913 research outputs found
A presentation for the mapping class group of a non-orientable surface from the action on the complex of curves
We study the action of the mapping class group M(F) on the complex of curves
of a non-orientable surface F. We obtain, by using a result of K. S. Brown, a
presentation for M(F) defined in terms of the mapping class groups of the
complementary surfaces of collections of curves, provided that F is not
sporadic, i.e. the complex of curves of F is simply connected. We also compute
a finite presentation for the mapping class group of each sporadic surface.Comment: 45 pages, accepted for publication in Osaka J. Mat
A finite generating set for the level 2 mapping class group of a nonorientable surface
We obtain a finite set of generators for the level 2 mapping class group of a
closed nonorientable surface of genus . This set consists of isotopy
classes of Lickorish's Y-homeomorphisms also called crosscap slides.Comment: 13 pages, 3 figure
Crosscap slides and the level 2 mapping class group of a nonorientable surface
Crosscap slide is a homeomorphism of a nonorientable surface of genus at
least 2, which was introduced under the name Y-homeomorphism by Lickorish as an
example of an element of the mapping class group which cannot be expressed as a
product of Dehn twists. We prove that the subgroup of the mapping class group
of a closed nonorientable surface N generated by all crosscap slides is equal
to the level 2 subgroup consisting of those mapping classes which act trivially
on H_1(N;Z_2). We also prove that this subgroup is generated by involutions.Comment: Final versio
Coloring directed cycles
Sopena in his survey [E. Sopena, The oriented chromatic number of graphs: A
short survey, preprint 2013] writes, without any proof, that an oriented cycle
can be colored with three colors if and only if ,
where is the number of forward arcs minus the number of
backward arcs in . This is not true. In this paper we show that can be colored with three colors if and only if
or does not contain three consecutive arcs going in the same
direction
On the commutator length of a Dehn twist
We show that on a nonorientable surface of genus at least 7 any power of a
Dehn twist is equal to a single commutator in the mapping class group and the
same is true, under additional assumptions, for the twist subgroup, and also
for the extended mapping class group of an orientable surface of genus at least
3.Comment: Two references and one paragraph added acknowledging the fact that
some results were known already. 6 page
On finite index subgroups of the mapping class group of a nonorientable surface
Let denote the mapping class group of a compact nonorientable
surface of genus and boundary components, and let
be the subgroup of generated by all Dehn twists. It
is known that is the unique subgroup of of index .
We prove that (and also ) contains a unique subgroup
of index up to conjugation, and a unique subgroup of index
up to conjugation, where . The other
proper subgroups of and have index greater than
. In particular, the minimum index of a proper subgroup of
is .Comment: To appear in Glas. Ma
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