## Rational homology cobordisms of plumbed manifolds.(English)Zbl 1446.57010

The slice-ribbon conjecture (Fox 1962) states that every slice knot is ribbon. P. Lisca proved [Geom. Topol. 11, 429–472 (2007; Zbl 1185.57006)] that the conjecture is true for 2-bridge knots, using an obstruction based on Donaldson’s diagonalization theorem to determine which lens spaces bound rational homology balls (if a knot $$K$$ is slice then its branched double cover is a rational homology sphere that bounds a rational homology ball; if $$K$$ is a 2-bridge knot then the branched double cover is a lens space). As the author notes, the starting point of the present work is an adaption of these ideas to the study of slice links with $$n > 1$$ components. Focusing on the case $$n = 2$$, this leads then to the following general problem: Which rational homology $$S^1 \times S^2$$’s bound rational $$S^1 \times D^3$$’s?
The author gives 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.”

### MSC:

 57K30 General topology of 3-manifolds

### Keywords:

rational homology cobordisms; plumbing

Zbl 1185.57006
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