View source for 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).

# Show that if <math>p_1\colon X_1\to B_1</math> and <math>p_2\colon X_2 \to B_2</math> are covering spaces, then so is their product <math>p_1\times p_2\colon X_1\times X_2\to B_1\times B_2</math>.
# Construct (i.e., describe in explicit terms) a simply-connected covering space of the space <math>X\subset\mathbb{R}^3</math> that is the union of a sphere and a diameter. Do the same when <math>X</math> is the union of a sphere and a circle intersecting it in two points.
# Do the same to the space <math>Y</math> of the term test: <math>Y=\{z\in{\mathbb C}\colon|z|\leq 1\}/(z\sim e^{2\pi i/3}z\mbox{ whenever }|z|=1)</math>.
# Find all the connected 2-sheeted and 3-sheeted covering spaces of the "figure eight space" <math>S^1\vee S^1</math> (two circles joined at a point), up to isomorphism of covering spaces without base points.
# Let <math>a</math> and <math>b</math> be the generators of <math>\pi_1(S^1\vee S^1)</math> corresponding to the two <math>S^1</math> summands. Draw a picture of the covering space of <math>S^1\vee S^1</math> corresponding to the normal subgroup generated by <math>a^2</math>, <math>b^2</math>, and <math>(ab)^4</math>, 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? [[0708-1300/Covering_Product|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: [[Media:0708-1300-Regular-Covering-Spaces.pdf|Regular Covering Spaces]] (PDF).