Covering map definition and elementary consequences

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Problem 2.2.26 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 2.2 "Homotopy Theory"

(a)(b)

(a)

Define the meaning of `` $p : E \to B$ is a covering map''.

(b)

Let $X$ be the familiar ``figure-8'' subspace of $\mathbb {R}^{2}$ : $X$ is the union of two circles with a single point $x_{0}$ in common. Give a covering space argument to show that $\pi_{1}(X, x_{0})$ contains elements $a$ and $b$ such that $(ab)^{2} \neq a^{2}b^{2}$ . Give a careful statement of any theorems from covering space theory that you use. Do not use results from the theory of free groups.

Problem 4.1.13 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 4.1 "Basic examples and properties"

Assuming the formula $\chi (A) = k\chi (B)$ for surfaces (Exercise 4.1.12), show that the only surfaces without boundary that can cover themselves with order $k > 1$ are $T$ (the torus) and $K$ (the Klein bottle).

Problem 4.1.19 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 4.1 "Basic examples and properties"

Suppose $r : I \to B$ is a path in $B$ with $r(0) = b_1$ and $r(1) = b_2$ . Define a map from $p^{-1}(b_1)$ to $p^{-1}(b_2)$ as follows: for each $a \in p^{-1}(b_1)$ , let $r_a$ be the unique path lifting of $r$ with $r_a(0) = a$ . Then send $a$ to $r_a(1)$ . Show that this defines a bijection between $p^{-1}(b_1)$ and $p^{-1}(b_2)$ .

References

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.

[Law06] Lawson T. Topology: A Geometric Approach. Oxford University Press, 2006.