Elements of Torelli topology: II. The extension problem

TL;DR

This paper investigates when a diffeomorphism of a subsurface can be extended to the entire surface acting trivially on homology, revealing that the answer depends on the subsurface's position and involves homology groups and a difference map.

Contribution

It provides a criterion for the extension problem in Torelli topology using homology groups and introduces the difference map, advancing understanding of subsurface diffeomorphisms.

Findings

Extension depends on subsurface position in a controlled way.

Homology groups and a difference map determine extendability.

Framework aids in analyzing Torelli groups and their abelian subgroups.

Abstract

By Torelli topology the author understands aspects of the topology of surfaces (potentially) relevant to the study of Torelli groups. The extension problem in Torelli topology is the problem of determining when a diffeomorphism of compact connected subsurface of a closed surface can be extended to a diffeomorphism of the whole surface acting trivially on its homology. The special case of extensions by the identity diffeomorphism was considered by A. Putman, whose work was one of the main sources of inspiration for the present paper. It turns out that the answer depends on the position of the subsurface, but only in a mild and controlled way discovered by A. Putman in the special case of extensions by the identity. The answer is stated in terms of two homology groups associated to the subsurface and a homomorphism between them, which we call the difference map of a diffeomorphism of the subsurface in question. The extension problem is motivated by the desire to extend to Torelli groups some well established tools and results from the theory of Teichm\"uller modular groups. In particular, in the next paper of this series the analysis of the extension problem in this paper will be applied to get a description of all abelian subgroups of Torelli groups. The paper is concluded by some speculations about what is the Torelli group of a surface with non-empty boundary.