Diffeomorphisms of 4-manifolds from graspers

TL;DR

This paper explores how degree one grasper families of embedded circles relate to various known constructions of diffeomorphisms in 4-manifolds, using a parameterised surgery map to connect embeddings to the mapping class group.

Contribution

It introduces a unified framework linking different diffeomorphism constructions via graspers and a parameterised surgery map in 4-manifold topology.

Findings

Establishes connections between graspers and theta clasper classes.

Relates barbell diffeomorphisms to graspers.

Provides a new perspective on twin twists through graspers.

Abstract

We relate degree one grasper families of embedded circles to various constructions of diffeomorphisms found in the literature -- theta clasper classes of Watanabe, barbell diffeomorphisms of Budney and Gabai, and twin twists of Gay and Hartman. We use a ``parameterised surgery map'' that for a smooth 4-manifold $M$ takes loops of framed embeddings of $S^1$ in the manifold obtained by surgery on some 2-sphere in $M$, to the mapping class group of $M$.