Extremal hypercuts and shadows of simplicial complexes (Q1717600)

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-face of $\mathbb{RP}^2_6$. Let $M_6 :=\mathbb{RP}^2_6 \setminus \{\beta\}$. Since $H_2(\mathbb{RP}^2; \mathbb{Q}) =\{0\}$, it follows that $\mathbb{RP}^2_6$ is a 2-hypertree over $\mathbb{Q}$ and hence $M_6$ is a 2-almost-hypertree over $\mathbb{Q}$ . \par In the paper under review, the authors show, among others, the following: \par (i) The $\mathbb{F}$-shadow ${\mathrm SH}(M_6; \mathbb{F})$ is non-empty if and only if $\mathbb{F}$ has characteristic 2. Thus, $M_6$ is a 2-almost-hypertree with empty shadow over $\mathbb{Q}$. \par (ii) For $n\geq 5$, if $\mathbb{Z}_n^{\ast}$ is generated by $\{-1, 2\}$ then there exists an $n$-vertex 2-dimensional simplicial complex $X_n$ which is 2-almost-hypertree over every field $\mathbb{F}$ and the $\mathbb{Q}$-shadow ${\mathrm SH}(X_n; \mathbb{Q})=\emptyset$. \par (iii) For $n\geq 5$, there does not exist any $n$-vertex 2-almost hypertree over $\mathbb{F}_2$ with empty $\mathbb{F}_2$-shadow. (The existence of the 2-dimensional simplicial complex $Y_4$ above shows that this result is not true for $n=4$.) \par (iv) If $d\geq 3$ is odd then every $d$-almost-hypertree has an $\mathbb{F}_2$-shadow of positive density. If $d\geq 4$ is even then every $d$-almost-hypertree has an $\mathbb{F}_2$-shadow of vanishingly small density. \par (v) For $n\geq 5$ odd, there exists an $n$-vertex 2-dimensional simplicial complex with $\binom{n-1}{2} - (n+1)$ facets and is shadowless over every field.