HW8

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Exams	Apr 24	Final, PM

Homework Assignment 8

Required 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! Also, read Hatcher's section 1.3.

Solve the following problems. (But submit only the underlined ones). In Hatcher's book, problems 1, 2, 3, 4, 5, 9, 10, 12, 14, 16, 17 and 18 in section 1.3.

Due date. This assignment is due at the end of our class on Thursday, February 16, 2006, though it will be wise for you to go over much of it before Term Exam 3.

Just for fun.

If this sentence is true, Klein bottles are orientable.

Well, assume this sentence is true. It says that if it is true then Klein bottles are orientable, and we've just assumed it is true. So Klein bottles are orientable. Notice that we have just proven that if the statement is true, then Klein bottles are orientable; but that's precisely what the statement says, so we have just proven it, so it is true and so Klein bottles are orientable. Surely you can't argue with that.

Above the line, edit only if you are sure

Feel free to edit below the line

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Problems for Submission

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Hatcher $\S$ 1.3, p. 79

2.: Show that if $p_1 \colon \tilde X_1 \to X_1$ and $p_2 \colon \tilde X_2 \to X_2$ are covering spaces, so is their product $p_1 \times p_2 \colon \tilde X_1 \times \tilde X_2 \to X_1 \times X_2$

4.: Construct 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.

10.: Find all the connected 2-sheeted and 3-sheeted covering spaces of $S^1 \vee S^1$ , up to isomorphism of covering spaces without basepoints.

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Hatcher $\S$ 1.3, p. 80

12.: 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 $S^1 \vee S^1$ corresponding to the normal subgroup generated by $a 2$ , $b 2$ , and $(a b) 4$ , and prove that this covering space is indeed the correct one.

18.: For a path-connected, locally path-connected, and semilocally simply-connected space $X$ , call a path-connected covering space $\tilde X \to X$ abelian if it is normal and has abelian deck transformation group. Show that $X$ has an abelian covering space that is a covering space of every other abelian covering space of $X$ , and that such a ‘universal’ abelian covering space is unique up to isomorphism. Describe this covering space explicitly for $X=S^1\vee S^1$ and $X=S^1\vee S^1\vee S^1$ .