Rational homology cobordisms of plumbed manifolds

Abstract

We investigate rational homology cobordisms of $3$–manifolds with nonzero 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 $3$–manifolds. We introduce a family of plumbed $3$–manifolds with $b_1=1$. By adapting an obstruction based on Donaldson’s diagonalization theorem we characterize all manifolds in our family 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. Our family 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.