Euler characteristic parity in dimensions congruent to two modulo four ($\chi (M)\equiv 0(\mathrm{mod}2)$)

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.

Take a connected sum of $r$ copies of Complex projective space ${\mathbb{CP}}^3$ and $s$ copies of $S^3\times S^3$, with the empty sum interpreted as $S^6$. Their Euler characteristics are four, zero and two, respectively. Removing a six-ball subtracts one from the Euler characteristic and gluing along its boundary $S^5$ subtracts zero, so each connected sum subtracts two. The result is $2+2r-2s$. For any even integer $n$, choose $r=\max (n/2-1,0)$ and $s=\max (1-n/2,0)$.