--- Problems on homomorphisms of mapping class groups,
Problems on
Mapping Class Groups and Related Topics, B.Farb Ed.,
Amer. Math. Soc., Proc. Symp.
Pure Math., 74 (2006), 85 --
94.
The purpose this note is to single out some of the
problems on the algebraic structure of the mapping class group. Most of our
problems are on homomorphisms from mapping class
groups. We also state a couple of others problems, such as those related to the
theory of Lefschetz fibrations.
ps file.
--- On sections of elliptic fibrations, (J.
with B. Ozbagci),
We find a new relation among right-handed Dehn twists in the mapping class group of a $k$-holed torus for $4 \leq k \leq 9$. This relation induces an elliptic Lefschetz pencil structure on the four-manifold \cp $#(9-k)$ \cpb $ $ with $k$ base
points and twelve singular fibers. By blowing up the base points we get an
elliptic Lefschetz fibration on the complex elliptic
surface $E(1)=$ \cp $#9$ \cpb
$ \to S^2$ with twelve singular fibers and $k$ disjoint sections. More
importantly we can locate these $k$ sections in a Kirby diagram of the induced
elliptic Lefschetz fibration. The $n$-th power of our relation gives an explicit description for
$k$ disjoint sections of the induced elliptic fibration on the complex elliptic
surface $E(n) \to S^2$ for $n \geq
1$. ps file. or pdf
file
--- Automorphisms of the
Hatcher-Thurston complex, (J. with
Let S be a compact, connected, orientable surface of positive
genus. Let HT(S) be the Hatcher-Thurston complex of $S$. We prove that Aut HT(S)
is isomorphic to the extended mapping class group of S modulo its
center. ps file.
--- Generating
the surface mapping class group by two elements, Transections
of Amer. Math. Soc. 357 (2005),
3299—3310.
Wajnryb proved that the mapping
class group of an orientable surface is generated by two elements. We prove
that one of these generators can be taken as a Dehn
twist. We also prove that the extended mapping class group is generated by two
elements, again one of which is a Dehn twist. Another result we prove is that
the mapping class groups are also generated by two elements of finite
order. ps file.
---
On stable torsion length of a Dehn twist, Mathematical Research Letters 12
(2005), 335—339.
(NEEDS TO BE UPDATED. PUBLISHED VERSION IS DIFFERENT ) In this note we prove that there is no
constant $C$, depending on the genus of the surface, such that every element in
the mapping class group can be written as a product of at most $C$ torsion
elements, answering a question of T. E. Brendle and
B. Farb in the negative. ps file.
--- Homomorphisms
from mapping class groups (J. with
This paper concerns rigidity of the mapping class groups. We
show that any homomorphism $\varphi:\mcg_g\to\mcg_h$
between mapping class groups of closed orientable surfaces with distinct genera
$g>h$ is trivial if $g\geq
3$ and has finite image for all $g\geq 1$. Some
implications are drawn for more general homomorphs of
these groups. ps file.
--- On cofinite subgroups
of mapping class groups, Proceedings
of 9th Gokova Geometry-Topology Conference, Turkish Journal of Mathematics 27 No. 1 (2003),
115-123.
For every positive integer $n$, we exhibit a cofinite subgroup $\Gamma_n$ of
the mapping class group of a surface of genus at most two such that $\Gamma_n$ admits an epimorphism
onto a free group of rank $n$. We conclude that $H^1(\Gamma_n;\Z)$
has rank at least $n$ and the dimension of the second bounded cohomology of
each of these mapping class groups is the cardinality of the continuum. In the
case of genus two, the groups $\Gamma_n$ can be
chosen not to
contain the Torelli group. Similarly for hyperelliptic mapping class groups. We also exhibit an automorphism of a subgroup of finite index in the mapping
class group of a sphere with four punctures (or a torus)
such that it is not the restriction of an endomorphism of the whole group.
--- Low-dimensional homology groups of mapping class
groups: a survey. Proceedings of 8th Gokova Geometry-Topology Conference, Turkish Journal of Mathematics 26 no. 1 (2002),
101-114.
In this survey paper, we give a complete list of known
results on the first and the second homology groups of surface mapping class
groups. Some known results on higher (co)homology are also
mentioned. ps file.
--- The second homology groups of mapping class groups
of orientable surfaces. (J. with A. Stipsicz) Math. Proc. Camb. Phil. Soc.
134 No.3
(2003), 479-489.
We first give an elementary computation for the second
homology groups of mapping class groups of closed orientable surfaces of genus
at least 4. This computation uses only the presentation of the mapping class
group and the Hopf theorem which gives the second
homology of a group from a given presentation. We then use Harer's
homology stability theorem and the Hochshild-Serre
spectral sqeuence for group extensions to give a new
proof of Harer's theorem, by extending to genus 4
case, on the second homology groups of mapping class groups. ps file.
--- Stable commutator length
of a Dehn twist.
It is proved that the stable commutator length of a Dehn twist
in the mapping class group is positive and the tenth power of a Dehn twist about a nonseparating
simple closed curve is a product of two commutators.
As an application a new proof of the fact that the growth rate of a Dehn twist is linear is given. ps file.
--- Commutators, Lefschetz fibrations and
signatures of surface bundles. (J. with H. Endo, D. Kotschick,
B. Ozbagci, A. Stipsicz) Topology 41 No.5 (2002), 961-977.
We construct examples of Lefschetz
fibrations with prescribed singular fibers. By taking
differences of pairs of such fibrations with the same
singular fibers, we obtain new examples of surface bundles over surfaces with
non-zero signature. From these we derive new upper bounds for the minimal genus
of a surface representing a given element in the second homology of a mapping
class group. ps file.
--- Mappping class groups of nonorientable
surfaces. Geometriae Dedicata 89 (2002),
109-133.
We obtain a finite set of generators for a nonorientable surface with punctures. We then compute the
first homology group of the mapping class group. As an application, we prove
that a homomorphism from the mapping class group of a nonorientable
surface of genus at least nine to the group of real-analytic diffeomorphisms of
the circle is either trivial or of order two. ps file.
--- Noncomplex smooth 4-manifolds with Lefschetz
fibrations. Internat.
Math. Res. Notices 2001 no. 3, 115-128.
Generalizing Matsumoto's relation in the mapping class
group of a surface of genus $2$, we obtain new relations in the higher genus
mapping class groups. By taking appropriate fiber sums of the corresponding Lefschetz fibrations, we
construct, for every $g\geq 2$, infinitely many pairwise nonhomeomorphic smooth
$4$-manifolds admitting genus-$g$ Lefschetz fibrations over the $2$-sphere $S^2$ but not carrying any
complex structure. The case of genus $2$ was obtained earlier by Ozbagci and Stipsicz based on Matsumoto's
relation. ps file.
--- Minimal number of singular fibers in a Lefschetz fibration. (J. with Burak
Ozbagci) Proc.
Amer. Math. Soc. 129 no. 5 (2001) 1545-1549.
There exists a (relatively minimal) genus
g Lefschetz fibration with only one singular
fiber over a closed (Riemann) surface of genus h iff
g>2 and h>1. The singular fiber can be chosen to be reducible or
irreducible. Other results are that every Dhen twist
on a closed surface of genus at least three is the product of two commutators and no Dehn twist on
any closed surface is equal to a single commutator. ps file.
--- On endomorphisms of surface
mapping class groups. Topology 40
no. 3 (2001), 463-467.
We prove in this paper that any endomorphism of the mapping
class group of an orientable surface onto a subgroup of finite index is an automorphism. ps file.
--- On the linearity of certain mapping class groups.
Turkish Journal of Mathematics 24 no.
4 (2000), 367-371.
S. Bigelow proved that the braid groups are linear.
That is, there is a faithful representation of the braid group into some
general linear group over a field. Using his result, we show that the mapping
class group of a sphere with punctures and that the hyperelliptis
mapping class groups are linear. In particular, the mapping class group of a
closed orientable surface of genus two is linear. ps file.
--- Surface mapping class groups are ultrahopfian.
(J. with John D. McCarthy). Math. Proc. Camb. Phil. Soc. 129 no. 1 (2000), 35-53.
A group G is called ulrahopfian
if every homomorphism F:G ---> G with F(G) normal in G and the
quotient G/F(G) abelian is an isomorphism. We prove
that the mapping class group of an oreintable surface
is ultrahopfian.
--- Automorphisms of complexes
of curves on punctured spheres and on punctured tori.
Topology and its Applications 95
no. 2 (1999), 85-111.
In this paper, we study the complexes of curves on
orientable surfaces of small genus in order to better understand the mapping
class groups of such surfaces. Our main result is that the group of automorphisms of the complex of curves of a surface is
isomorphic to the extended mapping class group of the surface, if the surface
is a sphere with at least five punctures or is a tori
with at least three punctures. As an application we prove that any isomorphism
between two finite index subgroups of the extended mapping class group is
induced by an inner automorphism of the extended
mapping class group. We conclude that the outer automorphism
group of a finite index subgroup of the extended mapping class group is finite.
--- First homology group of mapping class groups of nonorientable surfaces. Math. Proc. Camb. Phil. Soc.
123 no.3 (1998), 487-499.
In this paper, we compute the first homology group of
the mapping class group of a closed nonorientable
surface. It turns out that this group is cyclic of order two if the genus of
the surface is at least seven. Note that the genus of a nonorientable
surface is defined to be the number of real projective planes in a connected
sum decomposition. We also show that in this case the subgroup of the
mapping class group generated by Dehn twists is
perfect. As an algebraic application, we conclude that the group of isometries of a vector space of dimension $n\geq 7$ over the finite field of order two equipped with the
symmetric bilinear form $\langle \, , \rangle$ defined by $\langle v_i,v_j \rangle=\delta_{ij}$ on a basis $\{ v_1,v_2,\ldots,v_n \}$ is perfect.