# (Not) on a plane

A plane flies 100 km north, 100 km east, 100 km south and then 100 km west. Give a necessary and sufficient condition for it to land on the same spot where it took off.

# Euler & Lie

Any compact connected Lie group has Euler characteristic zero. What are the possible values for the Euler characteristic of a connected, but not necessarily compact Lie group?

# Your degree is a chance for expansion

Let $X$ and $Y$ be compact oriented smooth manifolds of the same dimension, with $Y$ connected. Let $f : X \to Y$ be a smooth function. Take $y \in Y$ a regular value of $f$ and, for $x \in f^{-1}(y)$, define $o(x)$ as $+1$ if $f$ preserves orientation at $x$ and as $-1$ if it reverses the orientation. We define the degree of $f$ as the number $d(f) = \sum_{x\in f^{-1}(y)}o(x)$.

Prove the following extension theorem: Assume $Y=S^k$ is a $k$-dimensional sphere and $X=\partial W$ is the boundary of a compact oriented $(k+1)$-manifold $W$. There exists a smooth extension $F:W\to Y$ of $f$ if and only if $d(f)=0$.

# “Representability” II

As we have seen in the last problem, we can identify a compact smooth manifold $M$ with $\mathrm{hom}_{\Bbb{R}\mathrm{-alg}}(C^\infty(M),\Bbb{R})$. Give a similar identification between the tangent bundle $TM$ and  $\mathrm{hom}_{\Bbb{R}\mathrm{-alg}}(C^\infty(M),\Bbb{R}[x]/(x^2))$.

# “Representability”

Let $M$ be a compact smooth manifold. Show that any $\Bbb{R}$-algebra morphism $C^\infty(M)\to\Bbb{R}$ is of the form $\mathrm{ev}_p$ for some $p\in M$, where $\mathrm{ev}_p(f)=f(p)$.