Rational homology cobordisms of plumbed 3-manifolds

TL;DR

This paper studies rational homology cobordisms of 3-manifolds with non-zero first Betti number, providing a construction method and characterizing certain manifolds that bound rational homology 4-manifolds, with implications for link concordance.

Contribution

It introduces a procedure to construct rational homology cobordisms between plumbed 3-manifolds and characterizes manifolds in a specific family that bound rational homology 4-manifolds.

Findings

Constructed rational homology cobordisms for a large family of plumbed 3-manifolds.

Characterized which manifolds in the family bound rational homology $S^1\times D^3$'s.

Included all Seifert fibered spaces over the 2-sphere with zero Euler invariant.

Abstract

We investigate rational homology cobordisms of 3-manifolds with non-zero 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-manifold. We introduce a family F of plumbed 3-manifolds with first Betti number equal to 1. By adapting an obstruction based on Donaldson's diagonalization theorem we characterize all manifolds in F 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. The family F 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.