This is a plan for a reading course in topology, first some point set topology, then some homotopy. The aims are:
- to understand how to use Tychonov's theorem to prove the de Bruijn-Erdős theorem
- to understand the classification of surfaces
- pick up a working knowledge of covering spaces of graphs and surfaces.
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.)
- Read Chapters 1-5 and do a good fraction of the exercises. This brings you to Tychonov's theorem.
- Chapters 6, 9. Do a good fraction of the exercises.
- 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.
- Study connectedness (James, ch 9).
- 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.)
- 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.
- Write up notes on topologies on function spaces, in particular: product, uniform, compact-open.
- What are the advantages of being Hausdorff? (A short essay, please.)
- 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.)
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.
- Write down definitions for simplicial complex, cell complex, manifold, smooth manifold. Provide interesting examples. In case, describe the natural maps.
- Write down definitions for: isotopy, ambient isotopy, homotopy. Knots may prove useful for examples.
- 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 $\pi_1(X,b)$.
- Determine the fundamental group of the circle. Go over a proof of the "fundamental" "theorem of algebra" based on this.
- 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.)
- Write up a definition of covering space. State the significant results relating paths in $X$ and paths in a covering space $Y$.
- 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 $\hat X$.
- (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.
- Determine the fundamental groups of the real projective plane and the torus.
- 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.