[Show abstract][Hide abstract] ABSTRACT: We explicitly construct genus-2 Lefschetz fibrations whose total spaces are
minimal symplectic 4-manifolds homeomorphic to complex rational surfaces $\CP
\# p \CPb$ for p=7, 8, 9, and to $3 \CP \#q \CPb$ for q =12,...,19.
Complementarily, we prove that there are no minimal genus-2 Lefschetz
fibrations whose total spaces are homeomorphic to any other simply-connected
4-manifold with b^+ at most 3, with one possible exception when b^+=3.
Meanwhile, we produce positive Dehn twist factorizations for several new
genus-2 Lefschetz fibrations with small number of critical points, including
the smallest possible example, which follow from a reverse engineering
procedure we introduce for this setting. We also derive exotic minimal
symplectic 4-manifolds in the homeomorphism classes of $\CP \# 4 \CPb$ and $3
\CP \# 6 \CPb$ from small Lefschetz fibrations over surfaces of higher genera.
[Show abstract][Hide abstract] ABSTRACT: In this paper, we study stable equivalence of exotically knotted surfaces in
4-manifolds, surfaces that are topologically isotopic but not smoothly
isotopic. We prove that any pair of embedded surfaces in the same homology
class become smoothly isotopic after stabilizing them by handle additions in
the ambient 4-manifold, which can moreover assumed to be attached in a standard
way (locally and unknottedly) in many favorable situations. In particular, any
exotically knotted pair of surfaces with cyclic fundamental group complements
become smoothly isotopic after a same number of standard stabilizations -
analogous to C.T.C. Wall's celebrated result on the stable equivalence of
simply-connected 4-manifolds. We moreover show that all constructions of exotic
knottings of surfaces we are aware of, which display a good variety of
techniques and ideas, produce surfaces that become smoothly isotopic after a
single stabilization.
[Show abstract][Hide abstract] ABSTRACT: In this article, we characterize isomorphism classes of Lefschetz fibrations
with multisections via their monodromy factorizations. We prove that two
Lefschetz fibrations with multisections are isomorphic if and only if their
monodromy factorizations in the relevant mapping class groups are related to
each other by a finite collection of modifications, which extend the well-known
Hurwitz equivalence. This in particular characterizes isomorphism classes of
Lefschetz pencils. We then show that, from simple relations in the mapping
class groups, one can derive new (and old) examples of Lefschetz fibrations
which cannot be written as fiber sums of blown-up pencils.
[Show abstract][Hide abstract] ABSTRACT: We study which surface mapping classes can be factorized into arbitrarily
large number of positive Dehn twists or only of a fixed number. In connection
to fundamental questions regarding the uniform topology of symplectic
4-manifolds / Stein fillings of contact 3-manifolds coming from the topology of
Lefschetz pencils / open books on them, we completely determine when a boundary
multitwist admits arbitrarily long positive Dehn twist factorizations along
nonseparating curves, and which mapping class groups contain elements with such
factorizations. In particular, we observe that only symplectic 4-manifolds of
general type can attain arbitrarily large topology regardless of the genus and
the number of base points of Lefschetz pencils on them.
[Show abstract][Hide abstract] ABSTRACT: The purpose of this note is to explain a combinatorial description of closed
smooth oriented 4-manifolds in terms of positive Dehn twist factorizations of
surface mapping classes, and further explore these connections. This is
obtained via monodromy representations of simplified broken Lefschetz
fibrations on 4-manifolds, for which we provide an extension of Hurwitz moves
that allows us to uniquely determine the isomorphism class of a broken
Lefschetz fibration. We furthermore discuss broken Lefschetz fibrations whose
monodromies are contained in special subgroups of the mapping class group;
namely, the hyperelliptic mapping class group and in the Torelli group,
respectively, and present various results on them which extend or contrast with
those known to hold for honest Lefschetz fibrations. Lastly, we show that there
are 4-manifolds admitting infinitely many pairwise nonisomorphic relatively
minimal broken Lefschetz fibrations with isotopic regular fibers.
[Show abstract][Hide abstract] ABSTRACT: We prove that any symplectic 4-manifold which is not a rational or ruled
surface, after sufficiently many blow-ups, admits an arbitrary number of
nonisomorphic Lefschetz fibrations of the same genus which cannot be obtained
from one another via Luttinger surgeries. This generalizes results of Park and
Yun who constructed pairs of nonisomorphic Lefschetz fibrations on knot
surgered elliptic surfaces. In turn, we prove that there are monodromy
factorizations of Lefschetz pencils which have the same characteristic numbers
but cannot be obtained from each other via partial conjugations by Dehn twists,
answering a problem posed by Auorux.
[Show abstract][Hide abstract] ABSTRACT: We give a short proof of a conjecture of Stipsicz on the minimality of fiber
sums of Lefschetz fibrations, which was proved earlier by Usher. We then
construct the first examples of genus g > 1 Lefschetz fibrations on minimal
symplectic 4-manifolds which admit unique decompositions as fiber sums.
Proceedings of the American Mathematical Society 07/2014; DOI:10.1090/proc/12835 · 0.68 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The purpose of this article is to initiate a study of positive multisections
of Lefschetz fibrations via positive factorizations in mapping class groups of
surfaces. We show that, using our methods, one can effectively capture various
interesting surfaces in symplectic 4-manifolds as multisections, such as
Seiberg-Witten basic classes or the curious 2-section of a genus two Lefschetz
fibration which leads to a counter-example to Stipsicz's conjecture on fiber
sum indecomposable Lefschetz fibrations.
[Show abstract][Hide abstract] ABSTRACT: We prove that there exists no a priori bound on the Euler characteristic of a
closed symplectic 4-manifold coming solely from the genus of a compatible
Lefschetz pencil on it, nor is there a similar bound for Stein fillings of a
contact 3-manifold coming from the genus of a compatible open book --- except
possibly for a few low genera cases. To obtain our results, we produce the
first examples of factorizations of a boundary parallel Dehn twist as
arbitrarily long products of positive Dehn twists along non-separating curves
on a fixed surface with boundary. This solves an open problem posed by Auroux,
Smith and Wajnryb, and a more general variant of it raised by Korkmaz, Ozbagci
and Stipsicz, independently.
[Show abstract][Hide abstract] ABSTRACT: For each \(g \ge 3\) and \(h \ge 2\), we explicitly construct (1) fiber sum indecomposable relatively minimal genus \(g\) Lefschetz fibrations over genus \(h\) surfaces whose monodromies lie in the Torelli group, (2) genus \(g\) Lefschetz fibrations over genus \(h\) surfaces that are not fiber sums of holomorphic ones, and (3) fiber sum indecomposable genus \(g\) surface bundles over genus \(h\) surfaces whose monodromies are in the Torelli group (provided \(g \ge 4\)). The last result amounts to finding explicit irreducible embeddings of surface groups into Torelli groups; in fact we find such embeddings into arbitrary terms of the Johnson filtration.
[Show abstract][Hide abstract] ABSTRACT: We prove that if a closed oriented 4-manifold X fibers over a 2- or
3-dimensional manifold, in most cases all of its virtual Betti numbers are
infinite. In turn, we show that a closed oriented 4-manifold X which is not a
tower of torus bundles and fibering over a 2- or 3-dimensional manifold does
not admit a torsion symplectic canonical class, nor is of Kodaira dimension
zero.
Proceedings of the American Mathematical Society 10/2012; 142(12). DOI:10.1090/S0002-9939-2014-12151-4 · 0.68 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We mostly determine which closed smooth oriented 4-manifolds fibering over
smaller non-zero dimensional manifolds are virtually symplectic, i.e. finitely
covered by symplectic 4-manifolds
[Show abstract][Hide abstract] ABSTRACT: For each pair of integers g at least 2 and h at least 1, we explicitly
construct infinitely many fiber sum and section sum indecomposable genus g
surface bundles over genus h surfaces whose total spaces are pairwise homotopy
inequivalent.
Journal of Topology and Analysis 09/2012; 5(2). DOI:10.1142/S179352531350009X · 0.33 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We show that there are vast families of contact 3-manifolds each member of
which admits infinitely many Stein fillings with arbitrarily big euler
characteristics and arbitrarily small signatures ---which disproves a
conjecture of Stipsicz and Ozbagci. To produce our examples, we set a framework
which generalizes the construction of Stein structures on allowable Lefschetz
fibrations over the 2-disk to those over any orientable base surface, along
with the construction of contact structures via open books on 3-manifolds to
spinal open books introduced in [24].
[Show abstract][Hide abstract] ABSTRACT: The purpose of this article is two-fold: We first give a more elementary
proof of a recent theorem of Korkmaz, Monden, and the author, which states that
the commutator length of the n-th power of a Dehn twist along a boundary
parallel curve on a surface with boundary S of genus g at least two is the
floor of (|n|+3)/2 in the mapping class group of S. The alternative proof we
provide goes through push maps and Morita's use of Milnor-Wood inequalities, in
particular it does not appeal to gauge theory. In turn, we produce infinite
families of pairwise non-homotopic 4-manifolds admitting genus g surface
bundles over genus h surfaces with distinguished sections which are flat but
admit no flat connections for which the sections are flat, for every fixed
pairs of integers g and h at least two. The latter result generalizes a theorem
of Bestvina, Church, and Souto, and allows us to obtain a simple proof of
Morita's non-lifting theorem (for an infinite family of non-conjugate
subgroups) in the case of marked surfaces.
The Michigan Mathematical Journal 06/2012; DOI:10.1307/mmj/1401973053 · 0.41 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The broken genera are orientation preserving diffeomorphism invariants of
closed oriented 4-manifolds, defined via broken Lefschetz fibrations. We study
the properties of the broken genera invariants, and calculate them for various
4-manifolds, while showing that the invariants are sensitive to exotic smooth
structures.
Geometry and Topology Monographs 05/2012; DOI:10.2140/gtm.2012.18.9
[Show abstract][Hide abstract] ABSTRACT: We show how certain stabilizations produce infinitely many closed oriented
4-manifolds which are the total spaces of genus g surface bundles (resp.
Lefschetz fibrations) over genus h surfaces and have non-zero signature, but do
not admit complex structures with either orientations, for "most" (resp. all)
possible values of g at least 3 and h at least 2 (resp. g at least 2 and h
non-negative).
Mathematical Research Letters 11/2011; 19(3). DOI:10.4310/MRL.2012.v19.n3.a5 · 0.41 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We investigate the possible self-intersection numbers for sections of surface
bundles and Lefschetz fibrations over surfaces. When the fiber genus g and the
base genus h are positive, we prove that the adjunction bound 2h-2 is the only
universal bound on the self-intersection number of a section of any such genus
g bundle and fibration. As a side result, in the mapping class group of a
surface with boundary, we calculate the precise value of the commutator lengths
of all powers of a Dehn twist about a boundary component, concluding that the
stable commutator length of such a Dehn twist is 1/2. We furthermore prove that
there is no upper bound on the number of critical points of genus-g Lefschetz
fibrations over surfaces with positive genera admitting sections of maximal
self-intersection, for g at least two.
Transactions of the American Mathematical Society 10/2011; DOI:10.1090/S0002-9947-2013-05840-0 · 1.12 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this article, we produce infinite families of 4-manifolds with positive first betti numbers and meeting certain conditions on their homotopy and smooth types so as to conclude the non-vanishing of the stable cohomotopy Seiberg-Witten invariants of their connected sums. Elementary building blocks used in the earlier work of Ishida and Sasahira are shown to be included in our general construction scheme as well. We then use these families to construct the first examples of families of closed smooth 4-manifolds for which Gromov's simplicial volume is nontrivial, Perelman's lambda-bar invariant is negative, and the relevant Gromov-Hitchin-Thorpe type inequality is satisfied, yet no non-singular solution to the normalized Ricci flow for any initial metric can be obtained. Fang, Zhang and Zhang conjectured that the existence of any non-singular solution to the normalized Ricci flow on smooth 4-manifolds with non-trivial Gromov's simplicial volume and negative Perelman's lambda-bar invariant implies the Gromov-Hitchin-Thorpe type inequality. Our results in particular imply that the converse of this fails to be true for vast families of 4-manifolds. Comment: 46 pages, 1 figure
[Show abstract][Hide abstract] ABSTRACT: In this article, we generalize the classification of genus one Lefschetz fibrations to genus one simplified broken Lefschetz fibrations, which have fibers of genera one and zero. We classify genus one Lefschetz fibrations over the 2-disk with certain non-trivial global monodromies using chart descriptions, and identify the 4-manifolds admitting genus one simplified broken Lefschetz fibrations. Comment: 19 pages, 12 figures
Journal of the Mathematical Society of Japan 10/2010; 67(3). DOI:10.2969/jmsj/06730877 · 0.62 Impact Factor