= Euler characteristic parity in dimensions congruent to two modulo four
{c}
{title2=$\chi(M)\equiv0\pmod2$}
For a closed oriented <manifold> of dimension $4r+2$, <Poincare duality> pairs all rational <Betti numbers> except the middle one, making their total contribution to the <Euler characteristic> even. The middle cohomology degree is odd, so its nondegenerate <Poincare duality pairing> is an <alternating bilinear form> and has even dimension. Hence the Euler characteristic is even. Nonorientable manifolds can have odd Euler characteristic, as <Real projective space> of even dimension demonstrates.
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