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Nermin Salepci

Publications and source records attributed to Nermin Salepci.

11 recordsLinked to original sources
arXiv2021

Morse shellings and compatible discrete Morse functions

We introduce a notion of Morse shellings (and tilings) on finite simplicial complexes which extends the classical one and its relation to discrete Morse theory.Skeletons and barycentric subdivisions of Morse shellable (or tileable) simplicial complexes are Morse shellable (or tileable). Moreover, every triangulated closed surface is Morse shellable while every closed three-manifold carries Morse shellable triangulations. Finally, any shelling encodes a class of discrete Morse functions whose critical points are in one-to-one correspondence, preserving the index, with the critical tiles of the shelling.

math.AT↗
arXiv2018

Tilings, packings and expected Betti numbers in simplicial complexes

Let $K$ be a finite simplicial complex. We prove that the normalized expected Betti numbers of a random subcomplex in its $d$-th barycentric subdivision $\text{Sd}^d (K)$ converge to universal limits as $d$ grows to $+ \infty$. In codimension one, we use canonical filtrations of $\text{Sd}^d (K)$ to upper estimate these limits and get a monotony theorem which makes it possible to improve these estimates given any packing of disjoint simplices in $\text{Sd}^d (K)$. We then introduce a notion of tiling of simplicial complexes having the property that skeletons and barycentric subdivisions of tileable simplicial complexes are tileable. This enables us to tackle the problem: How many disjoint simplices can be packed in $\text{Sd}^d (K)$, $d \gg 0$?

math.PR↗
arXiv2017

Asymptotic topology of random subcomplexes in a finite simplicial complex

We consider a finite simplicial complex $K$ together with its successive barycentric subdivisions $Sd^d(K), d\geq0,$ and study the expected topology of a random subcomplex in $Sd^d(K), d\gg0$. We get asymptotic upper and lower bounds for the expected Betti numbers of those subcomplexes, together with the average Morse inequalities and expected Euler characteristic.

math.PR↗
arXiv2017

Asymptotic measures and links in simplicial complexes

We introduce canonical measures on a locally finite simplicial complex $K$ and study their asymptotic behavior under infinitely many barycentric subdivisions. We also compute the face polynomial of the asymptotic link and dual block of a simplex in the $d^{th}$ barycentric subdivision $Sd^d(K)$ of $K$, $d\gg0$. It is almost everywhere constant. When $K$ is finite, we study the limit face polynomial of $Sd^d(K)$ after F.Brenti-V.Welker and E.Delucchi-A.Pixton-L.Sabalka.

math.GT↗
arXiv2015

Real open books and real contact structures

A real 3-manifold is a smooth 3-manifold together with an orientation preserving smooth involution, called a real structure. In this article we study open book decompositions on smooth real 3-manifolds that are compatible with the real structure. We call them real open book decompositions. We show that each real open book carries a real contact structure and two real contact structures supported by the same real open book decomposition are equivariantly isotopic. We also show that every real contact structure on a closed 3-dimensional real manifold is supported by a real open book. Finally, we conjecture that two real open books on a real contact manifold supporting the same real contact structure are related by positive real stabilizations and equivariant isotopy and that the Giroux correspondence applies to real manifolds as well namely that there is a one to one correspondence between the real contact structures on a real 3-manifold up to equivariant contact isotopy and the real open books up to positive real stabilization. Meanwhile, we study some examples of real open books and real Heegaard decompositions in lens spaces.

math.GT↗
arXiv2013

A real open book not fillable by a real 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.

math.GT↗
arXiv2011

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.

math.GT↗
arXiv2011

Invariants of real Lefschetz fibrations

In this note we introduce certain invariants of real Lefschetz fibrations. We call these invariants {\em real Lefschetz chains}. We prove that if the fiber genus is greater than 1, then the real Lefschetz chains are complete invariants of real Lefschetz fibrations with only real critical values. If however the fiber genus is 1, real Lefschetz chains are not sufficient to distinguish real Lefschetz fibrations. We show that by adding a certain binary decoration to real Lefschetz chains, we get a complete invariant.

math.GT↗
arXiv2010

Real elements in the mapping class group of $T^2$

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 Poincaré disk.

math.GT↗
arXiv2009

There is a unique real tight contact 3-ball

We prove that there is a unique real tight contact structure on the 3-ball with convex boundary up to isotopy through real tight contact structures. We also give a partial classification of the real tight solid tori with the real structure being antipodal map along longitudinal and the identity along meridional direction. For the proofs, we use the real versions of contact neighborhood theorems and the invariant convex surface theory in real contact manifolds.

math.GT↗