# 0708-1300/Homework Assignment 9

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## Contents

Read, reread and rereread your notes to this point, and make sure that you really, really really, really really really understand everything in them. Do the same every week!

## Doing

(Problems 1,2,4,5 below are taken with slight modifications from Hatcher's book, pages 79-80).

1. Show that if $p_1\colon X_1\to B_1$ and $p_2\colon X_2 \to B_2$ are covering spaces, then so is their product $p_1\times p_2\colon X_1\times X_2\to B_1\times B_2$.
2. Construct (i.e., describe in explicit terms) a simply-connected covering space of the space $X\subset\mathbb{R}^3$ that is the union of a sphere and a diameter. Do the same when $X$ is the union of a sphere and a circle intersecting it in two points.
3. Do the same to the space $Y$ of the term test: $Y=\{z\in{\mathbb C}\colon|z|\leq 1\}/(z\sim e^{2\pi i/3}z\mbox{ whenever }|z|=1)$.
4. Find all the connected 2-sheeted and 3-sheeted covering spaces of the "figure eight space" $S^1\vee S^1$ (two circles joined at a point), up to isomorphism of covering spaces without base points.
5. Let $a$ and $b$ be the generators of $\pi_1(S^1\vee S^1)$ corresponding to the two $S^1$ summands. Draw a picture of the covering space of $S^1\vee S^1$ corresponding to the normal subgroup generated by $a^2$, $b^2$, and $(ab)^4$, and prove that this covering space is indeed the correct one.

## Due Date

This assignment is due in class on Thursday February 28, 2008.

## Just for Fun

• What happens if in problem 1 we consider infinitely many covering spaces. That is, is the product of an infinite family of covering spaces a covering space? Here is an idea but don't look at it until you have think on the problem for a while.
• This raises another question. A "pathwise totally disconnected space" is a space in which every path is a constant path. How much of the theory of covering spaces can be generalized to "coverings" in which the fibers are pathwise totally disconnected, instead of discrete?
• Here is a short introduction to regular covering spaces and an application of them to the last problem on this homework: Regular Covering Spaces (PDF).