Classification of totally real elliptic Lefschetz fibrations via necklace diagrams

We show that totally real elliptic Lefschetz fibrations that admit a real section are classified by their "real loci" which is nothing but an $S^1$-valued Morse function on the real part of the total space. We assign to each such real locus a certain combinatorial object that we call a \emph{necklace diagram}. On the one hand, each necklace diagram corresponds to an isomorphism class of a totally real elliptic Lefschetz fibration that admits a real section, and on the other hand, it refers to a decomposition of the identity into a product of certain matrices in $PSL(2,\Z)$. Using an algorithm to find such decompositions, we obtain an explicit list of necklace diagrams associated with certain classes of totally real elliptic Lefschetz fibrations. Moreover, we introduce refinements of necklace diagrams and show that refined necklace diagrams determine uniquely the isomorphism classes of the totally real elliptic Lefschetz fibrations which may not have a real section. By means of necklace diagrams we observe some interesting phenomena underlying special feature of real fibrations.Comment: 25 pages, 30 figure

Real elements in the mapping class group of T2T^2

We present a complete classification of elements in the mapping class group of the torus which have a representative that can be written as a product of two orientation reversing involutions. Our interest in such decompositions is motivated by features of the monodromy maps of real fibrations. We employ the property that the mapping class group of the torus is identifiable with $SL(2,\Z)$ as well as that the quotient group $PSL(2,\Z)$ is the symmetry group of the {\em Farey tessellation} of the Poincar\'e disk.Comment: 15 pages, 11 figure

Morse shellability, tilings and triangulations

25 pages, 9 figuresWe introduce notions of tilings and shellings on finite simplicial complexes, called Morse tilings and shellings, and relate them to the discrete Morse theory of Robin Forman.Skeletons and barycentric subdivisions of Morse tileable or shellable simplicial complexes are Morse tileable or shellable. Moreover, every closed manifold of dimension less than four has a Morse tiled triangulation, admitting compatible discrete Morse functions, while every triangulated closed surface is even Morse shellable. Morse tilings extend a notion of $h$-tilings that we introduced earlier and which provides a geometric interpretation of $h$-vectors. Morse shellability extends the classical notion of shellability

Fibrations de Lefschetz réelles

Nous étudions les fibrations de Lefschetz réelles. Nous présentons des invariants de fibrations de Lefschetz réelles au dessus de D2 ou S2 n'ayant que des valeurs critiques réelles. Dans le cas où le genre des fibres est égal à 1, nous obtenons un objet cPas de résum

Reel lefschetz liflenmeleri.

In this thesis, we present real Lefschetz fibrations. We first study real Lefschetz fibrations around a real singular fiber. We obtain a classification of real Lefschetz fibrations around a real singular fiber by a study of monodromy properties of real Lefschetz fibrations. Using this classification, we obtain some invariants, called real Lefschetz chains, of real Lefschetz fibrations which admit only real critical values. We show that in case the fiber genus is greater then 1, the real Lefschetz chains are complete invariants of directed real Lefschetz fibrations with only real critical values. If the genus is 1, we obtain complete invariants by decorating real Lefschetz chains. For elliptic Lefschetz fibrations we define a combinatorial object which we call necklace diagrams. Using necklace diagrams we obtain a classification of directed elliptic real Lefschetz fibrations which admit a real section and which have only real critical values. We obtain 25 real Lefschetz fibrations which admit a real section and which have 12 critical values all of which are real. We show that among 25 real Lefschetz fibrations, 8 of them are not algebraic. Moreover, using necklace diagrams we show the existence of real elliptic Lefschetz fibrations which can not be written as the fiber sum of two real elliptic Lefschetz fibrations. We define refined necklace diagrams for real elliptic Lefschetz fibrations without a real section and show that refined necklace diagrams classify real elliptic Lefschetz fibrations which have only real critical values.Ph.D. - Doctoral Progra