Simplicial degree $d$ self-maps on $n$-spheres

Biplab Basak, Raju Kumar Gupta, Ayushi Trivedi
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Abstract

The degree of a map between orientable manifolds is a crucial concept in topology, providing deep insights into the structure and properties of the manifolds and the corresponding maps. This concept has been thoroughly investigated, particularly in the realm of simplicial maps between orientable triangulable spaces. In this paper, we concentrate on constructing simplicial degree $d$ self-maps on $n$-spheres. We describe the construction of several such maps, demonstrating that for every $d \in \mathbb{Z} \setminus {0}$, there exists a degree $d$ simplicial map from a triangulated $n$-sphere with $3|d| + n - 1$ vertices to $\mathbb{S}^n_{n+2}$. Further, we prove that, for every $d \in \mathbb{Z} \setminus {0}$, there exists a simplicial map of degree $3 d$ from a triangulated $n$-sphere with $6|d| + n$ vertices, as well as a simplicial map of degree $3d+\frac{d}{|d|}$ from a triangulated $n$-sphere with $6|d|+n+3$ vertices, to $\mathbb{S}^{n}_{n+2}$. Furthermore, we show that for any $|k| \geq 2$ and $n \geq |k|$, a degree $k$ simplicial map exists from a triangulated $n$-sphere $K$ with $|k| + n + 3$ vertices to $\mathbb{S}^n_{n+2}$. We also prove that for $d = 2$ and 3, these constructions produce vertex-minimal degree $d$ self-maps of $n$-spheres. Additionally, for every $n \geq 2$, we construct a degree $n+1$ simplicial map from a triangulated $n$-sphere with $2n + 4$ vertices to $\mathbb{S}^{n}_{n+2}$. We also prove that this construction provides facet minimal degree $n+1$ self-maps of $n$-spheres.
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n$球上的简单度$d$自映射
可定向流形之间的映射度是拓扑学中的一个重要概念,它能深入揭示流形和相应映射的结构与性质。这一概念已经得到了深入研究,尤其是在可定向三角空间之间的简单映射领域。在本文中,我们专注于在 $n$ 球体上构建度数为 $d$ 的单纯自映射。我们描述了几个这样的映射的构造,证明了对\mathbb{Z}中的每一个 $d\setminus{0}$中的每一个$d,都存在一个从具有$3|d| +n - 1$顶点的三角$n$球到$\mathbb{S}^n_{n+2}$的度$d$简单映射。此外,我们证明,对于 \mathbb{Z} 中的每一个 $d\\setminus{0}$中的每一个$d,都存在一个阶数为$3 d$的简单映射,从顶点为$6|d|+n$的三角$n$球到$\mathbb{Z}^{n}_{n+2}$,以及一个阶数为$3d+frac{d}{|d|}$的简单映射,从顶点为$6|d|+n+3$的三角$n$球到$\mathbb{S}^{n}_{n+2}$。此外,我们还证明,对于任意 $|k| \geq 2$ 和 $n \geq |k|$,都存在一个从具有 $|k| + n + 3$ 顶点的 $n$ 球形 $K$ 到 $mathbb{S}^{n_{n+2}$ 的度数为 $k$ 的简单映射。我们还证明,对于 $d = 2$ 和 3,这些构造会产生顶点最小度 $d$ 的 $n$ 球自映射。此外,每当 $n ≥geq 2$时,我们会构造一个度数为 $n+1$ 的简单映射,从顶点为 2n + 4$ 的阿特朗化 $n$ 球到 $\mathbb{S}^{n}_{n+2}$。我们还证明了这种构造提供了面最小度 $n+1$ 的 $n$ 球自映射。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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