Homeomorphisms of Bagpipes

TL;DR

This paper studies the structure of the mapping class group of a special class of surfaces called bagpipes, revealing how it decomposes into components related to the bag and pipes via group homomorphisms.

Contribution

It characterizes the subgroup of homeomorphisms fixing the boundary of the bag as a normal subgroup and describes its relation to the mapping class groups of the bag and pipes.

Findings

The subgroup fixing the boundary is a normal subgroup.

The subgroup is a homomorphic image of the product of mapping class groups.

Provides a structural understanding of the mapping class group for bagpipe surfaces.

Abstract

We investigate the mapping class group of an orientable $\omega$-bounded surface. Such a surface splits, by Nyikos's Bagpipe Theorem, into a union of a bag (a compact surface with boundary) and finitely many long pipes. The subgroup consisting of classes of homeomorphisms fixing the boundary of the bag is a normal subgroup and is a homomorphic image of the product of mapping class groups of the bag and the pipes.