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 $p_{1}\colon X_{1}\to B_{1}$ 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 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\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 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} is the union of a sphere and a circle intersecting it in two points.
3. Do the same to 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 Y} of the term test: 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 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 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} 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 b} be the generators 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 \pi_1(S^1\vee S^1)} corresponding to the two 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} 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 $(ab)^{4}$, 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, along with an application of them to the last problem on this homework. 0708-1300/Regular Covering Spaces.pdf