Euler characteristic of connected Lie group is zero or one

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Statement

Suppose G is a connected Lie group. Then G is either a Space with Euler characteristic zero (?) or a Space with Euler characteristic one (?). In other words, the Euler characteristic (?) of G is either 0 or 1.

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

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

Facts used

  1. Euler characteristic of compact connected nontrivial Lie group is zero