Homotopy motions of surfaces in $3$-manifolds

TL;DR

This paper introduces the concept of homotopy motions of surfaces in 3-manifolds and systematically studies their properties, addressing key problems in 3-manifold topology such as domination, loop behavior, and monodromies.

Contribution

It presents a new framework for understanding homotopy motions of surfaces in 3-manifolds, linking various natural problems in 3-manifold theory.

Findings

Defined homotopy motions of surfaces in 3-manifolds

Connected homotopy motions to domination and monodromy problems

Provided systematic analysis of homotopy motions in closed orientable 3-manifolds

Abstract

We introduce the concept of a homotopy motion of a subset in a manifold, and give a systematic study of homotopy motions of surfaces in closed orientable 3-manifolds. This notion arises from various natural problems in 3-manifold theory such as domination of manifold pairs, homotopical behavior of simple loops on a Heegaard surface, and monodromies of virtual branched covering surface bundles associated to a Heegaard splitting.