= Every even integer is the Euler characteristic of a closed oriented six-manifold
{title2=$\chi=2+2r-2s$}
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 $s=\max(1-n/2,0)$.
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