More generally, any compact parallelizable manifold, including any compact Lie group, has Euler characteristic 0.
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Changing to and between general tetrads is much similar and equally necessary ( except for parallelizable manifolds ).
3.
The Kervaire Milnor formula for the order of the cyclic group of diffeomorphism classes of exotic-spheres which bound parallelizable manifolds involves Bernoulli numbers.
4.
Let us note that a true translation gauge field ( i . e ., an affine connection which yields a flat linear connection on TX ) is well defined only on a parallelizable manifold X.
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