Catalogue listing Basic concepts of algebraic topology, including the fundamental group, plane curves, and a brief introduction to homology.
Prerequisites: 01:640:441 or permission of department.
| Week | Lecture dates | Material | HW assignment given (due Tuesday) |
|---|---|---|---|
| 1 | 1/21, 24 | CW complexes | Show that SX is a CW complex if X is; #0.2, #0.10 |
| 2 | 2/4, 7 | The fundamental group | Ch.1.1 #6,12,13,16(a,b,c),19 |
| 3 | 2/11, 14 | Van Kampen's Theorem | Ch.1.2 #6,7,11,15 |
| 4 | 2/18, 21 | Covering spaces | Ch.1.3 #1,4,9,14 |
| 5 | 2/25, 28 | Axioms for homology | Compute H* for the torus and RP2 over F=Z/2, F=R |
| 6 | 3/3, 6 | Simplicial homology | Ch.2.2 #4,7,12; 2.3 #3,4 |
| 7 | 3/10 | review, midterm | midterm on Tuesday 3/10 |
| 8 | 3/13, 17, 20 | Spring Break (no class) | move class to on-line |
| 9 | 3/24, 27 | Simplicial complexes | Ch.2.1 #1,2,8,17 Solutions |
| 10 | 3/31, 4/3 | Singular homology | Ch.2.1 #15,21,24,29 Solutions |
| 11 | 4/7, 4/10 | Cohomology | Ch.3.1 #8(a,c),9,12,13 (take G=Z/2) Solutions |
| 12 | 4/14, 4/17 | Cohomology | 2B#1, Ch.3,2 #3a,7 Prove the Jordan Curve Theorem Solutions |
| 13 | 4/21, 4/24 | Vector bundles | 1) if n is odd, show that the tangent bundle T to
Sn has a nowhere-zero section 2) If E→X has patching maps gij, show that the patching maps det(gij) define a line bundle 3) Show that a map f: X→ Grn detemines a vector bundle on X. Solutions |
| 14 | 4/28, 5/1 | Vector Bundles and K-theory | 1) Show that the clutching map for the tangent bundle
on the 2-sphere has degree 2 2) Compute KO(Sn) for n=0,1,2,3 3) Compute K(X ∨ Y ∨ Z), if X,Y,X are connected Solutions |
| 15 | 5/4 (Monday) | Homotopy theory | Review for Final |
| 16 | May 13 (Wednesday) | 8-11 AM | Final Exam (cumulative) |