Extremal hypercuts and shadows of simplicial complexes
type: Item
instance of: scholarly article
title: Extremal hypercuts and shadows of simplicial complexes
zbMATH Open document ID: 1410.05230
author: Ilan Newman
author: Yuval Peled
author: Yuri Rabinovich
author: Nathan Linial
published in: Israel Journal of Mathematics
publication date: 2019-02-07T00:00:00Z
review text: If each set of $d$ vertices of a $d$-dimensional simplicial complex $X$ form a $(d-1)$-face of $X$ then $X$ is said to be with full skeleton. A $d$-dimensional simplicial complex $X$ is called $d$-acyclic over a field $\mathbb{F}$ if the $d$-th homology $H_d(X; \mathbb{F})$ is trivial. An $n$-vertex $d$-dimensional simplicial complex $X$ with full skeleton is called a $d$-hypertree (over $\mathbb{F}$) if it is a maximal $n$-vertex $d$-acyclic (over $\mathbb{F}$) simplicial complex. A $d$-dimensional simplicial complex $Y$ is called a $d$-almost-hypertree (over $\mathbb{F}$) if $Y$ is of the form $X \setminus \{\beta\}$ for some $d$-face (facet) $\beta$ of a $d$-hypertree $X$ (over $\mathbb{F}$). For a $d$-dimensional simplicial complex $X$ with full skeleton and a field $\mathbb{F}$, the $\mathbb{F}$-shadow ${\mathrm SH}(X; \mathbb{F})$ of $X$ is the set of all $d$-simplices $\alpha\not\in X$ such that $H_d(X; \mathbb{F})$ is a proper subspace of $H_d(X\cup\{\alpha\};\mathbb{F})$ (equivalently, $H_d(X;\mathbb{F}) \neq H_d(X\cup \{\alpha\}; \mathbb{F})$). \par For a $(d+1)$-simplex $\sigma$, $\bar{\sigma}$ denotes the simplicial complex consisting of all the faces of $\sigma$ and $\partial\sigma$ denotes the boundary complex of $\bar{\sigma}$. Then trivially, $\bar{\sigma}$ is a $(d+1)$-hypertree and ${\mathrm SH}(\bar{\sigma}; \mathbb{F})=\emptyset$ for any field $\mathbb{F}$. For $d\geq 2$, let $\alpha \neq \beta$ be two facets of $\partial\sigma$, then $\partial\sigma\setminus \{\alpha\}$ is a $d$-hypertree and ${\mathrm SH}(\partial\sigma\setminus\{\alpha\}; \mathbb{F})= \{\alpha\}$ for any field $\mathbb{F}$. Let $Y_{d+2} := \partial\sigma\setminus\{\alpha, \beta\}$. Then $Y_{d+2}$ is a $d$-almost-hypertree. Clearly, ${\mathrm SH}(Y_{d+2}; \mathbb{F}) \subseteq \{\alpha, \beta\}$ and hence ${\mathrm SH}(Y_{d+2}; \mathbb{F})=\emptyset$ for any field $\mathbb{F}$. Let $\mathbb{RP}^2_6$ be the 6-vertex triangulation of the real projective plane and $\beta$ be a 2-f…
reviewed by: Basudeb Datta
Mathematics Subject Classification ID: 05E45
Mathematics Subject Classification ID: 05C65
zbMATH DE Number: 7015398
zbMATH Keywords: $d$-acyclic simplicial complex
zbMATH Keywords: full skeleton
zbMATH Keywords: $d$-hypertree
MaRDI profile type: Publication
full work available at URL: s11856-018-1803-0
OpenAlex ID: W2503224388
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DOI: 10.1007/S11856-018-1803-0
description: scientific article; zbMATH DE number 7015398