## MATH 422 Introduction to Geometric Topology, 2018 Spring

### Instructor: Professor Jae Choon Cha

Office hour: by an appointment
Instructor’s web page: http://gt.postech.ac.kr/~jccha/

### Classroom and hour

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

See POVIS LMS.

### 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 $\beta\simeq (\cdots(\alpha_1*\alpha_2)*\alpha_3 *\cdots )*\alpha_n \rel \{0,1\}$.