# Euler characteristic of connected Lie group is zero or one

From Topospaces

## Statement

Suppose is a connected Lie group. Then is either a Space with Euler characteristic zero (?) or a Space with Euler characteristic one (?). In other words, the Euler characteristic (?) of is either or .

The case of Euler characteristic one occurs if and only if is contractible. Otherwise, has Euler characteristic zero.

More generally, the Euler characteristic of a Lie group is equal to either zero or the number of path components.