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).

Homology : Overview

due by Monday, Feb 21, 2022

  1. For a topological space $X\neq\phi$, we define the suspension $\Sigma (X)$ of $X$ to be the quotient $\Sigma (X) = (X \times [-1, 1])/\sim$ where $\sim$ is the equivalence relation generated by $(x_1, t_1) \sim (x_2, t_2)$ if $t_1 = t_2$ and $t_1 \in \{-1, 1\}$. Let $i_X\colon\thinspace X \to \Sigma(X)$ be given by $i_X(x) = (x, 0)$.

    • (a) Show that $\widetilde{H}_0(\Sigma (X)) = 0$ and $\widetilde{H}_{n+1}(\Sigma (X))\cong \widetilde{H}_n(X)$ for $n \geq 0$.

    • (b) Show that, for all $n\geq 0$, $i_{X*}\colon\thinspace\widetilde{H}_n(X) \to \widetilde{H}_n(\Sigma (X))$ is the zero homomorphism.

    • (c) Define $\Sigma^n(X)$ inductively by $\Sigma^{n+1}(X)=\Sigma(\Sigma^n(X))$ for $n\geq 0$ and $\Sigma^0(X) = X$. Define $\Sigma^\infty(X)$ as the direct limit of the directed system $$X \overset{i_X}\longrightarrow \Sigma(X)\overset{i_{\Sigma (X)}}\longrightarrow \dots \to \Sigma^n(X) \overset{i_{\Sigma^n (X)}}\longrightarrow \dots .$$ What is $H_*(\Sigma^\infty(X))$ (prove your answer)?

  2. Let $X\subset \mathbb{R}^2$ be the two-holed disc $$X = D((0, 0), 1)\setminus\left(\overset{\circ} D((\frac{1}{2}, 0), \frac{1}{4}) \cup \overset{\circ} D((-\frac{1}{2}, 0), \frac{1}{4})\right),$$ where $D((x, y), r)$ and $\overset{\circ} D((x, y), r)$ denote the closed and open discs with centre $(x, y)$ and radius $r$, respectively.

    • (a) What is the homology of $X$ (prove your answer)?

    • (b) Let $f\colon\thinspace X\to X$ be a map that is homotopic to the identity on $X$. Show that $f$ has a fixed point.

    • (c) Prove or disprove: there is a map $f\colon\thinspace X \to X$ with no fixed point.

    • (a) Let $X = I_1 \coprod I_2$ be a disjoint union of two closed intervals and let $f\colon\thinspace X \to S^3$ be an embedding. Show that $$\widetilde{H}_n(S^3 \setminus f(X))= \begin{cases} \mathbb{Z}&\mbox{if } n = 2 \\ 0 & \mbox{if } n \neq 2 \end{cases}$$

    • (b) Let $Y = S^1\times I$ be an annulus and let $f\colon\thinspace Y \to S^3$ be an embedding. Show that $$\widetilde{H}_n(S^3 \setminus f(Y))= \begin{cases} \mathbb{Z}&\mbox{if } n = 1 \\ 0 & \mbox{if } n \neq 1 \end{cases}$$