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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

Citations:

Zbl 1185.57006
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