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Mehmetcik Pamuk

Publications and source records attributed to Mehmetcik Pamuk.

At least 19 recordsLinked to original sources
arXiv2026

Component-Perfect Multidimensional Discrete Morse Functions

We introduce the concept of component-perfect multidimensional discrete Morse (MDM) functions on finite regular CW complexes. While perfect discrete Morse functions in the classical sense require the number of critical cells of index $p$ to exactly equal the $p$-th Betti number, the multiparameter framework necessitates a topological dynamics perspective, where critical cells naturally cluster into connected components. Building upon the existing notion of strictly perfect MDM functions, we propose the more flexible definition of component-perfection. We also show how to restrict and extend MDM functions on finite regular CW complexes, laying the groundwork for more complex topological operations.

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arXiv2026

Small Sets of Generators for Handlebody Groups

The mapping class group of a $3$-dimensional handlebody of genus $g$, denoted by $\mathcal{M}(V_g)$, is a fundamental object of study in geometric topology. Building upon the initial generators introduced by Suzuki and their explicit formulation by Takahashi, Wajnryb established that $\mathcal{M}(V_g)$ is generated by exactly five elements for $g \ge 2$. Motivated by recent minimality results in related subgroups we investigate further reductions to this generating set. Through the use of the relations in Wajnryb's presentation, we show that for $g \geq 5$, the handlebody group $\mathcal{M}(V_g)$ is generated by three elements, and for $g \geq 3$, $\mathcal{M}(V_g)$ is generated by four elements, reducing Wajnryb's generating set of five elements by two and one respectively.

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arXiv2026

Small Sets of Topological Generators for Big Mapping Class Groups

Let $S(n)$ be the infinite-type surface with infinite genus and $n \in \mathbb{N}$ ends, all of which are accumulated by genus. The mapping class group of this surface, $\mathrm{Map}(S(n))$, is a Polish group that is not countably generated, but it is countably topologically generated. This paper focuses on finding minimal sets of generators for $\mathrm{Map}(S(n))$. We show that for $n \ge 8$, $\mathrm{Map}(S(n))$ is topologically generated by three elements, and for $n \ge 3$, it is topologically generated by four elements. We also establish a generating set of two elements for the Loch Ness Monster surface $S(1)$, and a generating set of three elements for the Jacob's Ladder surface $S(2)$.

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arXiv2026

Small Torsion Topological Generators for Big Mapping Class Groups

Let $S(n)$, for $n \in \mathbb{N}$, be the infinite-type surface of infinite genus with $n$ ends, each accumulated by genus. Although the mapping class groups of these surfaces are not countably generated,they are Polish groups and hence admit a countable topological generating set. We study minimal topological generating sets for $\mathrm{Map}(S(n))$ consisting entirely of torsion elements, with special attention to involutions. In particular, we prove that $\mathrm{Map}(S(n))$ is topologically generated by four involutions for all $n \geq 16$, and by three involutions for the Loch Ness Monster surface ($n = 1$) and the Jacob's Ladder surface ($n = 2$). We also establish that for even $n \geq 8$, $\mathrm{Map}(S(n))$ is topologically generated by four torsion elements of order $n$. For odd $n \geq 8$, it is topologically generated by three torsion elements of order $n$ and one torsion element of order $n - 1$.

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arXiv2025

Minimal Generation of Mapping Class Groups: A Survey of the Orientable Case

The mapping class group of an orientable surface, which records its symmetries up to isotopy, plays a central role in low-dimensional topology. This chapter explores the foundational problem of determining minimal generating sets for these groups. We chart the development of this area from classical results involving Dehn twist generators to more recent breakthroughs showing that mapping class groups can be generated by just two elements, pairs of torsion elements, or a small collection of involutions. This chapter contains a discussion of the most current results for punctured surfaces, including a new improvement showing that for an even number of punctures $p\geq 8$ the group $\mathrm{Mod}(Σ_{13,p})$ is generated by three involutions. Throughout, we highlight the rich interplay between the algebraic features of these generating sets and the underlying geometric structures they encode. The chapter aims to provide a comprehensive account of the pursuit of algebraic and geometric efficiency within one of topology's most intricate and influential groups.

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arXiv2025

Minimal Generation of Mapping Class Groups: A Survey of the Nonorientable Case

This chapter provides a comprehensive survey of foundational results and recent advances concerning minimal generating sets for the mapping class group of a nonorientable surface, $\mathrm{Mod}(N_{g})$, and its index-two twist subgroup, $\mathcal{T}_{g}$. Although the theory for orientable surfaces is well established, the nonorientable case presents unique challenges due to the presence of crosscaps, thus requiring generators beyond Dehn twists. We show that, for a sufficiently large genus $g$, both $\mathrm{Mod}(N_{g})$ and $\mathcal{T}_{g}$ are generated by two elements, which is the minimum possible number. The survey details various types of generating sets, including those composed of torsions, involutions, and commutators, illustrating the geometric and algebraic interplay. We unify foundational work with modern breakthroughs and extend results to punctured surfaces, $\mathrm{Mod}(N_{g,p})$, providing explicit generators, relations, and proof sketches with an emphasis on geometric intuition.

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arXiv2025

Elementary Methods for Persistent Homotopy Groups

We study the foundational properties of persistent homotopy groups and develop elementary computational methods for their analysis. Our main theorems are persistent analogues of the Van Kampen, excision, suspension, and Hurewicz theorems. We prove a persistent excision theorem, derive from it a persistent Freudenthal suspension theorem, and obtain a persistent Hurewicz theorem relating the first nonzero persistent homotopy group of a space to its persistent homology. As an application, we compute sublevelset persistent homotopy groups of alkane energy landscapes and show these invariants capture nontrivial loops and higher-dimensional features that comple- ment the information given by persistent homology.

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arXiv2023

On the Torsion Generators of the Mapping Class Groups

We study torsion generators for the (extended) mapping class group or the extended mapping class group of a closed connected orientable surface of genus g. We show that for every g is grater than or equal to 14, mapping class group can be generated by two torsion elements of order g+1 if g is even, and of orders g+1 and g+1 if g is odd. Also for g grater than or equal to 16, mapping class group can be generated by two torsion elements of orders g+1 if g+1 is not divisible by 3, and of orders g+1 and g+1 if g+1 is divisible by 3. Similarly, we obtain two torsion elements generating extended mapping class groups.

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arXiv2023

Involution Generators of the Big Mapping Class Group

Let S = S(n) denote the infinite surface with n ends, n \in N, accumulated by genus. For n \geq 6, we show that the mapping class group of S is topologically generated by five involutions. When n \geq 3, it is topologically generated by six involutions.

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arXiv2022

On the involution generators of the mapping class group of a punctured surface

Let Mod(Sigma_{g, p}) denote the mapping class group of a connected orientable surface of genus g with p punctures. For every even integer p \geq 10 and g \geq 14, we prove that Mod(Sigma_{g, p}) can be generated by three involutions. If the number of punctures p is odd and \geq 9, we show that Mod(Sigma_{g, p}) for g \geq 13 can be generated by four involutions. Moreover, we show that for an even integer p \geq 4 and 3 \leq g \geq 6, Mod(Sigma_{g, p}) can be generated by four involutions.

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arXiv2021

Generating the Extended Mapping Class Group by Three Involutions

We prove that the extended mapping class group, $\rm Mod^{*}(Σ_{g})$, of a connected orientable surface of genus $g$, can be generated by three involutions for $g\geq 5$. In the presence of punctures, we prove that $\rm Mod^{*}(Σ_{g,p})$ can be generated by three involutions for $g\geq 10$ and $p\geq 6$ (with the exception that for $g\geq 11$, $p$ should be at least $15$).

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arXiv2021

The Twist Subgroup is generated by two elements

We show that the twist subgroup $\mathcal{T}_g$ of a nonorientable surface of genus $g$ can be generated by two elements for every odd $g\geq27$ and even $g\geq42$. Using these generators, we can also show that $\mathcal{T}_g$ can be generated by two or three commutators depending on $g$ modulo $4$. Moreover, we show that $\mathcal{T}_g$ can be generated by three elements if $g\geq 8$. For this general case, the number of commutator generators is either three or four depending on $g$ modulo $4$ again.

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arXiv2020

Contiguity Distance between Simplicial Maps

We study properties of contiguity distance between simplicial maps. In particular, we show that simplicial versions of $LS$-category and topological complexity are particular cases of this more general notion.

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arXiv2020

Torsion Generators of the Twist Subgroup

We showed that the twist subgroup of the mapping class group of a closed connected nonorientable surface of genus $g\geq13$ can be generated by two involutions and an element of order $g$ or $g-1$ depending on whether $g$ is odd or even respectively.

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