# 0708-1300/Homework Assignment 9

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