21st) |
Surfaces Chapter 6 of Introduction to topological manifolds Armstrong: 7.1, 7.5 |
| Week 10 (April 2nd and 4th) |
Classification of surfaces. Seifert-van Kampen theorem Homology Armstrong: Section 7.5 Chapter 10 (pp 251-261, 264-267) of Introduction to topological manifolds John Conway's ZIP proof of the classification of compact surfaces. No HW this week |
| Week 11 (April 9th and 11th) |
Review of the material, Midterm #2: Armstrong: Sections 4.4 (but only discrete groups) 5.1, 5.2, 5.3 (no path/homotopy lifting lemma questions), 5.4, 5.5 7.1, 7.5 10.4 From Lee's book on top manifolds: Chapters 2 (from p38), 3 (from p73, only discrete groups), 6 (up to p178), 9 (for reference), 10 (pp 251-257, 264-267) Classification of 1-manifolds Descriptions/notations of some manifolds: projective space, spheres, tori, balls, surfaces No HW this week |
| Week 12 (April 16th and 18th) |
Knot theory. Armstrong: section 10.1 Wikipedia reference: 3-colourability (Optional) More details on knot theory: An Introduction to Knot Theory |
| Week 13 (April 23rd and 25th) |
Knot theory More useful links: https://katlas.math.toronto.edu/wiki/Main_Page |
| Week 14 (April 30th and May 2nd) |
Knot theory, Review Material for the final: Everything from Midterms 1, 2 and knot theory up to Lecture on April 23rd. |
| Week 15 |
RRR week |