1. Homotopy as Path of Paths (23/8/2023).
  2. Components & Inclusions (11/9/2023).
  3. Strange Covers (2/11/2023).
  4. Fundamental Groups of Graphs (20/11/2023).

Strange Covers

due by Thursday, Nov 2, 2023

Let $Z$ be the closed topologists sine curve $$Z=\left\{(x, y) \in \R^2: 0< x \leq 1, y = sin(\frac{2\pi}{x})\right\} \cup \left\{(0, y)\in \R^2 : -1\leq y\leq 1\right\}.$$ Let $X$ be the quotient of $Z$ obtained by identifying the points $(0, 0)\in Z$ and $(1, 0)\in Z$. Let $$\widetilde{X} = \bigcup_{n\in\Z}\left\{(x + n, y): (x, y)\in Z\right\}.$$

  1. Show that $X$ is path-connected but not locally path-connected.
  2. Show that the fundamental group of $X$ is trivial.
  3. Define a function $g: Z \to S^1$ by $g(x, y) = e^{2\pi ix}$ for all $(x, y)\in Z$. Show that $g$ induces a well-defined quotient function $f: X \to S^1$.
  4. Let $p_{S^1}: \R \to S^1$ be the universal covering map $p_{S^1}(t)= e^{2\pi i t}$. Show that there does not exist a map $F: X \to \R$ such that $f = p_{S^1} \circ F$.
  5. Show that $\widetilde{X}$ is connected.
  6. Show that there is a covering map $p: \widetilde{X} \to X$.
  7. Show that if $\widehat{p}: \widehat{X} \to X$ is a covering map and $\widehat{X}$ is connected, then there is a covering map $\widehat{p}':\widetilde{X} \to \widehat{X}$ with $p = \widehat{p} \circ \widehat{p}'$.