Euler characteristic of compact connected nontrivial Lie group is zero

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Statement

Suppose G is a nontrivial compact connected Lie group (?). Then, G is a Space with zero Euler characteristic (?), i.e., the Euler characteristic (?) of G is 0.

Related facts

Similar facts

Facts used

  1. Lefschetz fixed-point theorem
  2. Lie group implies polyhedron: Any Lie group can be given a simplicial complex structure, and is hence a polyhedron.
  3. Definition of Euler characteristic as the Lefschetz number of the identity map from a space to itself.
  4. Lefschetz number is homotopy-invariant: If f_1,f_2:X \to Y are homotopic maps, then the Lefschetz numbers of f_1 and f_2 are equal.

Proof

This proof uses a tabular format for presentation. Learn more about tabular proof formats|View all pages on facts with proofs in tabular format

Given: A compact connected nontrivial Lie group G.

To prove: The Euler characteristic of G is zero.

Proof: We denote by e the identity element of G.

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 G is a compact polyhedron, and hence a space with finitely generated homology Fact (2) G is compact and is a Lie group Given+Fact direct
2 Denote by m_e:G \to G the left multiplication map by e. m_e is the identity map and the Lefschetz number of m_e is \chi(G) Fact (3) G is a group, e its identity element Step (1) [SHOW MORE]
3 Let g be a non-identity element of G. -- G is nontrivial
4 Denote by m_g:G \to G the left multiplication map by g, a non-identity element of g. In other words, m_g(x) := gx. Then m_g has no fixed points in G -- G is a group Step (3) (basic group theory)
5 The Lefschetz number of m_g is zero. Fact (1) Steps (1), (4) [SHOW MORE]
6 m_e and m_g are homotopic maps G is connected and is a manifold (because it's a Lie group) [SHOW MORE]
7 The Lefschetz number of m_e equals the Lefschetz number of m_g Fact (4) -- Step (6) Step+Fact direct
8 The Euler characteristic of G is zero (final conclusion) -- -- Steps (2), (5), (7) [SHOW MORE]