SECTIONS OF SURFACE BUNDLES AND LEFSCHETZ FIBRATIONS

Download
2013-11-01
Baykur, R. Inanc
Korkmaz, Mustafa
Monden, Naoyuki
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 >= 2.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY

Suggestions

Liftable homeomorphisms of rank two finite abelian branched covers
Atalan, Ferihe; Medetoğulları, Elif; Ozan, Yıldıray (Springer Nature Switzerland AG, 2020)
We investigate branched regular finite abelian A-covers of the 2-sphere, where every homeomorphism of the base (preserving the branch locus) lifts to a homeomorphism of the covering surface. In this study, we prove that if A is a finite abelian p-group of rank k and Σ → S2 is a regular A-covering branched over n points such that every homeomorphism f : S2 → S2 lifts to Σ, then n = k+1. We will also give a partial classification of such covers for rank two finite p-groups. In particular, we prove that for a ...
On stable torsion length of a Dehn twist
Korkmaz, Mustafa (2005-03-01)
In this note we prove that the stable torsion length of a Dehn twist is positive. This, in particular, answers a question of T. E. Brendle and B. Farb in the negative. We also give upper bounds for this length.
Mapping class groups of nonorientable surfaces
Korkmaz, Mustafa (2002-02-01)
We obtain a finite set of generators for the mapping class group of a nonorientable surface with punctures. We then compute the first homology group of the mapping class group and certain subgroups of it. As an application we prove that the image of 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.
ON THE IDEAL TRIANGULATION GRAPH OF A PUNCTURED SURFACE
Korkmaz, Mustafa (2012-01-01)
We study the ideal triangulation graph T(S) of an oriented punctured surface S of finite type. We show that if S is not the sphere with at most three punctures or the torus with one puncture, then the natural map from the extended mapping class group of S into the simplicial automorphism group of T(S) is an isomorphism. We also show that, the graph T(S) of such a surface S. equipped with its natural simplicial metric is not Gromov hyperbolic. We also show that if the triangulation graph of two oriented punc...
From automorphisms of Riemann surfaces to smooth 4-manifolds
Beyaz, Ahmet; ONARAN, SİNEM; Park, B. Doug (2020-01-01)
Starting from a suitable set of self-diffeomorphisms of a closed Riemann surface, we present a general branched covering method to construct surface bundles over surfaces with positive signature. Armed with this method, we study the classification problem for both surface bundles with nonzero signature and closed simply connected smooth spin 4-manifolds.
Citation Formats
R. I. Baykur, M. Korkmaz, and N. Monden, “SECTIONS OF SURFACE BUNDLES AND LEFSCHETZ FIBRATIONS,” TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, pp. 5999–6016, 2013, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/48554.