On the inverse mapping class monoids

TL;DR

This paper introduces the inverse mapping class monoid for punctured surfaces, providing a presentation, an analogue of the Dehn-Nilsen-Baer theorem, and studying its properties and word problem, extending concepts from mapping class groups.

Contribution

It defines the inverse mapping class monoid, offers a presentation for punctured spheres, and extends classical theorems and presentations to this new algebraic structure.

Findings

Established the analogue of the Dehn-Nilsen-Baer theorem.

Provided a presentation of the inverse mapping class monoid.

Studied the word problem for the inverse monoid.

Abstract

Braid groups and mapping class groups have many features in common. Similarly to the notion of inverse braid monoid inverse mapping class monoid is defined. It concerns surfaces with punctures, but among given $n$ punctures several can be omitted. This corresponds to braids where the number of strings is not fixed. In the paper we give the analogue of the Dehn-Nilsen-Baer theorem, propose a presentation of the inverse mapping class monoid for a punctured sphere and study the word problem. This shows that certain properties and objects based on mapping class groups may be extended to the inverse mapping class monoids. We also give an analogues of Artin presentation with two generators.