We introduce the class Ξ£kβ(d) of k-stellated (combinatorial) spheres
of dimension d (0β€kβ€d+1) and compare and contrast it with the
class Skβ(d) (0β€kβ€d) of k-stacked homology d-spheres.
We have Ξ£1β(d)=S1β(d), and Ξ£kβ(d)βSkβ(d) for dβ₯2kβ1. However, for each kβ₯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 Wkβ(d) (and Kkβ(d)) of
simplicial complexes all whose vertex-links belong to Ξ£kβ(dβ1)
(respectively, Skβ(dβ1)). Thus, Wkβ(d)βKkβ(d) for dβ₯2k, while W1β(d)=K1β(d). Let
KΛkβ(d) denote the class of d-dimensional complexes all whose
vertex-links are k-stacked balls. We show that for dβ₯2k+2, there is a
natural bijection Mβ¦MΛ from Kkβ(d) onto KΛkβ(d+1) which is the inverse to the boundary map β:KΛkβ(d+1)βKkβ(d).Comment: Revised Version. Theorem 2.24 is new. 18 pages. arXiv admin note:
substantial text overlap with arXiv:1102.085