Deformations of circle-valued functions on $2$-torus

TL;DR

This paper provides an algebraic framework to understand how the fundamental groups of orbits of circle-valued smooth functions on a 2-torus behave under diffeomorphisms, focusing on functions with isolated singularities.

Contribution

It introduces an algebraic description of the fundamental groups of orbits of circle-valued functions on the 2-torus under diffeomorphism actions, with a focus on isolated singularities.

Findings

Algebraic characterization of orbit fundamental groups.

Description of the impact of diffeomorphisms on these groups.

Analysis of functions with isolated singularities on the 2-torus.

Abstract

In this paper we give an algebraic description of fundamental groups of orbits of circle-valued smooth functions from some subspace of the space of smooth functions with isolated singularities on $2$-torus $T^2$ with respect to the action of the group of diffeomorphisms of $T^2$.