Using rational homology circles to construct rational homology balls

Abstract

Motivated by Akbulut-Larson's construction of Brieskorn spheres bounding rational homology 4-balls, we explore plumbed 3-manifolds that bound rational homology circles and use them to construct infinite families of rational homology 3-spheres that bound rational homology 4-balls. Some of these rational homology 3-spheres are new examples of integer homology 3-spheres that bound rational homology 4-balls, but do not bound integer homology 4-balls (i.e. nontrivial elements of ker(ΘZ3→ΘQ3)). In particular, we find infinite families of torus bundles over the circle that bound rational homology circles, provide a simple method for constructing more general plumbed 3-manifolds that bound rational homology circles, and show that, for example, −1-surgery along any unknotting number one knot K with a positive crossing that can be switched to unknot K bounds a rational homology 4-ball.