### Instructor: Professor Jae Choon Cha

Office hour: by an appointment

Instructor’s web page: http://gt.postech.ac.kr/~jccha/

### Course home page

http://gt.postech.ac.kr/~jccha/intro-geometric-topology-2017-spring/

### Classroom and hour

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

### Syllabus

A brief version is available from POVIS. More detailed pdf version is available here.

### Lecture Notes

- Part 1: homotopy, fundamental group, reparametrization, free homotopy of loop, induced maps [revised 2017/03/13]
- Part 2: covering spaces, graphs, basic covering space theory with proofs [revised 2017/03/27]
- Part 3: free groups, group presentation, pushout, Seifert-van Kampen theorem [revised 2017/04/10]
- Part 4: fundamental group of surfaces, knots, Wirtinger presentation, Alexander polynomial [revised 2017/05/22]
- Part 5: smooth manifolds, tangent space, regular values, fundamental theorem of algebra [revised 2017/05/23]
- Part 6: Sard-Brown theorem, manifolds with boundary, orientation, degree, Brower’s fixed point theorem
- Appendix: proof of Sard’s theorem

### Final project

The final project of this semester is described here. Email your project to me as a single attached pdf file. (Scanning your handwriting is completely fine, as well as using software such as LaTeX.) It is due by June 14 6am.

### Homework Problems and announcements

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

**Problem Set #4** (due 5/29)

- Let $K$ be the trefoil knot. The following is another method to compute the fundamental group of $\R^3-K$. Recall that $K$ is a simple closed curve lying on the standard torus $T := \{(\sqrt{x^2+y^2}-2)^2 + z^2 = 1\}$. Let $A:= \{(\sqrt{x^2+y^2}-2)^2 + z^2 \le 1\}-K$ and $B := \{(\sqrt{x^2+y^2}-2)^2 + z^2 \ge 1\}-K$. Observe that $A\cup B=\R^3-K$ and $A\cap B=T-K$.
- Compute $\pi_1(A)$, $\pi_1(B)$ and $\pi_1(A\cap B)$
- Using (a) and Seifert-van Kampen, compute $\pi_1(\R^3-K)$ as a presentation.
- Find a group isomorphism between the presentation you obtained in (b) and the Wirtinger presentation we obtained in class. (Describe the image of each generator under your isomorphism.)
- Compute a Wirtinger presentation of the following knot $K$. (Show explicitly your choice of generators in the diagram!)

$K=$ - An $m$-component link in $\R^3$ is defined to be the union of $m$ disjoint simple closed curves embedded in $\R^3$. For example, a knot is a 1-component link.
- Let $L_1$ and $L_2$ be the 2-component links in $\R^3$ illustrated below.

$L_1$ = $L_2$ =

Find a presentation of $\pi_1(\R^3-L_1)$ and $\pi_1(\R^3-L_2)$, generalizing the Wirtinger presentation of knots. - Compute the abelianization of $\pi_1(\R^3-L_1)$ and $\pi_1(\R^3-L_2)$.
- In general, for an $m$-component of $L$ in $\R^3$, can you guess what the abelianization of $\pi_1(\R^3-L)$ must be? State and prove.
- Compute the Alexander polynomial of the knot $K$ illustrated in #2, and show that $K$ is not equivalent to any one of the unknot, trefoil and figure eight.
- Prove the following.
- For $X\subset\R^k$, a map $f\colon X \to \R^\ell$ given by $f(x_1,\ldots,x_k)=(f_1(x_1,\ldots,x_k), \ldots, f_\ell(x_1,\ldots,x_k))$ is smooth if and only if each $f_i\colon X \to \R$ is smooth.
- If $f, g\colon X\to \R$ is smooth, then $f+g$, $f-g$, $f\cdot g$ are smooth. In addition, if $g$ is nonzero on $X$, show that $f/g$ is smooth.
- Write an explicit formulas of the stereographic projection $h\colon S^2 – \{(0,0,1)\} \to \R^2=\mathbb{C}$, and its inverse. Describe a proof that $h$ is a diffeomorphism, using your formulas and #5 above.
- For two smooth manifolds $M\subset \R^k$ and $N\subset \R^l$, show that $M\times N \subset \R^{k+l}$ is a smooth manifold and $T(M\times N)_{(x,y)} = TM_x \times TM_y$ in $\R^{k+l}$.
- For a smooth map $f\colon M\to N$, the graph of $f$ is defined to be $G(f)=\{(x,f(x))\in M\times N \mid x\in M\}$. Show that $G(f)$ is a smooth manifold, and that the tangent space $TG(f)_{(x,f(x))}\subset TM_x \times TN_{f(x)}$ equals the graph of the linear map $df_x$. (It’s a good idea to think about the case of $f\colon \R\to \R$ as an example)

**Problem Set #3 **(due 4/24)

- Consider the subsets $A=\{(0,0,z)\in \R^3\}$, $K=\{(x,y,0)\in \R^3 \mid x^2+y^2=1\}$ and $J=\{(x+3,y,z)\in \R^3\mid (x,y,z)\in K\}$.
- Show that $\R^3-A$ is homotopy equivalent to $S^1$.
- Let $i \colon K\to \R^3-A$ be the inclusion. Compute the induced map $i_*\colon \pi_1(K) \to \pi_1(\R^3-A)$.
- Show that there is no map $f\colon D^2 \to \R^3-A$ such that $f|_{S^1}$ is a homeomorphism $S^1\to K$. (This means: $K$ does not bound any (singular) disk disjoint from $A$ in $\R^3$.)
- Show that there does not exist a homeomorphism $h\colon \R^3\to \R^3$ such that $h(A)=A$ and $h(K)=J$. (This means: the “link” $A\cup K$ in $\R^3$ is not equivalent to $A\cup J$.)
- Show that $G :=\langle x, y \mid xyx^{-1}y^{-1}, x^2 y \rangle$ is isomorphic to $\Z$. Find a word in $x,y$ which generates the group $G$.
- For continuous maps $i\colon C \to A$ and $j\colon C\to B$ of topological spaces $A$, $B$, $C$, define $X=A\cup B/\sim$ where $a\sim b$ ($a\in A$, $b\in B$) if and only if $a=i(c)$ and $b=j(c)$ for some $c\in C$. Show that $X$ endowed with the naturally defined maps $f\colon A\to A\cup B \to X$ and $g\colon B\to A\cup B \to X$ is a pushout in the following sense: if $\alpha\colon A\to Y$ and $\beta\colon B \to Y$ are maps satisfying $\alpha i = \beta j$, then there is a unique map $\phi\colon X \to Y$ satisfying $\phi f=\alpha$ and $\phi g=\beta$.
- Let $h\colon S^1\times S^1 \to S^1\times S^1$ be the map defined by $h(z, w)=(z^aw^b, z^cw^d)$ where $a,b,c,d\in \Z$.
- Compute the map $h_*\colon \pi_1(S^1\times S^1) \to \pi_1(S^1\times S^1)$, as a $2\times 2$ matrix over $\Z$ representing a linear map $\Z^2 \to \Z^2$, identifying $\pi_1(S^1\times S^1)$ with $\Z^2$.
- Show that $h$ is a homeomorphism if and only if $h_*$, as a matrix, has determinant $\pm 1$.
- Let $X=(D^2 \times S^1) \cup (D^2\times S^1)/(z,w)\sim h(z,w)$ (here $(z,w)\in S^1\times S^1\subset D^2\times S^1$). Show that $X$ is a 3-dimensional compact manifold if $h_*$ has determinant $\pm1$. Compute $\pi_1(X)$.
- Let $K$ be a bounded subset in $\R^n$. Viewing $K\subset \R^n \subset S^n=\R^n\cup\{\infty\}$, Show that $\pi_1(\R^n-K)$ is isomorphic to $\pi_1(S^n-K)$. (Hint: you may use the Seifert-van Kampen theorem.)
- In what follows, every graph is assumed to have finitely many vertices and edges.
- Recall that a tree is a connected graph with no cycle. (For instance, “X”, “Y”, “T” are trees, while “A” and “B” are not.) Show that a tree is simply connected.
- Let $X$ be a connected graph and $e$ be an edge of $X$ which is from a vertex $v$ to another. Let $Y$ be the graph obtained from $X$ by deleting $e$. Suppose $Y$ is connected. When a presentation of $\pi_1(Y,v)$ is given, compute a presentation of $\pi_1(X,v)$ in terms of that of $\pi_1(Y,v)$.
- Show that a graph $X$ contains a spanning tree $T$, that is, $T$ is a subgraph in $X$ which is a tree containing all vertices of $X$.
- Using (a), (b) and (c), show that the fundamental group of a connected graph is a free group.
- Let $X$ be the graph with one vertex $*$ and $n$ edges $x_1,\ldots,x_n$ attached to $*$. Each $x_i$ can be regarded as a loop.
- Show that $\pi_1(X,*)$ is isomorphic to the free group generated by the (path homotopy classes of) the loops $x_1,\ldots, x_n$.
- Given a word $w=x_{i_1}^{\epsilon_1}\cdots x_{i_r}^{\epsilon_r}$, let $f_w\colon (S^1,1) \to (X,*)$ be the loop representing $w$. Define $Y_w$ to be the space $(D^2 \cup X)\mathbin{/}z\sim f_w(x)$ for $z\in S^1$. Compute a presentation for $\pi_1(Y_w,*)$.S
- Show the following by generalizing (b): for a group $G=\langle x_1,\ldots,x_n\mid r_1,\ldots,r_m\rangle$, there is a path connected space $K$ such that $\pi_1(K)\cong G$.
- Let $X$ be the figure eight graph, which has one vertex $*$ and two edges $x$ and $y$. Now you know that $\pi_1(X,*)$ is the free group generated by $x$ and $y$.
- Describe a regular cover $p_1\colon (X_1,x_1)\to (X,*)$, as a graph, which has $G(X_1|X)\cong \Z_2\times\Z_2$.
- Describe a regular cover $p_2\colon (X_2,x_2)\to (X,*)$, as a graph, which satisfies $(p_2) _* \pi_1(X_2,x_2)$ is the normal subgroup of $\pi_1(X,*)$ generated by $x^2$, $y^2$, and $(xyx^{-1}y^{-1})^2$.
- Show that there is a covering map $q\colon X_2\to X_1$ such that $p_1 q = p_2$. Determine $G(X_2|X_1)$.

**Problem Set #2** (due 4/03)

- Let $X$ be the graph consisting of one vertex $x_0$ and two edges $a$, $b$. (You must be able to draw it.) Give an example of a pair of loops $\alpha, \beta\colon S^1\to X$ based at $x_0$ which are freely homotopic but not homotopic rel $\{1\}$. (Hint: to show that your loops are not homotopic rel $\{1\}$, you may compare endpoints of lifts, viewing them as paths, in an appropriate covering of $X$.)
- Suppose $p\colon E\to B$ and $f\colon X \to B$. Define $f^*(E)=\{(e,x)\in E\times X \mid p(e)=f(x)\}$. $f^*(E)$ is called the
*pullback*of $E$ under $f$. - Show that the projections of $E\times X$ onto $E$ and $X$ give rise to maps $q\colon f^*(E) \to X$ and $\tilde f\colon f^*(E)\to E$ making the following diagram commute: \[ \begin{array}{c@{}c@{}c}f^*(E) & \longrightarrow & E \\ \downarrow & & \downarrow\\ X & \longrightarrow & B \end{array} \]
- Show that for any map $g\colon Y\to X$ and $u\colon Y\to E$ such that $fg=pu$, there is a unique map $h\colon Y\to f^*(E)$ such that $\tilde f h = u$ and $qh=g$. (This is called the universal property of the pullback.)
- This is a continuation of #2. Suppose $p$ is a covering map. Prove the following.
- $q\colon f^*(E) \to X$ is a covering map.
- The fiber $p^{-1}(f(x))$ is in 1-1 correspondence with the fiber $q^{-1}(x)$.
- When $\tilde f(r_0)=e_0$, $q_*\pi_1(f^*(E),r_0) = f_*^{-1}(p_*\pi_1(E,e_0))$.
- The covering transformation groups $G(f^*(E)\mid X)$ and $G(E\mid B)$ are isomorphic if $f_*$ on $\pi_1$ is surjective. (Here, assume involved spaces are connected and locally path connected if necessary.)
- An $n$-manifold is defined to be a second countable Hausdorff topological space satisfying that for any point, there is an open neighborhood homeomorphic to an open subset of $\R^n$. Prove that any $n$-manifold has a universal cover.
- Recall that the $n$-dimensional real projective space is defined by $\R P^n = S^n/x\sim -x$.
- Show that the quotient map $S^n \to \R P^n$ is a covering map.
- It is known that $S^n$ is simply connected for $n\ge 2$. (We will show this later!) Using this, show that $\pi_1(\R P^n)$ is isomorphic to the cyclic group of order 2 for $n\ge 2$.
- Let $SO(n)$ be the space of $n\times n$ orthogonal matrices $A$ with $\det(A)=1$. $SO(3)$ acts on $\R^3$ by multiplication on the left, viewing $x\in \R^3$ as a column vector.
- Show that any $A\in SO(3)$ is a rotation around a line passing through the origin. (Hint: first show that $A$ must have 1 as an eigenvalue; you may also use that a matrix in $SO(2)$ is always a rotation.)
- Given a unit vector $x$ in $\R^3$ and $\theta\in [-\pi,\pi]$, a rotation $A\in SO(3)$ with axis $x$ and angle $\theta$ is associated. Prove that the correspondence $A \to (\theta/\pi)x$ induces a homeomorphism between $SO(3)$ and $D^3/x\sim -x \; (x\in S^2)$.
- Show that $\R P^3 \cong D^3/x\sim -x \; (x\in S^2)$; using this, and assuming $S^3$ is simply connected, compute $\pi_1(SO(3))$.
- Find a loop $t\mapsto A_t\in SO(3)$ ($0\le t \le 1$) representing a nontrivial element in $\pi_1(SO(3))$.
- Let $X$ be the graph in #1. Define $f\colon (X,x_0)\to (S^1,1)$ be the map sending $a$ to the loop $\alpha(s)=e^{2\pi is}\; (0\le s\le 1)$ and $b$ to the constant loop $c_1$. Note that there is the induced map $f_*\colon\pi_1(X,x_0)\to \pi_1(S^1,1)=\Z$. Let $h$ be the projection $\Z\to \Z/n$.
- What is the covering $p\colon (E,e_0)\to (X,x_0)$ satisfying $p_*\pi_1(E,e_0)=\Ker(hf_*)$? Describe $E$ as a graph and $p$ as a map between graphs.
- Do (a) for the map $g\colon (X,x_0)\to (S^1,1)$, in place of $f$, which takes both $a$ and $b$ to $\alpha$.

**Problem Set #1** (due 3/20)

- Give a detailed proof that homotopy is an equivalence relation.
- 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.
- Recall that a subset $X$ in $\R^n$ is
*star-shaped*if there is $x_0 \in X$ such that for any $x\in X$, the line segment with endpoints $x$ and $x_0$ lies in $X$. - Show that a star-shaped space is contractible.
- Give an example of a subset in $X$ which is contractible but not star-shaped, with proofs.
- Fix $*$ in $S^2 = \{x\in \R^3\mid |x|=1\}$ and $x_0$ in a path connected space $X$. Define \[[S^2,X]_0 = \{\phi\colon (S^2,*)\to (X,x_0)\} / \simeq \text{ rel } *.\]Let $D^2=\{x\in \R^2 \mid |x|\le 1\}$ and $\partial D^2 = \{x\in D^2\mid |x|=1\}$. Define \[\pi_2(X,x_0) = \{f\colon (D^2,\partial D^2) \to (X,x_0)\}/\simeq \text{ rel } \partial D^2.\]
- Show that $\pi_2(X,x_0)$ is in 1-1 correspondence with $[S^2,X]_0$.
- For $\alpha\colon (I,\{0,1\}) \to (X,x_0)$ and $f\colon (D^2,\partial D^2)\to (X,x_0)$, define $\alpha\cdot f\colon (D^2,\partial D^2) \to (X,x_0)$ by \[(\alpha\cdot f)(x) = \begin{cases} f(2x) &\text{for }|x|\le \frac12, \\ \alpha(1-2|x|) &\text{for } |x|\ge \frac12.\end{cases}\]Show that $([\alpha],[f]) \mapsto [\alpha\cdot f]$ is a well-defined function $\pi_1(X,x_0)\times \pi_2(X,x_0) \to \pi_2(X,x_0)$. (Indeed it is a group action of $\pi_1(X,x_0)$ on $\pi_2(X,x_0)$.)
- Define $\sim$ on $\pi_2(X,x_0)$ by $[f] \sim [g]$ iff $[g]=[\alpha\cdot f]$ for some $[\alpha]\in\pi_1(X,x_0)$. Show that $\sim$ is an equivalence relation.
- Show that $[S^2,X]$ is in 1-1 correspondence with $\pi_2(X,x_0)/\sim$.