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).

Map Lifting and Classification of Covers

due by Monday, Nov 16, 2020

  1. For the topological space $$X=\{(x, y)\in\mathbb{R}^2: x\in\mathbb{Q} \textrm{ or }y=0\},$$ prove or disprove each of the following.

    • (a) $X$ is connected.
    • (b) $X$ is path-connected.
    • (c) $X$ is locally path-connected.

  2. Let $p: (Y, y_0)\to (S^1, 1)$ be a connected cover of the circle, let $G = \pi_1(S^1, \mathbb{Z})$ and let $H = p_*(\pi_1(Y, y_0))\subset G$. Then prove or disprove that each of the following conditions guarantee that the based cover is isomorphic to some based cover of the form $$p_n: (S^1, 1)\to (S^1, 1),\\ p_n(z) = z^n$$

    • (a) the group $H$ is infinite.
    • (b) the group $H$ has finite index in the group $G$.
    • (c) the group $H$ is isomorphic as a group to $G$.