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).

Whitehead Theory

due by Monday, Dec 28, 2020

  1. For which integers $n\geq 0$ does every map from the $n$-sphere $S^n$ to the torus $S^1\times S^1$ extend to a map to the $(n+1)$-disc $D^{n+1}$. Note that the $1$-sphere is the circle and the $0$-sphere consists of $2$ points.

  2. Prove or disprove the following.

    • (a) If $X$ and $Y$ are Eilenberg-MacLane spaces, so is their product $X\times Y$.
    • (b) If $X$ is an Eilenberg-MacLane space and $Y$ is a connected cover of $X$, then $Y$ is an Eilenberg-MacLane space.
    • (c) If $X$ is an Eilenberg-MacLane space and $X$ is a connected cover of $Y$, then $Y$ is an Eilenberg-MacLane space.

  3. Let $X$ be a CW complex with fundamental group $\Z/2\Z$. Determine the number of maps from $X$ to $S^1$ up to homotopy (with proof).