Algebraic & Geometric Topology

Rational homology cobordisms of plumbed manifolds

Paolo Aceto

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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.

Article information

Algebr. Geom. Topol., Volume 20, Number 3 (2020), 1073-1126.

Received: 24 March 2015
Revised: 30 April 2019
Accepted: 9 September 2019
First available in Project Euclid: 5 June 2020

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57M12: Special coverings, e.g. branched 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

rational homology cobordisms plumbing


Aceto, Paolo. Rational homology cobordisms of plumbed manifolds. Algebr. Geom. Topol. 20 (2020), no. 3, 1073--1126. doi:10.2140/agt.2020.20.1073.

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