arXiv subjects

Sainkupar Marwein Mawiong

Publications and source records attributed to Sainkupar Marwein Mawiong.

5 recordsLinked to original sources

arXiv2026

On Weakly Contractible Non-Contractible Finite Topological Spaces of Ten Points

Cianci and Ottina proved that a homotopically trivial non-contractible finite $T_0$-space has at least nine points, and classified such spaces with exactly nine points. We complete the classification for ten points. The main tool is the notion of a naked pair: a relation $c<a$ between a minimal and a maximal point that does not factor through a middle point. We show that minimal homotopically trivial spaces with at most ten points have no naked pairs. This turns minimality into simple conditions on the incidences between the three levels of the space and, together with two lemmas of Cianci and Ottina and an Euler characteristic count, reduces the classification to short case analyses. No such space exists with one or two middle points; there are exactly six with three middle points, forming three types and their order-duals, and exactly four with four middle points. All ten spaces have height two, an antichain of middle points and collapsible order complexes, and the count is confirmed by an exhaustive computer enumeration. We also show that the absence of naked pairs persists up to seventeen points but not beyond, and that height three already occurs at eleven points.

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arXiv2026

Minimum Size of a Poset Realizing $\Z_{2}\times\Z_{2^{n}}$ as its Automorphism Group

We study the realization of finite groups as automorphism groups of finite posets. Given a finite group $G$, let $β(G)$ denote the smallest number of elements in a poset $P$ with $\Aut(P)\cong G$. While $β(G)$ is known for several cyclic and small abelian groups, the non-cyclic abelian case is largely open. In this paper we prove that $β(\Z_{2}\times\Z_{2^{n}})=2^{\,n+1}+2$ for every $n\ge 3$.

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arXiv2026

The Minimum Size of a Poset Realizing $\mathbb{Z}_2 \times \mathbb{Z}_4$ as its Automorphism Group

For a finite group $G$, let $β(G)$ denote the minimum cardinality $|P|$ among finite posets $P$ whose automorphism group $\Aut(P)$ is isomorphic to $G$. While every finite group is realizable as the automorphism group of some finite poset, exact values of $β(G)$ are known only in special cases, most notably for cyclic groups. In this paper we prove that $β(\mathbb{Z}_2 \times \mathbb{Z}_4) = 14$; in particular, the product bound $β(G \times H) \le β(G) + β(H)$ is sharp in this case. The upper bound is realized by an explicit $14$-element poset $P_{14}$, whose automorphism group is computed by a height-function argument together with a rigidity analysis of its covering relations. The lower bound, which constitutes the substantive part of the proof, is established by a case analysis of the orbit decompositions of a hypothetical poset on at most $13$ points under a faithful $G$-action, organized according to the largest orbit size; in each case we construct an order-automorphism outside the given copy of $G$, contradicting $\Aut(P) \cong G$. Among non-cyclic groups, to our knowledge this is the first exact determination of $β(G)$ whose lower bound requires a structural analysis of this kind: for the other non-cyclic abelian groups of order at most $8$, namely $\mathbb{Z}_2 \times \mathbb{Z}_2$ and $\mathbb{Z}_2^3$, the value of $β$ is elementary. The arguments are closely adapted to the subgroup lattice of $\mathbb{Z}_2 \times \mathbb{Z}_4$.

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arXiv2025

Minimal Finite Model of Wedge Sum of Spheres

We classify minimal finite models of the Möbius band and several wedge sums of spheres. In particular, we show that the minimal finite model of the Möbius band coincides with that of the circle $S^{1}$. Furthermore, we prove that both $S^{2}\vee S^{1}$ and $S^{2}\vee S^{2}$ admit minimal finite models on exactly seven points, and that each of $S^{1}\vee S^{1}\vee S^{2}$, $S^{1}\vee S^{1}\vee S^{1}\vee S^{2}$, $S^{1}\vee S^{2}\vee S^{2}$, $S^{2}\vee S^{2}\vee S^{2}$, and $S^{2}\vee S^{2}\vee S^{2}\vee S^{2}$ admits a minimal finite model on exactly eight points.

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arXiv2025

Multiple Cylinder of Relations for Finite Spaces and Nerve Theorem for Strong-Good Cover

In this paper, we develop the concept of multiple cylinder of relations which is a generalization of the relation cylinder, extending the multiple non-Hausdorff mapping cylinder to sequences of finite T0-spaces linked by a series of relations. This construction is important in capturing complex homotopical structures across chains of finite spaces and, when the relations are induced by maps, it serves as a third space that collapses to two distinct finite spaces. Additionally, we introduce the concept of a strong-good cover for simplicial complexes and finite spaces, char acterized by collapsible (rather than merely contractible) intersections. This leads to a strengthened version of the Nerve Theorem, which we develop for simplicial complexes as well as for finite spaces with strong-good covers, demonstrating that these complexes and spaces and their associated nerves maintain the same simple homotopy type, thereby refining classical results for finite simplicial complexes and finite topological structures.

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