The link surgery of $S^2\times S^2$ and Scharlemann's manifolds

TL;DR

This paper investigates the effects of knot surgery on certain elliptic fibrations, demonstrating that under specific conditions, the resulting manifolds are standard, providing new insights into Scharlemann's manifolds.

Contribution

It shows that knot surgery on elliptic fibrations with particular vanishing circle configurations yields standard manifolds, offering an alternative proof regarding Scharlemann's manifolds.

Findings

Knot surgery on specific elliptic fibrations results in standard manifolds.

The diffeomorphism provides an alternative proof that Scharlemann's manifold is standard.

Conditions involve elliptic fibrations with two parallel, oppositely oriented vanishing circles.

Abstract

Fintushel-Stern's knot surgery gave many pairs of exotic manifolds, which are homeomorphic but non-diffeomorphic. We show that if an elliptic fibration has two parallel, oppositely oriented vanishing circles (for example $S^2\times S^2$ or Matsumoto's $S^4$), then the knot surgery gives rise to standard manifolds. The diffeomorphism can give an alternative proof that Scharlemann's manifold is standard (originally by Akbulut [Ak1]).