Every even integer is the Euler characteristic of a closed oriented six-manifold ($\chi =2+2r-2s$)

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 $s=\max (1-n/2,0)$.