scholarly journals Products of pairs of Dehn twists and maximal real Lefschetz fibrations

2013 ◽  
Vol 210 ◽  
pp. 83-132
Author(s):  
Alex Degtyarev ◽  
Nermin Salepci
Keyword(s):  
Existence And Uniqueness ◽  
Modular Group ◽  
Lefschetz Fibrations ◽  
Dehn Twists ◽  
Lefschetz Fibration ◽  
Geometric Application

AbstractWe address the problem of existence and uniqueness of a factorization of a given element of the modular group into a product of two Dehn twists. As a geometric application, we conclude that any maximal real elliptic Lefschetz fibration is algebraic.

scholarly journals Products of pairs of Dehn twists and maximal real Lefschetz fibrations

2013 ◽  
Vol 210 ◽  
pp. 83-132 ◽  
Author(s):  
Alex Degtyarev ◽  
Nermin Salepci
Keyword(s):  
Existence And Uniqueness ◽  
Modular Group ◽  
Lefschetz Fibrations ◽  
Dehn Twists ◽  
Lefschetz Fibration ◽  
Geometric Application

AbstractWe address the problem of existence and uniqueness of a factorization of a given element of the modular group into a product of two Dehn twists. As a geometric application, we conclude that any maximal real elliptic Lefschetz fibration is algebraic.

Fukaya A∞-structures associated to Lefschetz fibrations. V

2021 ◽  
pp. 1-70
Author(s):  
Paul Seidel
Keyword(s):  
Fukaya Category ◽  
Lefschetz Fibrations ◽  
Direct Identification ◽  
Lefschetz Fibration

We (re)consider how the Fukaya category of a Lefschetz fibration is related to that of the fiber. The distinguishing feature of the approach here is a more direct identification of the bimodule homomorphism involved.

scholarly journals A NOTE ON LEFSCHETZ FIBRATIONS ON COMPACT STEIN 4-MANIFOLDS

2012 ◽  
Vol 14 (05) ◽  
pp. 1250035 ◽  
Author(s):  
SELMAN AKBULUT ◽  
M. FIRAT ARIKAN
Keyword(s):  
Alternative Proof ◽  
Lefschetz Fibrations ◽  
Lefschetz Fibration

Loi–Piergallini and Akbulut–Ozbagci showed that every compact Stein surface admits a Lefschetz fibration over the disk D2 with bounded fibers. In this note we give a more intrinsic alternative proof of this result.

scholarly journals A real open book not fillable by a real Lefschetz fibration

2014 ◽  
Vol 25 (04) ◽  
pp. 1450031
Author(s):  
Ferit Öztürk ◽  
Nermin Salepci
Keyword(s):  
Real Structure ◽  
Manifolds With Boundary ◽  
Open Book ◽  
Lefschetz Fibrations ◽  
Open Book Decompositions ◽  
Lefschetz Fibration

A real 3- or 4-manifold has by definition an orientation preserving smooth involution acting on it. We consider Lefschetz fibrations of 4-dimensional manifolds-with-boundary and open book decompositions on their boundary in the existence of a real structure. We prove that there is a real open book which cannot be filled by a real Lefschetz fibration, although it is filled by non-real Lefschetz fibrations.

scholarly journals Classification of Boundary Lefschetz Fibrations over the Disc

2018 ◽  
pp. 399-418
Author(s):  
Stefan Behrens ◽  
Gil R. Cavalcanti ◽  
Ralph L. Klaasse
Keyword(s):  
Single Component ◽  
Stable Structure ◽  
Complex Structures ◽  
Lefschetz Fibrations ◽  
Lefschetz Fibration ◽  
Generalized Complex Structures ◽  
Type Change ◽  
Generalized Complex

This chapter shows that a 4-manifold admits a boundary Lefschetz fibration over the disc if and only if it is diffeomorphic to S1×S3#nCP¯2,#mCP2#nCP¯2 or #m(S2×S2). Given the relation between boundary Lefschetz fibrations and stable generalized complex structures, the chapter concludes that the 4-manifolds S1×S3#nCP¯2,#(2m+1)CP2#nCP¯2 and #(2m+1)S2×S2 admit stable generalized complex structures whose type change locus has a single component and are the only 4-manifolds whose stable structure arises from boundary Lefschetz fibrations over the disc.

scholarly journals Towards quantized complex numbers: $q$-deformed Gaussian integers and the Picard group

2021 ◽  
Vol Volume 1 ◽  
Author(s):  
Valentin Ovsienko
Keyword(s):  
Chebyshev Polynomials ◽  
Existence And Uniqueness ◽  
Modular Group ◽  
Picard Group ◽  
Complex Numbers ◽  
Gaussian Integers ◽  
Group I

This work is a first step towards a theory of "$q$-deformed complex numbers". Assuming the invariance of the $q$-deformation under the action of the modular group I prove the existence and uniqueness of the operator of translations by~$i$ compatible with this action. Obtained in such a way $q$-deformed Gaussian integers have interesting properties and are related to the Chebyshev polynomials. Comment: 21 pages

Hypothesis on the Solvability of Parabolic Equations with nonlocal Condition

2002 ◽  
Vol 7 (1) ◽  
pp. 93-104 ◽  
Author(s):  
Mifodijus Sapagovas
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Nonlocal Condition ◽  
Nonlocal Conditions ◽  
Numerical Algorithms ◽  
Uniqueness Of The Solution

Numerous and different nonlocal conditions for the solvability of parabolic equations were researched in many articles and reports. The article presented analyzes such conditions imposed, and observes that the existence and uniqueness of the solution of parabolic equation is related mainly to ”smallness” of functions, involved in nonlocal conditions. As a consequence the hypothesis has been made, stating the assumptions on functions in nonlocal conditions are related to numerical algorithms of solving parabolic equations, and not to the parabolic equation itself.

Pseudo-Anosov Theory

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Braid Groups ◽  
Intersection Numbers ◽  
Branched Covers ◽  
Algebraic Integers ◽  
Dehn Twists ◽  
Mapping Classes ◽  
Dynamical Properties

This chapter focuses on the construction as well as the algebraic and dynamical properties of pseudo-Anosov homeomorphisms. It first presents five different constructions of pseudo-Anosov mapping classes: branched covers, constructions via Dehn twists, homological criterion, Kra's construction, and a construction for braid groups. It then proves a few fundamental facts concerning stretch factors of pseudo-Anosov homeomorphisms, focusing on the theorem that pseudo-Anosov stretch factors are algebraic integers. It also considers the spectrum of pseudo-Anosov stretch factors, along with the special properties of those measured foliations that are the stable (or unstable) foliations of some pseudo-Anosov homeomorphism. Finally, it describes the orbits of a pseudo-Anosov homeomorphism as well as lengths of curves and intersection numbers under iteration.

Sign in / Sign up

Export Citation Format

Share Document