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Topological Methods in Nonlinear Analysis

A characterization of $g_2$-minimal normal 3-pseudomanifolds with at most four singularities
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A characterization of $g_2$-minimal normal 3-pseudomanifolds with at most four singularities

Authors

  • Biplab Basak https://orcid.org/0000-0002-4978-7022
  • Raju Kumar Gupta https://orcid.org/0000-0002-3970-8343
  • Sourav Sarkar https://orcid.org/0000-0002-6283-7135

DOI:

https://doi.org/10.12775/TMNA.2024.012

Keywords

Normal pseudomanifolds, vertex folding, edge folding, one-vertex suspension

Abstract

Let $\Delta$ be a $g_2$-minimal normal 3-pseudomanifold. A vertex in $\Delta$ whose link is not a sphere is called a singular vertex. When $\Delta$ contains at most two singular vertices, its combinatorial characterization is known \cite{BasakSwartz}. In this article, we present a combinatorial characterization of such a $\Delta$ when it has three singular vertices, including one $\mathbb{RP}^2$-singularity, or four singular vertices, including two $\mathbb{RP}^2$-singularities. In both cases, we prove that $\Delta$ is obtained from a one-vertex suspension of a surface, and some boundary complexes of $4$-simplices by applying the combinatorial operations of types connected sums, vertex foldings, and edge foldings

References

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B. Basak and R.K. Gupta, A characterization of normal 3-pseudomanifolds with g2 ≤ 4 (2022), 13 pp., arXiv: 2202.06638v1.

B. Basak, R.K. Gupta and S. Sarkar, A characterization of normal 3-pseudomanifolds with at most two singularities, Discrete Math. 346 (2023), no. 12, paper no. 113588, 15 pp.

B. Basak and S. Sarkar, On a construction of some homology d-manifolds (2023), 22 pp.

B. Basak and E. Swartz, Three-dimensional normal pseudomanifolds with relatively few edges, Adv. Math. 365 (2020), 107035, 1–25.

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E. Nevo and E. Novinsky, A characterization of simplicial polytopes with g2 = 1, J. Combin. Theory Ser. A 118 (2011), 387–395.

I. Novik and E. Swartz, Face numbers of pseudomanifolds with isolated singularities, Math. Scan. 110 (2012), 198–212.

E. Swartz, Topological finiteness for edge-vertex enumeration, Adv. Math. 219 (2008), 1722–1728.

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H. Zheng, A characterization of homology manifolds with g2 ≤ 2, J. Combin. Theory Ser. A 153 (2018), 31–45.

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Published

2024-09-21

How to Cite

1.
BASAK, Biplab, GUPTA, Raju Kumar and SARKAR, Sourav. A characterization of $g_2$-minimal normal 3-pseudomanifolds with at most four singularities. Topological Methods in Nonlinear Analysis. Online. 21 September 2024. Vol. 64, no. 2, pp. 479 - 491. [Accessed 20 May 2025]. DOI 10.12775/TMNA.2024.012.
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Issue

Vol 64, No 2 (December 2024)

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Articles

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Copyright (c) 2024 Biplab Basak, Raju Kumar Gupta, Sourav Sarkar

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This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.

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