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

Final Project

The final project is writing a short article on the notion of the linking number.

Due date is June 12, by 11:59pm. Hand in your homework at the MA422 collection box on the first floor of Math Building.

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

Problem Set #6 (due 06/01)

Hand in your homework at the MA422 collection box on the first floor of Math Building, by 1:00pm on June 1st.

  1. 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 $\R^{k+\ell}$.
  2. 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)
  3. 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.
  4. Show that the following surfaces in $\R^3$ are smooth manifolds of dimension 2.  (Assume $a,b,c>0$.)
    1. Ellipse: $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$
    2. Hyperboloid: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z^2}{c^2}\pm 1$
    3. Hyperbolic paraboloid: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=\frac{z}{c}$
  5. Suppose $A(t)=[a_{ij}(t)]$ is an $n\times m$ matrix whose entries $a_{ij}(t)$ are real-valued continuous maps 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 $n$.
  6. Suppose $f\colon M\to N$ is a smooth map.
    1. 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 $f$.
    2. 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 $f$.
  7. 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.
  8. Suppose $M$, $N$ and $P$ are compact connected smooth manifolds of the same dimension, and $f\colon M\to N$, $g\colon N\to P$ are smooth maps. Show that $\deg gf = \deg g\cdot \deg f$.

Problem Set #5 (due 05/14)

  1. 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.
  2. 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.)
  3. 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 $G’/G^{\prime\prime}\otimes \Q$.
    1. Express each of $t^{-1}y_2$, $y_2$, $ty_2$, $t^2y_2$ as a linear combination of $y_1$ and $y_3$.
    2. Why are $y_1$ and $y_3$ linearly independent? Provide a detailed proof.
  4. Compute a Wirtinger presentation of the following knot $K$. (Show explicitly your choice of generators in the diagram!)
  5. 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.
  6. 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.
  7. This is a continuation of #6. Generalize the relation (P) to a theorem for an arbitrary knot diagram.  Give a proof.
  8. 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.
    1. 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.
    2. Compute the abelianization of $\pi_1(\R^3-L_1)$ and $\pi_1(\R^3-L_2)$.
    3. 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.
  9. Suppose $U$ is the trivial knot in $S^3$.
    1. Show that the set of free homotopy classes of maps $S^1\to \R^3\sm U$ is in 1-1 correspondence with $\Z$.
    2. 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)

  1. 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 $G$.
  2. 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 $\phi g=\beta$.
  3. Give detailed proofs of the following facts. In what follows, all graphs are assumed to be connected and with finitely many vertices and edges.
    1. A tree is contractible. (Here a tree is a graph with no cycle; use induction on the number of edges.)
    2. 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.)
  4. 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.
    1. Show that $\pi_1(X,*)$ is isomorphic to the free group generated by (the classes of) the loops $x_1,\ldots, x_n$.
    2. 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 $\pi_1(Y_w,*)$.
    3. 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 $\pi_1(K)\cong G$.
  5. 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.)
    1. 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 $\Z^2$.
    2. Show that $h$ is a homeomorphism if and only if $h_*$, as a matrix, has determinant $\pm 1$.
    3. 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 $\pi_1(X)$.
  6. 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 $\pi_1(S^n-K)$.
  7. Define the $n$-dimensional real projective space by $\R P^n = S^n/x\sim -x$.
    1. Show that the quotient map $S^n \to \R P^n$ is a covering map.
    2. Show that $\pi_1(\R P^n)\cong \Z_2$ for $n\ge 2$.
  8. 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.
    1. 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.
    2. 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 $D^3/x\sim -x \; (x\in S^2)$.
    3. Using #7 and (b), compute $\pi_1(SO(3))$.
    4. 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.
    5. 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)

  1. 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]
    1. $p\colon E\to B$ is regular.
    2. For a loop $\alpha$ in $B$, either all the lifts of $\alpha$ are loops, or all the lifts of $\alpha$ are not loops.
  2. 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.
  3. 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]
  4. 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 $\{0,1\}$.
  5. 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.
  6. 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.
  7. 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.
    1. The quotient map $p\colon X\to B$ is a covering map.
    2. $p$ is regular.
    3. The covering transformation group is isomorphic to $G$.
    4. If $X$ is locally path connected and simply connected, then $\pi_1(B,*)\cong G$.

Problem Set #2 (due 03/28)

  1. The $n$-dimensional real projective space is a quotient space of $S^n$ which is defined by $\R P^n = S^n/x\sim -x$.
    1. Show that the quotient map $S^n \to \R P^n$ is a covering map.
    2. 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 $n\ge 2$.
  2. Suppose $p\colon E\to B$ and $f\colon X \to B$.  Define $f^*(E)=\{(e,x)\in E\times X \mid p(e)=f(x)\}$.
    1. 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:
      f^*(E) & \longrightarrow & E \\
      \downarrow & & \downarrow\\
      X & \longrightarrow & B
    2. Show that if $p$ is a covering map, then $f^*(E) \to X$ is a covering map.
    3. Show that  $p^{-1}(f(x))$ is in 1-1 correspondence with $q^{-1}(x)$.
  3. 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 $\pi_1(B,b_0)$.
  4. Let $f\colon E\to B$ be the covering described at the end of Lecture Note #4. See the following diagram.
    1. Show that $[x],\, [y]\in \pi_1(B,*)$ are nontrivial elements, by applying the above #3.
    2. Show $[x]\ne [y]$ in $\pi_1(B,*)$. Show $[x]\ne [x]^2$ in $\pi_1(B,*)$.
    3. 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.
    4. 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 $B \to S^1$.
  5. Let $B$ be the base space of the covering in the above #4.
    1. Show that $[x][y]\ne [y][x]$ in  $\pi_1(B,*)$, using lifts in a covering map.
    2. 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)

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