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

We introduce the class ${\mathrm{\Sigma }}_k(d)$ 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 ${\mathrm{\Sigma }}_1(d)={\mathcal{S}}_1(d)$, and $\Sigma_k(d) \subseteq {\cal S}_k(d)$for$d \geq 2k - 1$\mathrm{.}However,foreach$k \geq 2$thereare$k$-stackedsphereswhicharenot$k$-stellated\mathrm{.}Theexistenceof$k$-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 ${\mathrm{\Sigma }}_k(d-1)$ (respectively, ${\mathcal{S}}_k(d-1)$). Thus, ${\cal W}_k(d) \subseteq {\cal K}_k(d)$for$d \geq 2k$,while${\cal W}_1(d) = {\cal K}_1(d)$\mathrm{.}Let$\bar{{\cal K}}_k(d)$denotetheclassof$d$-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 $\bar{{\cal K}}_k(d + 1)$whichistheinversetotheboundarymap$\partial \colon \bar{{\cal K}}_k(d + 1) \to {\cal K}_k(d)$.