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 Groups

due by Monday, Nov 9, 2020

  1. Prove or disprove: the free group on the empty set is the trivial group.

  2. Let $S$ be a set and let $W$ be the group of finite words in $S$, i.e., $$W = \{(x_1, \dots, x_n): n\geq 0, x_i\in S\ \forall i\}$$ and define a binary operation $*$ on $W$ by $$(x_1, \dots, x_n) * (y_1, \dots, y_m) = (x_1, \dots, x_n, y_1, \dots, y_m) .$$ Prove or disprove the following.

    • (a) $(W, *)$ is a Semigroup.
    • (b) $(W, *)$ is a Monoid.
    • (c) $(W, *)$ is a Group.

  3. Given a set $S$, consider words in $W = S\cup \bar{S}$ as in the construction of the free group, with the given equivalence relation. Show that every word is equivalent to a reduced word.