#### Volume 20, issue 3 (2020)

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Rational homology cobordisms of plumbed manifolds

### Paolo Aceto

Algebraic & Geometric Topology 20 (2020) 1073–1126
##### 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 ${S}^{1}×{S}^{2}$’s bound rational homology ${S}^{1}×{D}^{3}$’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 ${S}^{1}×{D}^{3}$’s. For all these manifolds a rational homology cobordism to ${S}^{1}×{S}^{2}$ 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.

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##### Keywords
rational homology cobordisms, plumbing
##### Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 57M12, 57M25