On the equivariant triangulation of some small covers
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866914328321458176 |
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| author | Gupta, Raju Kumar Sarkar, Soumen |
| author_facet | Gupta, Raju Kumar Sarkar, Soumen |
| contents | In this paper, we study certain properties of $\mathbb{Z}_2^n$-equivariant triangulations of small covers. We show that any $\mathbb{Z}_2^n$-equivariant triangulation of a small cover naturally induces a triangulation of the orbit space. Then, we explicitly construct the minimal $\mathbb{Z}_2^3$-equivariant triangulation of $\mathbb{RP}^3$, which contains $11$ vertices and prove that this is the unique $\mathbb{Z}_2^3$-equivariant triangulation of $\mathbb{RP}^3$ with $11$ vertices. For a finite group $G$, we give a method for constructing some $G$-equivariant triangulations of connected sums of manifolds from their respective $G$-equivariant triangulations. In particular, we construct a $\mathbb{Z}_2^3$-equivariant triangulation of $\mathbb{RP}^3 \# \mathbb{RP}^3$ with $17$ vertices, which is the best known yet. This triangulation of $\mathbb{RP}^3 \# \mathbb{RP}^3$ provides another minimal $g$-vector improving one of the result of Lutz in \cite{LS}. Moreover, we prove that a $\ZZ_2^4$-equivariant triangulation of $\mathbb{RP}^4$ requires at least $18$ vertices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12857 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the equivariant triangulation of some small covers Gupta, Raju Kumar Sarkar, Soumen Algebraic Topology 57Q15, 52B12, 05E45 In this paper, we study certain properties of $\mathbb{Z}_2^n$-equivariant triangulations of small covers. We show that any $\mathbb{Z}_2^n$-equivariant triangulation of a small cover naturally induces a triangulation of the orbit space. Then, we explicitly construct the minimal $\mathbb{Z}_2^3$-equivariant triangulation of $\mathbb{RP}^3$, which contains $11$ vertices and prove that this is the unique $\mathbb{Z}_2^3$-equivariant triangulation of $\mathbb{RP}^3$ with $11$ vertices. For a finite group $G$, we give a method for constructing some $G$-equivariant triangulations of connected sums of manifolds from their respective $G$-equivariant triangulations. In particular, we construct a $\mathbb{Z}_2^3$-equivariant triangulation of $\mathbb{RP}^3 \# \mathbb{RP}^3$ with $17$ vertices, which is the best known yet. This triangulation of $\mathbb{RP}^3 \# \mathbb{RP}^3$ provides another minimal $g$-vector improving one of the result of Lutz in \cite{LS}. Moreover, we prove that a $\ZZ_2^4$-equivariant triangulation of $\mathbb{RP}^4$ requires at least $18$ vertices. |
| title | On the equivariant triangulation of some small covers |
| topic | Algebraic Topology 57Q15, 52B12, 05E45 |
| url | https://arxiv.org/abs/2602.12857 |