  Let $\pi=\pi_1(M)$ for a compact 3-manifold $M$, and let $\chi_4$, $p$ and
$q^*$ be the invariants of Hausmann-Weinberger, Kotschick and Hillman
respectively. For certain class of compact 3-manifolds $M$, including all those
not containing two-sided $RP^2$, we determine $\chi_4(\pi)$. We address when
does $\chi_4(\pi)=p(\pi)$, when does $\chi_4(\pi)=q^*(\pi)$, and answer a
question raised by Hillman.
