MATH 752: INTRODUCTORY TOPOLOGY, II
MWF 8:50–9:40 AM in Van Vleck B211 with Autumn Kent Office: Van Vleck 615 Office hours: Monday 2–4, by appointment, or try stopping by. Exams (timed take–homes): Exam 1, 3/5 – 3/12Problem Sessions: TBDMain text: Algebraic topology, by Allen Hatcher. Supplementary materials:
Grades: There will be two midterm exams (designed as miniature qualifying exams), and regular homework (on the right side of this page). Your final grade in the course will be based on your performance on these three items, weighted roughly as Homework (20%), Midterm #1 (30%), Midterm #2 (50%). |
Homework:
HW5, due Friday, 05.02.2014.
Hatcher: 4.1: 12, 15, 16. HW4, due Wednesday, 04.16.2014.
Hatcher: 3.3: 28, 29, 32, 33. HW3, due Wednesday, 04.02.2014.
1. Show that \(\mathbb{Q}[x,y]/(x^3,y^3,x^2y^2)\) is not the rational cohomology ring of any closed orientable \(195\)-manifold. 2. ("Half lives, half dies") Prove that if \(M\) is a compact orientable \(3\)-manifold with \(\partial M \neq \emptyset\), then \( \mathrm{dim}_\mathbb{Q} \mathrm{ker}(H_1(\partial M;\mathbb{Q}) \to H_1(M; \mathbb{Q})) = \frac{1}{2} \mathrm{dim}_\mathbb{Q}(H_1(\partial M ; \mathbb{Q}))\).Hatcher: 3.3: 1, 3, 5, 7, 8, 11, 24, 25. HW2, due Friday, 03.07.2014.
1. Let \(X\) be a projective space over \(\mathbb{R}\) or \(\mathbb{C}\). Give a geometric description of the cohomology ring in terms of homology. In other words, what does the cup product mean when thought of as a product on \(\oplus_{i=0}^\infty H_i(X;R)\), where \(R = \mathbb{F}_2\) when \(X\) is the real projective space and \(R=\mathbb{Z}\) for the complex one.Hatcher: 3.2: 1, 3, 4, 7, 8, 9, 11, 18. HW1, due Friday, 02.21.2014.
0. AHatcher: 3.1: 2, 5, 6, 7, 8, 9, 11 |