| SES # | TOPICS | READINGS |
|---|---|---|
| 1 | Organizational Meeting | |
| 2 | n-manifolds and Orientability, Compact, Connected 2-manifolds | Lecture 1 Massey: Chapter 1, Sec. 2-3 Lecture 2 Massey: Chapter 1, Sec. 4 |
| 3 | Classification Theorem for Compact Surfaces, Triangulation | Lecture 3 Massey: Chapter 1, Sec. 5 Lecture 4 Massey: Chapter 1, Sec. 6, and "Step 1" of the proof in Sec. 7 |
| 4 | Classification Theorem for Compact Surfaces (cont.), Euler Characteristic | Lecture 5 Massey: Chapter 1, the rest of section Sec. 7 Lecture 6 Massey: Chapter 1, Sec. 8 |
| 5 | Review of Group Theory, Homotopy and the Fundamental Group | Lecture 7 Review group theory using the notes on Basic Group Theory (PDF), or the online Group Theory notes by J. S. Miline Lecture 8 Massey: Chapter 2, Sec. 2, 3 |
| 6 | The Fundamental Group (cont.), Homotopy Equivalence and Homotopy Type | Lecture 9 Massey: Chapter 2, end of Sec. 3, and Sec. 4 Lecture 10 Massey: Chapter 2, Sec. 8 |
| 7 | The Fundamental Group of a Circle, Retracts, Brower Fixed-Point Theorem | Lecture 11 Massey: Chapter 2, Sec. 5 Lecture 12 Massey: Chapter 2, part of Sec. 4 after thm 4.1, Sec. 6 |
| 8 | Weak Product of Groups, The Fundamental Group of a Torus, Free Abelian Groups | Lecture 13 Massey: Chapter 3, Sec. 2, and Chapter 2, Sec. 7 Lecture 14 Massey: Chapter 3, Sec. 3 |
| 9 | Free Products, Free Groups, Presentations of Groups | Lecture 15 Massey: Chapter 3, Sec. 4, Sec. 5 Lecture 16 Massey: Chapter 3, Sec. 6, maybe a bit from Sec. 5 |
| 10 | Siefert-Van Kampen Theorem and its Generalization | Lecture 17 Massey: Chapter 4, first half of Sec. 2 Lecture 18 Massey: Chapter 4, second part of Sec. 2 |
| 11 | Applications of the Siefert-Van Kampen Theorem, Structure of the Fundamental Group of a Compact Surface | Lecture 19 Massey: Chapter 4, Sec. 3 Lecture 20 Massey: Chapter 4, Sec. 4, beginning of Sec. 5 |
| 12 | Fundamental Groups on Closed Surfaces, Application to Knot Theory | Lecture 21 Massey: Chapter 4, second part of Sec. 5 Lecture 22 Massey: Chapter 4, Sec. 6 |
| 13 | Covering Spaces, Path Lifting Lemma, Homotopy Lifting Lemma | Lecture 23 Massey: Chapter 5, Sec. 2 Lecture 24 Massey: Chapter 5, Sec. 3 |
| 14 | Fundamental Group of a Covering Space, Lifting of Arbitrary Maps to a Covering Space | Lecture 25 Massey: Chapter 5, Sec. 4 Lecture 26 Massey: Chapter 5, Sec. 5 |
| 15 | Homomorphisms and Isomorphisms of Covering Spaces, Action of the Fundamental Group on Fibers of Covering Spaces | Lecture 27 Massey: Chapter 5, Sec. 6 Lecture 28 Massey: Chapter 5, Sec. 7 |
| 16 | Regular Covering Spaces and Quotient Spaces, Borsuk-Ulam Theorem for the 2-sphere | Lecture 29 Massey: Chapter 5, Sec. 8 Lecture 30 Massey: Chapter 5, Sec. 9 |
| 17 | The Existence Theorem for Covering Spaces, Induced Covering Space over a Subspace | Lecture 31 Massey: Chapter 5, Sec. 10 Lecture 32 Massey: Chapter 5, Sec. 11 |
| 18 | Graphs, Trees, Fundamental Group of a Graph | Lecture 33 Massey: Chapter 6, Sec. 2-3 Lecture 34 Massey: Chapter 6, Sec. 4-5 |
| 19 | Euler Characteristic and Coverings of Graphs, Generators of Subgroups of Free Groups | Lecture 35 Massey: Chapter 6, Sec. 6-7 Lecture 36 Massey: Chapter 6, Sec. 8 |
| 20 | Delta Complex, Singular Chains, Homology | Lecture 37 Hatcher: Chapter 2, Sec. 1 (pp. 102-105) Lecture 38 Hatcher: Chapter 2, Sec. 1 (pp. 105-107) |
| 21 | Singular Homology, The Homomorphism pi_1(X) -> H_1(X) | Lecture 39 Hatcher: Chapter 2, Sec. 1 (p. 108 - top of p. 110) Lecture 40 Hatcher: Chapter 2, appendix A (pp. 166-168) |
| 22 | Degree of a Map and its Applications, Higher Homotopy Groups | Lecture 41 Hatcher: Chapter 2, Sec. 2 (pp. 134-135) Lecture 42 Hatcher: Chapter 4, Sec. 1 (pp. 340-343) |
| 23 | Cell Complex, Whitehead's Theorem | Lecture 43 Hatcher: Chapter 0, Sec. 1 (pp. 5-6), bit from Chapter 4 |
| 24 | Presentations of Final Projects | |
| 25 | Presentations of Final Projects (cont.) |
Table of Contents
Study Materials
Readings
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The primary book for the course is:
Massey, W. S. "Algebraic Topology: An Introduction." Graduate Texts in Mathematics. Vol. 56. New York, NY: Springer-Verlag, 1977. ISBN: 0387902716.
Students should note that there are two other books in the GTM series (GTM 70 and GTM 127) by Massey, which are different books.
The material covered in this course is also contained in the beginning of:
Hatcher, Allen. Algebraic Topology. Cambridge, UK: Cambridge University Press, 2001. ISBN: 0521795400 (paperback.)
This is an excellent text with many examples and pictures, and it can by found online at Allen Hatcher's Homepage.
The text by Massey is our primary source because it spends more time on the material we plan on covering, and gives a very careful exposition.
Finally, for instructions on how to present a lecture for this class, please read Guidelines for Lectures (PDF)