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).

Functoriality and Applications

due by Monday, Nov 2, 2020

  1. Let $f: (X, x_0)\to (Y, y_0)$ be a map between based topological spaces and let $f_*: \pi_1(X, x_0)\to \pi_1(Y, y_0)$ be the induced map on fundamental groups. Prove or disprove the following statements:

    • (a) If $f$ is injective then so is $f_*$.
    • (b) If $f$ is surjective then so is $f_*$.
    • (c) If $f$ is a homeomorphism then $f_*$ is an isomorphism.

  2. For each of the following subsets of the $2$-sphere $S^2 = \{(x, y, z)\in \mathbb{R}^3 : x^2 + y^2 + z^2 = 1\}$, prove or disprove that $S^2$ retracts onto the given subset. Note: we do not mean deformation retracts.

    • (a) The north pole $N = \{(0, 0, 1)\}$
    • (b) The equator E = $\{(x, y, 0)\in \mathbb{R}^3 : x^2 + y^2 = 1\}$.
    • (c) The closed northern hemisphere $H = \{(x, y, z)\in \mathbb{R}^3 : x^2 + y^2 + z^2 = 1, z \geq 0\}$