### 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

### Syllabus

See POVIS LMS.

### Lecture Notes

- #1 Homotopy and fundamental group
- #2 Change of basepoint and free homotopy of maps $S^1\to X$
- #3 Induced homomorphisms and products
- #4 Covering spaces and graphs
- #5 Fundamental theorems on covering spaces, part I
- #6 Fundamental theorems on covering spaces, part II
- #7 Free groups and group presentations
- #8 Seifert van-Kampen theorem
- #9 Surfaces and abelianization
- #10 Knots
- #11 Wirtinger presentation
- #12 Alexander polynomial
- #13 Smooth manifolds
- #14 Regular values and fundamental theorem of algebra
- #15 Manifolds with boundary and Brouwer’s fixed point theorem

### 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. $\def\C{\mathbb{C}} \def\R{\mathbb{R}} \def\Z{\mathbb{Z}} \def\Q{\mathbb{Q}} \def\Ker{\operatorname{Ker}} \def\rel{\text{ rel }} \def\sm{\smallsetminus}$

**Announcement: make-up class**

May 25 Friday, 2:00pm

Math Building 206

**Problem Set #6** (due 06/01)

- For two smooth manifolds $M\subset \R^k$ and $N\subset \R^\ell$, show that $M\times N \subset \R^{k+\ell}$ is a smooth manifold and $T(M\times N)_{(x,y)} = TM_x \times TM_y$ in $\R^{k+\ell}$.
- 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)
- For a smooth manifold $M\subset \R^k$, define the
*tangent bundle*of $M$ by $TM := \{(x, v)\subset \R^k\times \R^k\mid x\in M,\ v\in TM_x\}$ and the*normal bundle*of $M$ by $\nu M:=\{(x, v)\subset \R^k\times \R^k\mid x\in M,\ v\in (TM_x)^\perp\}$. Here, $(TM_x)^\perp:=\{v\in \R^k \mid v \perp TM_x\}$ designates the orthogonal complement. Show that $TM$ and $\nu M$ are smooth manifolds. - Show that the following surfaces in $\R^3$ are smooth manifolds of dimension 2. (Assume $a,b,c>0$.)
- Ellipse: $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$
- Hyperboloid: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z^2}{c^2}\pm 1$
- Hyperbolic paraboloid: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=\frac{z}{c}$
- Suppose $A(t)=[a_{ij}(t)]$ is an $n\times m$ matrix whose entries $a_{ij}(t)$ are real-valued functions defined on an open subset $U\subset\R$. Suppose $A(t_0)$ has rank $n$. Show that there is an open neighborhood of $t_0$ on which $A(t)$ has rank $n$.
- Suppose $f\colon M\to N$ is a smooth map.
- Suppose $x\in M$ is a regular point for $f$. Show that there is an open neighborhood of $x$ in $M$ which consists of regular points for $f$.
- Suppose $M$ is compact, and $y\in N$ is a regular value for $f$. Show that there is an open neighborhood of $y$ in $N$ which consists of regular values for $f$.
- Suppose $M$ is a smooth manifold of dimension $n$ with boundary. Recall the definition of $\partial M$: for $x\in M$, $x\in \partial M$ if and only if there is a parametrization $g\colon U\cap \R^n_+ \to M$ such that $x\in g(U\cap \partial \R^n_+)$. Prove that $\partial M$ is well-defined, independent of the choice of a parametrization.

More problems will be posted soon!

**Problem Set #5** (due 05/14)

- Draw a sequence of knot diagrams, in such a way that the knot given by each diagram is obviously seen to be equivalent to the next one, to show that the following knot is trivial.

- Let $K$ be the figure eight knot. Recall that we have computed a Wirtinger presentation for $K$. See Lecture Note #11. Use the Wirtinger presentation to show that there is an epimorphism of $G=\pi_1(\R^3\sm K)$ onto the dihedral group $D_10$. Conclude that the figure eight knot $K$ is nontrivial. (Hint: try the map $x\mapsto (1\,2)(3\,4)$, $w\mapsto (2\,3)(4\,5)$, where $x$ and $w$ are the generators used in Lecture Note #11.)
- Recall from Lecture Note #12 that we computed the Alexander polynomial of the figure eight knot. In the computation, we claimed that the vectors $y_1$ and $y_3$ form a basis for the vector space $G’/G^{\prime\prime}\otimes \Q$.
- Express each of $t^{-1}y_2$, $y_2$, $ty_2$, $t^2y_2$ as a linear combination of $y_1$ and $y_3$.
- Why are $y_1$ and $y_3$ linearly independent? Provide a detailed proof.
- Compute a Wirtinger presentation of the following knot $K$. (Show explicitly your choice of generators in the diagram!)

- Compute the Alexander polynomial of the above knot $K$. Show that $K$ is not equivalent to any one of the trivial knot, trefoil knot and figure eight knot.
- Recall from Lecture Note #11 that certain conjugates of the four Wirtinger relators of the figure eight knot diagram satisfy the relation (P). For the trefoil knot diagram in Lecture Note #12, find conjugates of the three Wirtinger relators which satisfies an analogous relation. Explain how you found them, using a picture of some loops on the knot diagram. Do the same for the knot in the above #4.
- This is a continuation of #6. Generalize the relation (P) to a theorem for an arbitrary knot diagram. Give a proof.
- 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.
- Suppose $U$ is the trivial knot in $S^3$.
- Show that the set of free homotopy classes of maps $S^1\to \R^3\sm U$ is in 1-1 correspondence with $\Z$.
- Using (a), prove that there is no homeomorphism $h\colon \R^3\to \R^3$ such that $h(L_1)=L_2$, where $L_1$ and $L_2$ are the links in #8. In other words, show that the links $L_1$ and $L_2$ are not equivalent.

**Problem Set #4** (due 04/25)

- 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 is a generator of the group $G$.
- For two given 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 $c\in C$. $X$ is a topological space endowed with the quotient topology. 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 $\phi g=\beta$.
- Give detailed proofs of the following facts. In what follows, all graphs are assumed to be connected and with finitely many vertices and edges.
- A tree is contractible. (Here a tree is a graph with no cycle; use induction on the number of edges.)
- For a graph $X$, $\pi_1(X)$ is a free group with finitely many generators. (Take a maximal tree of $X$, and use induction on the number of edges not contained in the maximal tree.)
- 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 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 $z\in S^1 \subset D^2$. Compute a presentation for $\pi_1(Y_w,*)$.
- 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 $h\colon S^1\times S^1 \to S^1\times S^1$ be the map defined by $h(z, w)=(z^aw^b, z^cw^d)$ where $a,b,c,d\in \Z$. (As usual, regard $S^1$ as the space of unit length complex numbers.)
- 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 $\R^n$ with $n\ge 3$. Viewing $K\subset \R^n \subset S^n=\R^n\cup\{\infty\}$, Show that $\pi_1(\R^n-K)$ is isomorphic to $\pi_1(S^n-K)$.
- Define the $n$-dimensional real projective space by $\R P^n = S^n/x\sim -x$.
- Show that the quotient map $S^n \to \R P^n$ is a covering map.
- Show that $\pi_1(\R P^n)\cong \Z_2$ for $n\ge 2$.
- Let $SO(n)$ be the space of $n\times n$ matrices $A$ satisfying $A^T A=I$ and $\det(A)=1$. We regard $SO(n)$ as a subspace of $\R^{n^2}$, by viewing each entry of a matrix as a coordinate. $SO(n)$ acts on $\R^n$ by multiplication on the left, viewing $x\in \R^n$ as a column vector.
- Show that any $A\in SO(3)$ has 1 as an eigenvalue. Using this, show that $A$ is a rotation around a line passing through the origin.
- Given a unit vector $x$ in $\R^3$ and $\theta\in [0,\pi]$, associate a rotation $A\in SO(3)$ with axis $x$ and angle $\theta$ along the right hand orientation. Prove that the correspondence $A \to (\theta/\pi)x$ induces a homeomorphism between $SO(3)$ and $D^3/x\sim -x \; (x\in S^2)$.
- Using #7 and (b), compute $\pi_1(SO(3))$.
- Give an alternative computation of $\pi_1(SO(3))$, by applying the Seifert-van Kampen theorem directly to the quotient space $D^3/x\sim -x \; (x\in S^2)$. Here, use $U=q(\operatorname{int} D^3)$ and $V=q(D^3\sm \{0\})$, where $q\colon D^3 \to D^3/x\sim -x \; (x\in S^2)$ is the quotient map.
- Find a loop $s\mapsto A_s\in SO(3)$ ($0\le s \le 1$) representing a nontrivial element in $\pi_1(SO(3))$. Describe your matrix $A_s$ explicitly.

**Problem Set #3** (due 04/11)

- We say that a covering $p\colon E\to B$ with $E$ connected is
*regular*if $p_*(\pi_1(E,e_0))$ is a normal subgroup of $\pi_1(B,b_0)$ for every $b_0\in B$ and $e_0\in p^{-1}(b_0)$. Otherwise, we say that $p\colon E\to B$ is*irregular*. Prove that the following are equivalent. [Typo fixed on 04/09] - $p\colon E\to B$ is regular.
- For a loop $\alpha$ in $B$, either all the lifts of $\alpha$ are loops, or all the lifts of $\alpha$ are not loops.
- Explain why every connected covering of $S^1$ is regular. Find an example of an irregular cover of the graph with one vertex and two edges, with a proof.
- Suppose $B$ is connected, locally path connected and semi-locally simply connected. Prove the following: for every surjective homomorphism $\pi_1(B,b_0)\to G$ onto a group $G$, there is a regular covering $p\colon E\to B$ such that $E$ is connected and the covering transformation group is isomorphic to $G$. [Typo fixed on 04/09]
- Prove that a path connected space $X$ is simply connected if and only if for every pair of paths $\alpha$, $\beta\colon I\to X$ such that $\alpha(0)=\beta(0)$ and $\alpha(1)=\beta(1)$, $\alpha$ and $\beta$ are homotopic rel $\{0,1\}$.
- Recall, from the proof of the realization part of the classification of coverings, that we defined the subsets of the form $\langle\alpha,V\rangle$. (See Lecture Note #6.) Give a fully detailed proof that they form a base for a topology.
- Consider the complex exponential map $\exp\colon \C \to C\sm \{0\}$ defined by $\exp(z)=e^z$. As usual, for a domain $D\subset \C\sm\{0\}$, we say that a continuous map $L\colon D\to \C$ is a logarithm if $\exp(L(z))=z$ for every $z$. Give a topological proof that there does not exist a logarithm on $\C\sm \{0\}$, using the lifting criterion. In addition, for a given domain $D\subset \C\sm \{0\}$, find a necessary and sufficient condition for a logarithm $L\colon D\to \C$ to exist, in terms of the fundamental groups.
- Suppose a group $G$ acts on a connected space $X$, on the left: $G\times X \to X$. (Recall our convention that maps are assumed to be continuous unless; here $G$ has discrete topology.) Let $B$ be the orbit space with the quotient topology, that is, $B=X/\sim$ where $x\sim y$ if $y=gx$ for some $g\in G$. Suppose, for each $x\in X$, there is an open neighborhood $U$ such that $gU\cap U=\emptyset$ for every $g\in G$ which is not the identity. Prove the following.
- The quotient map $p\colon X\to B$ is a covering map.
- $p$ is regular.
- The covering transformation group is isomorphic to $G$.
- If $X$ is locally path connected and simply connected, then $\pi_1(B,*)\cong G$.

**Problem Set #2** (due 03/28)

- The $n$-dimensional real projective space is a quotient space of $S^n$ which 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, that is, $\pi_1(S^n)$ is trivial for any choice of a basepoint. (We will prove this later!) Using this, show that $\pi_1(\R P^n)$ is isomorphic to the cyclic group of order 2 for $n\ge 2$.
- Suppose $p\colon E\to B$ and $f\colon X \to B$. Define $f^*(E)=\{(e,x)\in E\times X \mid p(e)=f(x)\}$.
- 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 if $p$ is a covering map, then $f^*(E) \to X$ is a covering map.
- Show that $p^{-1}(f(x))$ is in 1-1 correspondence with $q^{-1}(x)$.
- Suppose $p\colon E\to B$ is a covering map and $b_0\in B$, $e_0\in E$ are basepoints such that $p(e_0)=b_0$. For two loops $\alpha$, $\beta$ in $B$ based at $b_0$, let $\tilde\alpha$, $\tilde\beta$ be lifts in $E$ starting from $e_0$. Show that $\tilde\alpha(1)=\tilde\beta(1)$ if $[\alpha]=[\beta]$ in $\pi_1(B,b_0)$.
- Let $f\colon E\to B$ be the covering described at the end of Lecture Note #4. See the following diagram.

- Show that $[x],\, [y]\in \pi_1(B,*)$ are nontrivial elements, by applying the above #3.
- Show $[x]\ne [y]$ in $\pi_1(B,*)$. Show $[x]\ne [x]^2$ in $\pi_1(B,*)$.
- Explain why it is not possible to prove that $[x]^2$ is nontrivial in $\pi_1(B,*)$ by applying the above #3 to this covering. Find the maximal number of elements in $\pi_1(B,*)$ which can be seen to be mutually different by applying #3 to this covering. Explain why.
- Show that $[x]^2$ is nontrivial in $\pi_1(B,*)$. Try to find two different proofs, one by applying the above #3 to this covering, and another by using an appropriate map $B \to S^1$.
- Let $B$ be the base space of the covering in the above #4.
- Show that $[x][y]\ne [y][x]$ in $\pi_1(B,*)$, using lifts in a covering map.
- Find two maps $f,g\colon S^1\to B$ with $f(1)=*=g(1)$ such that $f$ and $g$ are freely homotopic but represent different elements in $\pi_1(B,*)$.

**Problem Set #1** (due 03/12)

- 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.
- Prove that the union of the $x$-axis, $y$-axis and $z$-axis in $\R^3$ is contractible.
- Show that if $X$ and $Y$ are homotopy equivalent, then $X\times Z$ and $Y\times Z$ are homotopy equivalent.
- Prove that each of the following pairs are homotopy equivalent.
- $D^n=\{x\in \R^n \mid |x|\le 1\}$ and $I^n=I\times\cdots\times I$.
- $S^n$ and $\R^{n+1}-\{0\}$.
- $S^1$ and $\R^3-\{z$-axis$\}$.
- 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.)
- 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.
- 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 $\beta\simeq (\cdots(\alpha_1*\alpha_2)*\alpha_3 *\cdots )*\alpha_n \rel \{0,1\}$.