Minimal Euler characteristics of 4-manifolds with 3-manifold groups

Let π = π1(M) for a compact 3-manifold M, and χ4, p and q* be the invariants of Hausmann and Weinberger (1985), Kotschick (1994) and Hillman (2002), respectively. For a certain class of compact 3-manifolds M, including all those not containing two-sided RP2’s, we determine χ4(π). We address when p(π) equals χ4(π) and when q*(π) equals χ4(π), and answer a question raised by Hillman (2002).