1. Introduction (12/10/2020).
  2. Paths and Homotopies (19/10/2020).
  3. Path and Homotopy Lifting (26/10/2020).
  4. Functoriality and Applications (2/11/2020).
  5. Free Groups (9/11/2020).
  6. Map Lifting and Classification of Covers (16/11/2020).
  7. Galois Theory of Covering Spaces (23/11/2020).
  8. Free Products and the Seifert-Van Kampen theorem (7/12/2020).
  9. Graphs and Two-Complexes (14/12/2020).
  10. CW-Complexes and Higher Homotopy Groups (21/12/2020).
  11. Whitehead Theory (28/12/2020).
  12. Simplicial Complexes and Homology (11/1/2021).

Path and Homotopy Lifting

due by Monday, Oct 26, 2020

  1. Consider the projection map $p : S^1 \to [-1, 1]$, $p(x, y) = x$. Prove or disprove that the point $1 \in [-1, 1]$ has an evenly covered neighbourhood for $p$.

  2. Prove or disprove: given $\epsilon > 0$, there exists a sequence of closed subsets $J_1, J_2, \dots, J_n\subset S^2$ of the $2$-sphere $S^2$ with the following properties.

    1. $J_i$ has diameter less that $\epsilon$ for all $i$, $1 \leq i \leq n$.
    2. $\bigcup_{i = 1}^n J_i = S^2$.
    3. Each set $J_i$ is connected.
    4. $J_{k+1}\cap \bigcup_{i = 1}^k J_i$ is connected for all $k$, $1 < k \leq n$.

  3. Let $\alpha: [0, 1]\to S^1$ be a path and let $\widetilde{\alpha'}, \widetilde{\alpha''}: [0, 1]\to \mathbb{R}$ be two lifts of $\alpha$ with respect to the usual covering map. Then prove or disprove that the function $\Phi: [0, 1]\to \mathbb{R}$ given by $\Phi(t) = \widetilde{\alpha'}(t) - \widetilde{\alpha''}(t)$ must be a constant function.