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).

CW-Complexes and Higher Homotopy Groups

due by Monday, Dec 21, 2020

  1. Let $X$ be a CW complex with two $0$-cells and three $1$-cells, so that each of the attaching maps for the $1$-cells (to the $0$-skeleton) are injective. Determine the fundamental group of $X$ (with proof).

  2. Let $\Gamma$ be a graph and $X$ a CW-complex. Consider a map $f: |\Gamma|\to X$. Then prove or disprove the following.

    • (a) $f$ must be homotopic to a map with image in the $0$-skeleton $X^{(0)}$ of $X$.
    • (b) $f$ must be homotopic to a map with image in the $1$-skeleton $X^{(1)}$ of $X$.

  3. Fix $n\geq 2$. Let $D=[0, 1]^n$ and let $A$ and $B$ be subsets of $D$ given by $$A = \{(x_1, x_2, \dots, x_n)\in D : \exists i, 1 \leq i \leq n, x_i \in \{0, 1\}\}$$ and $$B = (\{1\}\times [0, 1]^{n- 1})\cup\{(x_1, x_2, \dots, x_n)\in D : \exists i, 1 < i \leq n, x_i \in \{0, 1\}\}$$ Let $X = D / A$ and $Y = D/ B$ be the quotients of $D$ with $A$ and $B$ identified to points, respectively. Prove or disprove the following

    • (a) $X$ is homeomorphic to the $n$-sphere $S^n$.
    • (b) $Y$ is homeomorphic to $D$.