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).

Paths and Homotopies

due by Monday, Oct 19, 2020

  1. Fix a space $X$ and consider the relation on $X$ where points $p$ and $q$ are related if and only if there is an injective, continuous function $f : [0, 1] \to X$ such that $f(0) = p$ and $f(1) = q$. For each of the following properties for this relation, prove or disprove that the property must hold.

    • (a) Refelexivity.
    • (b) Symmetry.
    • (c) Transitivity.

  2. Fix a topological space X. Given a homotopy $H: [0,1]\times [0, 1]\to X$, we get a function $\varphi_H$ to the space of functions $X^{[0, 1]} = \{f: [0, 1]\to X\}$ from the interval to X with the product topology given by $$\varphi_H(t) = s \mapsto H(s, t).$$ Prove or disprove the following statements.

    • (a) If $H$ is continuous then $\varphi_H$ must be continuous.
    • (b) If $\varphi_H$ is continuous then $H$ must be continuous.