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]\displaystyle{ p_1\colon X_1\to B_1 }[/math] and [math]\displaystyle{ p_2\colon X_2 \to B_2 }[/math] are covering spaces, then so is their product [math]\displaystyle{ 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]\displaystyle{ X\subset\mathbb{R}^3 }[/math] that is the union of a sphere and a diameter. Do the same when [math]\displaystyle{ X }[/math] is the union of a sphere and a circle intersecting it in two points.
- Do the same to the space [math]\displaystyle{ Y }[/math] of the term test: [math]\displaystyle{ 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]\displaystyle{ S^1\vee S^1 }[/math] (two circles joined at a point), up to isomorphism of covering spaces without base points.
- Let [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] be the generators of [math]\displaystyle{ \pi_1(S^1\vee S^1) }[/math] corresponding to the two [math]\displaystyle{ S^1 }[/math] summands. Draw a picture of the covering space of [math]\displaystyle{ S^1\vee S^1 }[/math] corresponding to the normal subgroup generated by [math]\displaystyle{ a^2 }[/math], [math]\displaystyle{ b^2 }[/math], and [math]\displaystyle{ (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 [math]\displaystyle{ 1 }[/math] we consider infinitely many covering spaces. This is, is the product of an infinite family of covering spaces a covering space?
