--- Lefschetz
fibrations on 4-manifolds (J. with Andras
Stipsicz), in Handbook of Teichmüller spaces
II. Editor:
Athanase Papadopoulos European Mathematical
Society 2009, 271--296. pdf.
--- Lefschetz
fibrations and an invariant of finitely presented groups. Internat. Math. Res.
Notice 2009, No.9 1547-157. (old version. arXiv: math.GT/0703196)
--- 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 E. Irmak), Israel Journal of
Mathematics 162 (2007)
183—196.
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 W. Harvey), Bulletin of
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.