MATH 422 Introduction to Geometric Topology, 2018 Spring

Instructor: Professor Jae Choon Cha

Office hour: by an appointment
Instructor’s web page:

Classroom and hour

Monday and Wednesday 14:00-15:15, Science Building II 105



Lecture Notes

Homework Problems and announcements

Homework problems will be posted on this webpage, and will be collected in class on the due date. Turn it in BEFORE the lecture starts.  $\def\R{\mathbb{R}}\def\Z{\mathbb{Z}}\def\Ker{\operatorname{Ker}}\def\rel{\text{ rel }}$

Problem Set #1 (due 03/12)

  1. Give a detailed proof that homotopy is an equivalence relation.
  2. Show that a homotopy inverse of a homotopy equivalence $f\colon X\to Y$ is unique up to homotopy, that is, two homotopy inverses of $f$ are homotopic.
  3. Prove that the union of the $x$-axis, $y$-axis and $z$-axis in $\R^3$ is contractible.
  4. Show that if $X$ and $Y$ are homotopy equivalent, then $X\times Z$ and $Y\times Z$ are homotopy equivalent.
  5. Prove that each of the following pairs are homotopy equivalent.
    1. $D^n=\{x\in \R^n \mid |x|\le 1\}$ and $I^n=I\times\cdots\times I$.
    2. $S^n$ and $\R^{n+1}-\{0\}$.
    3. $S^1$ and $\R^3-\{z$-axis$\}$.
  6. Suppose $X$ is a space, and $\alpha\colon I\to X$ be a path. Let $Y$ be the quotient space $X\cup (I\times I)/\alpha(s)\sim(s,0)$ where $0\le s\le 1$. That is, for two points $p$, $q$ in the disjoint union $X\cup (I\times I)$, $p\sim q$ if and only if either (i) $p=q$, or (ii) $\{p,q\}=\{(s,0),\alpha(s)\}$ for some $s\in I$, or (iii) $p=(s,0)$ and $q=(t,0)$ for some $s,t\in I$ such that $\alpha(s)=\alpha(t)$. Show that $X$ and $Y$ are homotopy equivalent. (Hint: try to prove that the inclusion $X\hookrightarrow Y$ is a homotopy equivalence.  Here a homotopy on $I\times I$ which sends $I\times I$ onto $I\times [0,1-t]$ at time $t$ may be useful.)
  7. Suppose $X$ is a path connected space and $x_0,x_1\in X$.  Show that if $\pi_1(X,x_0)$ is abelian, then for any two paths $\alpha$ and $\beta$ in $X$ from $x_0$ to $x_1$, the isomorphisms $\alpha_*$, $\beta_*\colon \pi_1(X,x_1)\to \pi_1(X,x_0)$ are equal.
  8. Suppose $\alpha_1$, $\ldots$, $\alpha_n\colon I\to X$ are paths satisfying $\alpha_i(1)=\alpha_{i+1}(0)$ for $i=1,\ldots,n-1$. Define $\beta\colon I\to X$ by $\beta(s)=\alpha_i(n\cdot s-i+1)$ for $\frac{i-1}{n} \le s \le \frac{i}{n}$, $i=1,\ldots,n$. Show that $\beta\simeq (\cdots(\alpha_1*\alpha_2)*\alpha_3 *\cdots )*\alpha_n \rel \{0,1\}$.