On homology planes and contractible 4‐manifolds

Abstract

AbstractWe call a non‐trivial homology sphere a Kirby–Ramanujam sphere if it bounds both a homology plane and a Mazur or Poénaru 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.