The minimal genus problem for elliptic surfaces

TL;DR

This paper addresses a specific case of the minimal genus problem for embedded surfaces in elliptic 4-manifolds, demonstrating that the minimal genus aligns with the bounds set by the adjunction inequality.

Contribution

It provides a solution for a particular case of the minimal genus problem in elliptic 4-manifolds using a novel approach involving the diffeomorphism group's action.

Findings

Achieves the minimal genus predicted by the adjunction inequality for the case studied.

Introduces a restricted transitivity property of the diffeomorphism group on second homology.

Advances understanding of embedded surfaces in elliptic 4-manifolds.

Abstract

We solve a certain case of the minimal genus problem for embedded surfaces in elliptic 4-manifolds. The proofs involve a restricted transitivity property of the action of the orientation preserving diffeomorphism group on the second homology. In the case we consider we get the minimal possible genus allowed by the adjunction inequality.