  We investigate rational homology cobordisms of 3-manifolds with non-zero
first Betti number. This is motivated by the natural generalization of the
slice-ribbon conjecture to multicomponent links. In particular we consider the
problem of which rational homology $S^1\times S^2$'s bound rational homology
$S^1\times D^3$'s. We give a simple procedure to construct rational homology
cobordisms between plumbed 3-manifold. We introduce a family F of plumbed
3-manifolds with first Betti number equal to 1. By adapting an obstruction
based on Donaldson's diagonalization theorem we characterize all manifolds in F
that bound rational homology $S^1\times D^3$'s. For all these manifolds a
rational homology cobordism to $S^1\times S^2$ can be constructed via our
procedure. The family F is large enough to include all Seifert fibered spaces
over the 2-sphere with vanishing Euler invariant. In a subsequent paper we
describe applications to arborescent link concordance.
