# Topology

This is a plan for a reading course in topology, first some point set topology, then some homotopy. The aims are:

1. to understand how to use Tychonov's theorem to prove the de Bruijn-Erdős theorem
2. to understand the classification of surfaces
3. pick up a working knowledge of covering spaces of graphs and surfaces.
A long-term goal is to learn about Lie groups, which requires some minimal understanding of manifolds.

## Part I: James

We're using I. M. James "Topological and Uniform Spaces", Springer 1987.(Which I like because it introduces you to ultrafilters, and its short.)

1. Read Chapters 1-5 and do a good fraction of the exercises. This brings you to Tychonov's theorem.
2. Chapters 6, 9. Do a good fraction of the exercises.
3. Write out proofs that the separation conditions collapse on topological groups (i.e., $T_0=T_1=T_2$). Here are some articles that may be useful.
4. Study connectedness (James, ch 9).
5. Arrow's Theorem: This has very little to do with topology, but it's very interesting and has nice proofs using ultrafilters. Some quick finds on the web:
There's also a blog article by Tao. (Let me know which of these is most useful.)
6. Look up the Zariski topology on $\fld^d$, for an algebraically closed field $\fld$. Assuming it is compact (true), prove that any ideal in the ring $\fld[\seq x1d]$ is finitely generated.
7. Write up notes on topologies on function spaces, in particular: product, uniform, compact-open.
8. What are the advantages of being Hausdorff? (A short essay, please.)
9. James's terminology is not always standard. Find another reasonable book on topology and prepare a translation table.

## Part II: Homotopy, Covers, Surfaces

Any introductory text you find on algebraic topology that starts with homotopy will probably be useful. Two that do work are:

• Hatcher "Algebraic Topology" CUP 2002. on-line
• Lee "Introduction to Topological Manifolds". (You can get a pdf through the library. This also covers a lot of the stuff in James.)
One goal is to be able to explain the sentences "the covering graphs of a finite connected graph correspond to the permutation representations of its fundamental group" and "an embedding of a graph in a surface determines a homomorphism from its fundamental group into the fundamental group of the surface". This is covered in Chapter 0 and in Sections 1.1 and 1.3 of Hatcher. (Note that this is much denser than James's book.)

I will give a lecture on free groups at some point.

We start by studying fundamental groups and covering spaces. As a group, provide a set of notes dealing with the following.

1. Write down definitions for simplicial complex, cell complex, manifold, smooth manifold. Provide interesting examples. In case, describe the natural maps.
2. Write down definitions for: isotopy, ambient isotopy, homotopy. Knots may prove useful for examples.
3. Let $X$ be a topological space and assume $a\in X$. A loop in $X$ based at $a$ is a map $f:[0,1]\to X$ such that $f(0)=f(1)=a$. The loops in $X$ based at $a$ form a monoid. Homotopy is an equivalence relation on loops (proof), and the equivalence classes of loops at $a$ form a group (proof), denoted $\pi_1(X,a)$ and called the first fundamental group of $X$. [For us, it will be the first and only, so we will usually drop the "first".] If $X$ is connected in some appropriate sense (which you should specify) and $b$ is a second point in $X$, determine the relation between $\pi_1(X,a)$ and $\pi_1(X,b)$.
4. Determine the fundamental group of the circle. Go over a proof of the "fundamental" "theorem of algebra" based on this.
5. Suppose $\cC$ is a cell complex and the graph $X$ be its 1-skeleton. Each walk in the graph determines a path in the complex. When are two walks homotopic? (Here and elsewhere, two paths are homotopic if they have the same end points and there is a homotopy from one to the other that fixes the end points.)
6. Write up a definition of covering space. State the significant results relating paths in $X$ and paths in a covering space $Y$.
7. Assume $X$ is not too evil and let $a$ be a point in $X$. Let $\hat{X}$ denote the set of homotopy classes of paths in $X$ that start at $a$. Show that $\hat{X}$ is a covering space for $X$. Show that $\pi_1(X,a)$ acts freely on $\hat{X}$, and that the orbits of $\pi_1$ are the fibres of the cover. Show that if $X$ is a manifold, so is $\hat X$.
8. (Continuing from the previous.) Show that if $Y$ covers $X$, then $\hat{X}$ covers $Y$ and the composition of the two covering maps is the map with the orbits of $\pi_1(X)$ as its fibres.
9. Determine the fundamental groups of the real projective plane and the torus.
10. Suppose $Y_1$ and $Y_2$ cover $X$. By considering the map from $Y_1\times Y_2$ to $X\times X$, show that there is subspace of $Y_1\times Y_2$ that covers $Y_1$ and $Y_2$. Show that the obvious diagram commutes.