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Rational homology cobordisms of plumbed manifolds

Paolo Aceto

Algebraic & Geometric Topology 20 (2020) 1073–1126

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 S1 × S2’s bound rational homology S1 × D3’s. We give a simple procedure to construct rational homology cobordisms between plumbed 3–manifolds. We introduce a family of plumbed 3–manifolds with b1 = 1. By adapting an obstruction based on Donaldson’s diagonalization theorem we characterize all manifolds in our family that bound rational homology S1 × D3’s. For all these manifolds a rational homology cobordism to S1 × S2 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.

rational homology cobordisms, plumbing
Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 57M12, 57M25
Received: 24 March 2015
Revised: 30 April 2019
Accepted: 9 September 2019
Published: 27 May 2020
Paolo Aceto
Mathematical Institute
University of Oxford
United Kingdom