Regular homotopic deformation of compact surface with boundary and mapping class group

TL;DR

This paper establishes an algebraic criterion for when a surface diffeomorphism in the 3-sphere can be realized through regular homotopic deformation, providing a formula for the required moves.

Contribution

It introduces a necessary and sufficient algebraic condition and a formula for the number of pass moves for regular homotopies of surfaces in the 3-sphere.

Findings

Algebraic condition for regular homotopic deformation

Formula for signed pass moves needed

Characterization of surface diffeomorphisms in 3-sphere

Abstract

A necessary and sufficient algebraic condition for a diffeomorphism over a surface embedded in the 3-sphere to be induced by a regular homotopic deformation is discussed, and a formula for the number of signed pass moves needed for this regular homotopy is given.