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.

# differential geometry

# 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 and be compact oriented smooth manifolds of the same dimension, with connected. Let be a smooth function. Take a regular value of and, for , define as if preserves orientation at and as if it reverses the orientation. We define the degree of as the number .

Prove the following extension theorem: Assume is a -dimensional sphere and is the boundary of a compact oriented -manifold . There exists a smooth extension of if and only if .

# “Representability” II

As we have seen in the last problem, we can identify a compact smooth manifold with . Give a similar identification between the tangent bundle and .

# “Representability”

Let be a compact smooth manifold. Show that any -algebra morphism is of the form for some , where .