0708-1300/Homework Assignment 9

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Reading

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_1\colon X_1\to B_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S^1\vee S^1} corresponding to the normal subgroup generated by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^2} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^2} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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?