0708-1300/Homework Assignment 9

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Contents

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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!

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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 S1 summands. Draw a picture of the covering space of S^1\vee S^1 corresponding to the normal subgroup generated by a2, b2, and (ab)4, and prove that this covering space is indeed the correct one.
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Due Date

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

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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).
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