## Algebraic & Geometric Topology

### Rational homology cobordisms of plumbed manifolds

Paolo Aceto

#### 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 $S1×S2$’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

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

Dates
Revised: 30 April 2019
Accepted: 9 September 2019
First available in Project Euclid: 5 June 2020

https://projecteuclid.org/euclid.agt/1591374776

Digital Object Identifier
doi:10.2140/agt.2020.20.1073

Mathematical Reviews number (MathSciNet)
MR4105549

Zentralblatt MATH identifier
07207571

#### Citation

Aceto, Paolo. Rational homology cobordisms of plumbed manifolds. Algebr. Geom. Topol. 20 (2020), no. 3, 1073--1126. doi:10.2140/agt.2020.20.1073. https://projecteuclid.org/euclid.agt/1591374776

#### References

• P Aceto, Arborescent link concordance, in preparation
• A Donald, Embedding Seifert manifolds in $S^4$, Trans. Amer. Math. Soc. 367 (2015) 559–595
• D Eisenbud, W Neumann, Three-dimensional link theory and invariants of plane curve singularities, Annals of Mathematics Studies 110, Princeton Univ. Press (1985)
• L H Kauffman, L R Taylor, Signature of links, Trans. Amer. Math. Soc. 216 (1976) 351–365
• A G Lecuona, On the slice-ribbon conjecture for Montesinos knots, Trans. Amer. Math. Soc. 364 (2012) 233–285
• R Lee, S H Weintraub, On the homology of double branched covers, Proc. Amer. Math. Soc. 123 (1995) 1263–1266
• P Lisca, Lens spaces, rational balls and the ribbon conjecture, Geom. Topol. 11 (2007) 429–472
• P Lisca, Sums of lens spaces bounding rational balls, Algebr. Geom. Topol. 7 (2007) 2141–2164
• W D Neumann, A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Amer. Math. Soc. 268 (1981) 299–344
• W D Neumann, On bilinear forms represented by trees, Bull. Austral. Math. Soc. 40 (1989) 303–321
• W D Neumann, F Raymond, Seifert manifolds, plumbing, $\mu$–invariant and orientation reversing maps, from “Algebraic and geometric topology” (K C Millett, editor), Lecture Notes in Math. 664, Springer (1978) 163–196