Take home exam, 10/23 – 10/26
Written exam, 12/10 at 4:30pm

10/1 at 4:30pm in Van Vleck B219

Algebraic topology, by Allen Hatcher.

A Basic Course in Algebraic Topology, by William S. Massey

Classical Topology and Combinatorial Group Theory, by John Stillwell

Midterm 1 (.pdf) (.tex)

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 (30%), Midterm #1 (30%), Midterm #2 (40%).

The purpose of the course is to prepare you for 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.

Section 2.1: 4, 5, 8, 9, 11, 22
Section 2.2: 9, 12, 28, 29, 32

1. Show that action of \(\Gamma[2] = \mathrm{ker}(\mathbb{P}\mathrm{SL}_2(\mathbb{Z}) \to \mathbb{P}\mathrm{SL}_2(\mathbb{Z}/2\mathbb{Z}))\) on the upper half–plane \(\mathbb{H}\) is a covering space action.

2. Consider the set \( X = \{a, b, x, y\} \) equipped with the topology \( T = \{ \emptyset, X, \{x\}, \{y\}, \{x,y\}, \{a,x,y\}, \{b,x,y\} \}\). Show that this space is path connected and that its fundamental group is isomorphic to \(\mathbb{Z}\).

Section 1.3: 12, 14, 16, 18, 20, 25, 29

1. Prove Lemma 1: The only compact connected surface with boundary that possesses no nonseparating arc is the disk.

2. Prove Lemma 2: The only compact connected surfaces with boundary whose essential arcs are all nonseparating are the annulus, the Möbius band, the torus with an open disk removed, and the Klein bottle with an open disk removed.

3. Build a \(3\)–manifold whose fundamental group is isomorphic to the dyadic rationals (the additive group of \(\mathbb{Z}[\frac{1}{2}]\)).

Section 1.2: 16, 19
Section 1.3: 3, 4, 9, 10

Section 1.1: 16, 17
Section 1.2: 5, 6, 8, 9, 10, 11, 20

Chapter 0: 22
Section 1.1: 1, 3, 5, 6, 8, 12, 13

Chapter 0: 1, 2, 3, 5, 6, 9, 10, 12, 14, 16, 20, 23