Rational homology cobordisms of plumbed manifolds

Aceto, Paolo

doi:10.2140/agt.2020.20.1073

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Rational homology cobordisms of plumbed manifolds

Paolo Aceto

Algebraic & Geometric Topology 20 (2020) 1073–1126

DOI: 10.2140/agt.2020.20.1073

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.

Keywords

rational homology cobordisms, plumbing

Mathematical Subject Classification 2010

Primary: 57M27

Secondary: 57M12, 57M25

References

Publication

Received: 24 March 2015

Revised: 30 April 2019

Accepted: 9 September 2019

Published: 27 May 2020

Authors

Paolo Aceto
Mathematical Institute University of Oxford Oxford United Kingdom

Volume 20, issue 3 (2020)