  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.
