arXiv subjects

Ponaki Das

Publications and source records attributed to Ponaki Das.

5 recordsLinked to original sources

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

math.CO↗

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 $\beta(G)$ 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 $\beta(G)$ are known only in special cases, most notably for cyclic groups. In this paper we prove that $\beta(\mathbb{Z}_2 \times \mathbb{Z}_4) = 14$; in particular, the product bound $\beta(G \times H) \le \beta(G) + \beta(H)$ 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 $\beta(G)$ 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 $\beta$ is elementary. The arguments are closely adapted to the subgroup lattice of $\mathbb{Z}_2 \times \mathbb{Z}_4$.

math.CO↗

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 cannot have fewer than nine points and classified all such spaces with exactly nine points. The present paper completes the classification for spaces with exactly ten points. No such space exists when the number of middle elements is one or two; this is established by Euler-characteristic arithmetic, beat-point arguments, and an analysis of forced naked edges. For exactly three middle elements there are precisely six spaces up to homeomorphism, forming three explicit types and their order-duals; for exactly four middle elements there are precisely four such spaces. The ten valid spaces are each shown to have a contractible order complex: seven explicit elementary collapse sequences are given, one for each of Types~I through~VII, and the three remaining spaces, the order-duals of Types~I, II, and~III, inherit contractibility from the identity $\mathcal{K}(X^{\mathrm{op}})=\mathcal{K}(X)$ of simplicial complexes, since chains in $X$ and $X^{\mathrm{op}}$ coincide as sets and any collapse sequence for $\mathcal{K}(X)$ is simultaneously one for $\mathcal{K}(X^{\mathrm{op}})$.

math.AT↗

arXiv2024

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.

math.AT↗

arXiv2024

Minimal Finite Model of Wedge Sum of Spheres

We classify minimal finite models of the M\"{o}bius band and several wedge sums of spheres. In particular, we show that the minimal finite model of the M\"{o}bius 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.

math.AT↗