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).

Free Products and the Seifert-Van Kampen theorem

due by Monday, Dec 7, 2020
  1. Let $G$ be a group. Prove or disprove that the free product $G * \{e\}$ is isomorphic to $G$.
  2. Let $G = (\mathbb{Z}/2\mathbb{Z}) *(\mathbb{ Z}/2\mathbb{Z})$. Prove or disprove that $G$ contains a subgroup of finite index isomorphic to $\mathbb{Z}$.
  3. Suppose the topological space $X$ is the union of two path-connected open subsets whose intersection is simply-connected and each of which has fundamental group isomorphic to $\mathbb{Z}$. Prove or disprove that the fundamental group of $X$ must be isomorphic to the free group on two generators.