Exam 1, 3/5 – 3/12
Exam 2, 4/28 – 5/14

TBD

Algebraic topology, by Allen Hatcher.

Will go here

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%).

The purpose of the 751–752 sequence is to train you in the fundamental machinery of Algebraic Topology, ultimately tested by the Qualifying Exam in Geometry and Topology. Final letter grades will be allocated according to a crude scheme based on your numerical performance: an A means that it is likely that you will pass the qualifying exam, a B means that is not likely that you will pass the qualifying exam, and an F means that you did little to no work in the course. I will not assign any other letter grades.

4.1: 12, 15, 16.
4.2: 1, 8, 9, 15, 31, 32.

3.3: 28, 29, 32, 33.
4.1: 2, 3, 4, 8, 10, 11.

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 \( \mathrm{dim}_\mathbb{Q} \mathrm{ker}(H_1(\partial M;\mathbb{Q}) \to H_1(M; \mathbb{Q})) = \frac{1}{2} \mathrm{dim}_\mathbb{Q}(H_1(\partial M ; \mathbb{Q}))\).

3.3: 1, 3, 5, 7, 8, 11, 24, 25.

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.

3.2: 1, 3, 4, 7, 8, 9, 11, 18.

0. A triangulation \(T\) of a space \(X\) is a simplicial complex \(T\) and a homeomorphism \(T \to X\). Two simplicial complexes are isomorphic if there are homeomorphic via a map that takes simplices to simplices via linear homeomorphisms. Two triangulations \(f: S \to X\) and \(g: T \to X\) of a space \(X\) are said to be equivalent if \(S\) and \(T\) have isomorphic subdivisions.
Show that any two triangulations of the disk are equivalent.

3.1: 2, 5, 6, 7, 8, 9, 11