On $k$-stellated and $k$-stacked spheres

Abstract

We introduce the class ${\Sigma }_k(d)$ of $k$-stellated (combinatorial) spheres of dimension $d$ ($0\le k\le d+1$) and compare and contrast it with the class ${\mathcal{S}}_k(d)$ ($0\le k\le d$) of $k$-stacked homology $d$-spheres. We have ${\Sigma }_1(d)={\mathcal{S}}_1(d)$, and ${\Sigma }_k(d)\subseteq {\mathcal{S}}_k(d)$ for $d\ge 2k-1$. However, for each $k\ge 2$ there are $k$-stacked spheres which are not $k$-stellated. The existence of $k$-stellated spheres which are not $k$-stacked remains an open question. We also consider the class ${\mathcal{W}}_k(d)$ (and ${\mathcal{K}}_k(d)$) of simplicial complexes all whose vertex-links belong to ${\Sigma }_k(d-1)$ (respectively, ${\mathcal{S}}_k(d-1)$). Thus, ${\mathcal{W}}_k(d)\subseteq {\mathcal{K}}_k(d)$ for $d\ge 2k$, while ${\mathcal{W}}_1(d)={\mathcal{K}}_1(d)$. Let ${\bar{\mathcal{K}}}_k(d)$ denote the class of $d$-dimensional complexes all whose vertex-links are $k$-stacked balls. We show that for $d\ge 2k+2$, there is a natural bijection $M\mapsto \bar{M}$ from ${\mathcal{K}}_k(d)$ onto ${\bar{\mathcal{K}}}_k(d+1)$ which is the inverse to the boundary map $\partial :{\bar{\mathcal{K}}}_k(d+1)\to {\mathcal{K}}_k(d)$.

arXiv commentRevised Version. Theorem 2.24 is new. 18 pages. arXiv admin note: substantial text overlap with arXiv:1102.0856

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