1. Categories and Chain Complexes (31/1/2022).
  2. Simplicial Complexes and Simplicial Homology (7/2/2022).
  3. CW-Complexes (14/2/2022).
  4. Homology : Overview (21/2/2022).

CW-Complexes

due by Monday, Feb 14, 2022

  1. Let $\Gamma$ be a $1$-dimensional CW complex and let $X$ be 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 $X$.

  2. 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 $D$.