Euler characteristic
Arbitrary dimensions

The n dimensional sphere has singular homology groups equal to

hence has Betti number 1 in dimensions 0 and n, and all other Betti numbers are 0. Its Euler characteristic is then χ = 1 + (−1) ; that is, either 0 if n is odd, or 2 if n is even.

The n dimensional real projective space is the quotient of the n sphere by the antipodal map. It follows that its Euler characteristic is exactly half that of the corresponding sphere – either 0 or 1.

The n dimensional torus is the product space of n circles. Its Euler characteristic is 0, by the product property. More generally, any compact parallelizable manifold, including any compact Lie group, has Euler characteristic 0.

The Euler characteristic of any closed odd-dimensional manifold is also 0. The case for orientable examples is a corollary of Poincaré duality. This property applies more generally to any compact stratified space all of whose strata have odd dimension. It also applies to closed odd-dimensional non-orientable manifolds, via the two-to-one orientable double cover.
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