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