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).

Galois Theory of Covering Spaces

due by Monday, Nov 23, 2020

  1. Let $p: (Y, y_0)\to (X, x_0)$ be a connected cover, let $G =\pi_1(X, x_0)$ and let $H = p_*(\pi_1(Y, y_0))\subset G$. Then for each of the following conditions, prove or disprove that the condition guarantees that for every point $y_1\in p^{-1}(x_0)$ the based cover $p: (Y, y_1)\to (X, x_0)$ is isomorphic to the based cover $p: (Y, y_0)\to (X, x_0)$.

    • (a) the group $H$ is abelian.
    • (b) the group $H$ is normal in $G$.
    • (c) the group $G$ is abelian.

  2. Let $(X, x_0)$ be a path-connected topological space. Let $G = \pi_1(X, x_0)$ and consider the space $Y$ and the map $p: Y\to X$ constructed in the lectures corresponding to the subgroup $H = G$ of $G$ (i.e., when the subgroup of $G$ considered is the group $G$ itself). Prove or disprove each of the following.

    • (a) $p$ is always a bijection.
    • (b) $p$ is always continuous.
    • (c) $p$ is continuous if $X$ is locally path-connected.
    • (d) the inverse of $p$ exists and is continuous if $X$ is locally path-connected.
    • (e) the inverse of $p$ always exists and is continuous.

  3. Let $(X, x_0)$ be a path-connected based topological space and consider the space $Y$ and the map $p: Y\to X$ constructed in the lectures corresponding to the some subgroup $H \subset G$. Let $U$ be a path-connected open subset of $X$ and let $U_{\gamma}$ be an associated set as in the lectures. Prove or disprove each of the following.

    • (a) $p\vert_{U_\gamma}: U_\gamma \to U$ is bijective.
    • (b) $p\vert_{U_\gamma}: U_\gamma \to U$ is continuous.
    • (c) $p\vert_{U_\gamma}: U_\gamma \to U$ is surjective.
    • (d) $p\vert_{U_\gamma}: U_\gamma \to U$ has a continuous inverse.
    • (e) $p\vert_{U_\gamma}: U_\gamma \to U$ is injective.