On homology planes and contractible $4$-manifolds

TL;DR

This paper introduces new examples and infinite families of Kirby-Ramanujam spheres, which are special homology spheres bounding both homology planes and contractible 4-manifolds, expanding understanding of their structure and relationships.

Contribution

It provides the first additional examples and three infinite families of Kirby-Ramanujam spheres, including a diffeomorphic family to splice of Brieskorn spheres, advancing the classification of these objects.

Findings

Discovered three infinite families of Kirby-Ramanujam spheres.

Showed one family is diffeomorphic to splice of Brieskorn spheres.

Proved these spheres bound contractible 4-manifolds and are trivial in homology cobordism group.

Abstract

We call a non-trivial homology sphere a Kirby-Ramanujam sphere if it bounds both a homology plane and a Mazur or Po\'enaru manifold. In 1980, Kirby found the first example by proving that the boundary of the Ramanujam surface bounds a Mazur manifold and it has remained a single example since then. By tracing their initial step, we provide the first additional examples and we present three infinite families of Kirby-Ramanujam spheres. Also, we show that one of our families of Kirby-Ramanujam spheres is diffeomorphic to the splice of two certain families of Brieskorn spheres. Since this family of Kirby-Ramanujam spheres bound contractible $4$-manifolds, they lie in the class of the trivial element in the homology cobordism group; however, both splice components are separately linearly independent in that group.