Euler characteristics of closed oriented four-manifolds ($\chi (M)=2-2b_1(M)+b_2(M)$)

Poincare duality gives the displayed formula for connected closed oriented four-manifolds. Every integer occurs. A connected sum of $r$ copies of the Complex projective plane and $s$ copies of $S^1\times S^3$ has Euler characteristic $2+r-2s$, because the summand characteristics are three and zero. Given any integer $h$, choose $s$ large enough that $r=h-2+2s\ge 0$; when both counts are zero use $S^4$. If simple connectivity is additionally required, the possible values instead begin at two, since $b_1=0$; connected sums of projective planes realize every such value.