Differentiaaltopologie

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Examenvragen

Exam June 24, 2013

In 2013, the course was taught by Nansen Petrosyan. Classes were in English, but the exam could be taken in Dutch or in English. The theory part was closed book with a short oral discussion later, while the exercises part was with the book of Milnor and notes. For each parts we had 2 hours approximately.

Theory

1. Consider the projective space $\mathbb{ℝ}{P}^n$, defined as the quotient of $S^n$ by the equivalence relation $x\sim y$ if and only if $y=-x$. Recall that the topology of $\mathbb{ℝ}{P}^n$ is given by the following: a subset $U$ of $\mathbb{ℝ}{P}^n$ is open if and only if $q(U)$ is open in $S^n$ where $q:\mathbb{ℝ}{P}^n\to S^n:x\mapsto \{x,-x\}$.
   1. Show that $\mathbb{ℝ}{P}^n$ is a smooth manifold. You can use the following theorem (which you do not need to prove): A second countable Hausdorff space $M$ is an $n$-dimensional smooth manifold if there are charts $\varphi :U\subseteq \mathbb{ℝ}{P}^n\to U^{\prime }\subseteq S^n$ such that: the charts are homeomorphisms between open sets; each element of $M$ belongs to the domain of a chart; and if $\varphi :U\to U^{\prime }$ and $\psi :V\to V^{\prime }$ are charts, $\varphi {\psi }^{-1}:\psi (U\cap V)\to \varphi (U\cap V)$ is a smooth map.
   2. Prove that $\mathbb{ℝ}{P}^n$ is not diffeomorphic to $S^n$.
   3. Show that there does not exist a nowhere vanishing smooth vector field on $S^n$ when $n$ is even. Deduce that there does not exist a nowhere vanishing smooth vector field on $\mathbb{ℝ}{P}^n$.
2. Let $M$ be a compact smooth manifold without boundary in ${\mathbb{ℝ}}^n$. Define the normal bundle $N(M,\epsilon )=\{(x,\alpha )\in M\times {\mathbb{ℝ}}^n\mid \alpha \mathrm{\perp }{T}_{x}M\}$.
   1. Prove that the normal bundle is an $n$-dimensional smooth submanifold of $M\times {\mathbb{ℝ}}^n$.
   2. Show that an $\epsilon >0$ exists such that $N(M,\epsilon )\to {\mathbb{ℝ}}^n:(x,\alpha )\mapsto x+\alpha$ is a diffeomorphism onto $N_{\epsilon }=\{y\in {\mathbb{ℝ}}^n\mid d(M,y)<\epsilon \}$. [This result, the tubular neighborhood theorem, was used in class but its proof was left as an exercise].
   3. Give an example that shows that the conclusion is false when $M$ is not compact.

Exercises

1. For a smooth function $f:U\to \mathbb{ℝ}$ on an open subset $U$ of ${\mathbb{ℝ}}^n$ we define the gradient at $p\in U$ to be the vector $\mathrm{\nabla }f(p)=⟨\frac{\partial f}{\partial x_1}(p),\dots ,\frac{\partial f}{\partial x_n}(p)⟩$. Suppose $M$ is a smooth manifold in ${\mathbb{ℝ}}^n$. Show that if $f$ is constant on $U\cap M\ni p$, then $\mathrm{\nabla }f(p)$ is orthogonal to $T_{p}M$.
2. Denote by $\mathrm{S}\mathrm{L}(2,\mathbb{ℝ})$ the set of $2\times 2$ matrices with real entries and determinant $1$. Explain how $\mathrm{S}\mathrm{L}(2,\mathbb{ℝ})$ can be given the structure of a connected smooth manifold and determine its dimension. [Something similar (for ${\mathrm{G}\mathrm{L}}^{+}(n,\mathbb{ℝ})$ and ${\mathrm{G}\mathrm{L}}^{-}(n,\mathbb{ℝ})$) appeared in an exercise given in class.]
3. Consider the complex polynomial $p(z)=z^n+a_1{z}^{n-1}+\cdots +a_{n-1}z+a_n$ of positive degree. Let $D^2=\{z\in \mathbb{ℂ}\mid |z|\le 1\}$.
   1. Show that $q(z)=(1-|z{|}^2{)}^{n}p(z/(1-|z{|}^2))$ determines a smooth function $D^2\to {\mathbb{ℝ}}^2$.
   2. Prove that the smooth vector field $v:D^2\to {\mathbb{ℝ}}^2:z\mapsto \overrightarrow{q(z)}$ has a zero in the interior of $D^2$. Deduce the Fundamental Theorem of Algebra.