Double point surgery and configurations of surfaces

TL;DR

This paper introduces double point surgery, a new operation on immersed surfaces in 4-manifolds, enabling the construction of knotted surface configurations and exotic group actions with complex fixed points.

Contribution

It presents a novel surgical technique on surfaces in 4-manifolds and demonstrates its use in creating knotted configurations and exotic symmetries.

Findings

Constructed knotted surface configurations in 4-manifolds.

Produced exotic Z/m x Z/n actions with complex fixed-point sets.

Established a new method for manipulating surfaces in 4-manifolds.

Abstract

We introduce a new operation, double point surgery, on immersed surfaces in a 4-manifold, and use it to construct knotted configurations of surfaces in many 4-manifolds. Taking branched covers, we produce smoothly exotic actions of Z/m x Z/n on simply connected 4-manifolds with complicated fixed-point sets.