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).

Graphs and Two-Complexes

due by Monday, Dec 14, 2020

  1. An edge path of length between distinct vertices $p$ and $q$ of a graph $\Gamma$ is said to be minimal if every edge path between $p$ and $q$ in has length at least k. Prove or disprove the following.

    • (a) Every minimal path is reduced.
    • (b) Every reduced path is minimal.

  2. Let $\Gamma$ be a graph that contains no (non-constant) reduced loops. Prove or disprove that its geometric realization $|\Gamma|$ is homotopy equivalent to a discrete topological space.

  3. Let $X$ be a topological space obtained by attaching a finite number of $1$-cells to a torus. Prove or disprove the following.

    • (a) $X$ must be path-connected.
    • (b) $X$ must be simply-connected.
    • (c) $X$ cannot be simply-connected.

  4. Let $X$ be the topological space obtained byattaching two 2-cells to the circle, with the attaching maps $z\mapsto z^2$ and $z\mapsto z^3$, respectively. Determine the fundamental group of $X$.