  In math.DG/9903083 (henceforth referred to as EA) we defined an integer
invariant $h(Y)$ for oriented integral homology 3-spheres $Y$ which only
depends on the rational homology cobordism class of $Y$ and is additive under
connected sums. In this paper we establish lower bounds for $h(Y)$ when $Y$ is
the boundary of a smooth, compact, oriented 4-manifold with $b_2^+=1$. As
applications, we give an upper bound for how much $h$ changes under -1 surgery
on knots in terms of the slice genus of the knot, and compute $h$ for a family
of Brieskorn spheres.
  This paper contains, in revised form, most of the material from v1 of EA that
was left out in the final version of that paper. In particular, Theorem 1 of
the present paper is virtually the same as Theorem 1 of v1 of EA. The proof is
also essentially the same, but the exposition has been improved, with more
details.
