Syllabus for Math 221B, Winter 2019

Instructor: Stephen Bigelow
Office hours: South Hall 6514 at M3-4, T2-3, W3-4, or any time you can find me, or by appointment.
Textbook: Algebraic Topology, by A. Hatcher, Cambridge University Press. It is available online at http://www.math.cornell.edu/~hatcher/.

We will cover all of Chapters 0 and 1 in Hatcher. The main topics are the fundamental group, van Kampen's Theorem, and covering spaces.

Your grade will be determined by 50% homework and 50% final exam. The final exam will probably be take-home, details to be announced.

Homework

Jan 16: Chapter 0, questions 1,2,3,4,11.
Jan ~~23, or 25, or~~ 28: §1.1, questions 1, 2, 3, 4, ~~5, 6~~.
Feb 6 11: §1.1, questions 5–12.
Feb 20: §1.2, questions 3, 4, 7, 8.
Feb 27: §1.2, questions 9, 10.
Mar 6: §1.3, questions 1, 2, 3, 4.
Mar 15: §1.3, questions 9, 10, 12, 14.

Everything

Mostly for my own benefit, here is a list of every numbered example, lemma, proposition, theorem, and corollary in Chapter 1 of Hatcher.

eg. linear homotopy.
path homotopy is an equivalence relation.
$\pi_1(X,x_0)$ is a group.
eg. convex sets.
$\beta_h$ is an isomorphism.
Why it is called "simply connected".
$\pi_1(S^1)$.
The fundamental theorem of algebra.
The Brouwer fixed point theorem.
Borsuk-Ulam.
Corollary to Borsuk-Ulam. (Skip it.)
$\pi_1(X \times Y)$. (Skip the proof.)
eg. the torus.
$\pi_1(S^n)$. (Prove it later by van Kampen's theorem.)
...technical lemma about writing a loop as a product of small loops...
Invariance of domain.
Retractions, deformation retractions, and $i_*$.
Homotopy equivalence induces an isomorphism.
...technical lemma about homotopies not rel basepoint...
Van Kampen. (Just for two open sets.)
eg. wedge sums.
eg. a graph.
eg. linking of circles.
eg. torus knots. (I just talked about the trefoil.)
eg. the shrinking wedge of circles. (Skip it.)
$\pi_1$ of cell complexes. (Just for attaching a single cell.)
$\pi_1$ of closed surfaces.
Every group is a $\pi_1$. (Just do finitely presented groups.)
eg. cyclic groups.
Homotopy lifting. (Just path lifting and path-homotopy lifting.)
$p_*$ is injective, and you can describe the image.
The number of sheets is the index of the subgroup.
Lifting criterion. (Skip the proof.)
Unique lifting.
eg. a quotient of a cylinder. (Skip it.)
Existence of a covering space for every subgroup. (Skip the proof.)
Uniqueness of covering spaces up to isotopy.
The Galois correspondence, and the effect of change of basepoint.
Deck transformations. (Just for normal covering spaces.)
Covering space actions and orbit spaces.

Here are the most important things you should be able to do.

Define everything, especially ``fundamental group'' and ``covering space''.
Understand that the fundamental group is functorial.
State the Galois correspondence, and know the covering space concepts that correspond to subgroup, coset, index, normal subgroup, and conjugate subgroup.
Use van Kampen's theorem to compute the fundamental group in some simple examples.

The final exam will probably be light on proofs. You should be able to prove that the fundamental group is a group. Maybe the path lifting property is not too long for me to ask for. It would be nice if you knew proofs of big theorems like van Kampen's theorem. Maybe I could ask you to prove a simpler thing that uses similar tricks, like the Lebesgue number, or inserting trips back to the basepoint.

Van Kampen Lemmas

There are three key ideas in the proof of van Kampen's theorem. One of them is to convert a path to a concatenation of loops, by inserting trips to the basepoint and back again. The other two can be encapsulated in the following lemmas.

Let $\{U_\alpha\}$ be an open cover of $X$. We do not need any assumptions about basepoints or path-connectedness. Let $f,g \colon I \to X$ be paths. Use the convention that $$f_1 \cdot \ldots \cdot f_n$$ is a path that follows each $f_i$ at $n$-times the speed.

Lemma. For all sufficiently large $n$, $$f = f_1 \cdot \ldots \cdot f_n,$$ where each $f_i$ has image in some $U_\alpha$ (with $\alpha$ depending on $i$).

Lemma. If $f \simeq g$ then for all sufficiently large $n$, you can get between the two expressions $$f = f_1 \cdot \ldots \cdot f_n, \ g = g_1 \cdot \ldots \cdot g_n$$ by a sequence of changes of the form $$h_1 \cdot \ldots h_i \cdot h_{i+1} \cdot \ldots \cdot h_n \mapsto h_1 \cdot \ldots h'_i \cdot h'_{i+1} \cdot \ldots \cdot h_n$$ where $h_i \cdot h_{i+1} \simeq h'_i \cdot h'_{i+1}$ by a homotopy rel endpoints that lies entirely in some $U_\alpha$.