Rational homology cobordisms of plumbed manifolds

Aceto, Paolo (2020) Rational homology cobordisms of plumbed manifolds. ALGEBRAIC AND GEOMETRIC TOPOLOGY, 20 (3). pp. 1073-1126. ISSN 1472-2739

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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 x S-2 's bound rational homology S-1 x 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 x D-3 's. For all these manifolds a rational homology cobordism to S-1 x 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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