Euler characteristic of compact connected nontrivial Lie group is zero
(Redirected from Compact connected nontrivial Lie group implies zero Euler characteristic)
- Lefschetz fixed-point theorem
- Lie group implies polyhedron: Any Lie group can be given a simplicial complex structure, and is hence a polyhedron.
- Definition of Euler characteristic as the Lefschetz number of the identity map from a space to itself.
- Lefschetz number is homotopy-invariant: If are homotopic maps, then the Lefschetz numbers of and are equal.
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Given: A compact connected nontrivial Lie group .
To prove: The Euler characteristic of is zero.
Proof: We denote by the identity element of .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||is a compact polyhedron, and hence a space with finitely generated homology||Fact (2)||is compact and is a Lie group||Given+Fact direct|
|2||Denote by the left multiplication map by . is the identity map and the Lefschetz number of is||Fact (3)||is a group, its identity element||Step (1)||[SHOW MORE]|
|3||Let be a non-identity element of .||--||is nontrivial|
|4||Denote by the left multiplication map by , a non-identity element of . In other words, . Then has no fixed points in||--||is a group||Step (3)||(basic group theory)|
|5||The Lefschetz number of is zero.||Fact (1)||Steps (1), (4)||[SHOW MORE]|
|6||and are homotopic maps||is connected and is a manifold (because it's a Lie group)||[SHOW MORE]|
|7||The Lefschetz number of equals the Lefschetz number of||Fact (4)||--||Step (6)||Step+Fact direct|
|8||The Euler characteristic of is zero (final conclusion)||--||--||Steps (2), (5), (7)||[SHOW MORE]|