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\large{2015 Spring Math 5043 \quad Assignment 3. \qquad Due: March 2, 2015}
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1. Hatcher Exercise 1.3.3

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2. Hatcher Exercise 1.3.4

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3. Hatcher Exercise 1.3.8

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4. Hatcher Exercise 1.3.9

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5. Hatcher Exercise 1.3.14

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6. Let $G$ be a path connected and locall path connected topological group. Let $p: \widetilde{G} \to G$ be a covering map. Show that $\widetilde{G}$ may be given a structure of topological group such that $p$ is a group morphism.

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7. A $G$ action on $X$ is discrete if fany point in $X$ has a neighbourhood $U$ such that $gU, g \in G$ are pairwise disjoint.

Let $G$ acts freely and dicretely on a topolocal space $X$. Show that the natural projection $X \to X / G$ is a covering map.

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Fruit for thought, not to be graded:

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I. Hatcher Exercise 1.3.10 and 1.3.12

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Note: Unlike the previous friut for thought problems, these are easy and mechanic exericises.
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