Rational homology cobordisms of plumbed manifolds

Aceto, Paolo¹

¹Mathematical Institute, University of Oxford, Oxford, United Kingdom

Algebraic and Geometric Topology, Tome 20 (2020) no. 3, pp. 1073-1126

Résumé

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.

Détail

DOI : 10.2140/agt.2020.20.1073

Classification : 57M27, 57M12, 57M25
Keywords: rational homology cobordisms, plumbing

Bibliographie

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