On kk-stellated and kk-stacked spheres

We introduce the class $\Sigma_k(d)$ of $k$-stellated (combinatorial) spheres of dimension $d$ ($0 \leq k \leq d + 1$) and compare and contrast it with the class ${\cal S}_k(d)$ ($0 \leq k \leq d$) of $k$-stacked homology $d$-spheres. We have $\Sigma_1(d) = {\cal S}_1(d)$, and $\Sigma_k(d) \subseteq {\cal S}_k(d)$ for $d \geq 2k - 1$. However, for each $k \geq 2$ 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 ${\cal W}_k(d)$ (and ${\cal K}_k(d)$) of simplicial complexes all whose vertex-links belong to $\Sigma_k(d - 1)$ (respectively, ${\cal S}_k(d - 1)$). Thus, ${\cal W}_k(d) \subseteq {\cal K}_k(d)$ for $d \geq 2k$, while ${\cal W}_1(d) = {\cal K}_1(d)$. Let $\bar{{\cal K}}_k(d)$ denote the class of $d$-dimensional complexes all whose vertex-links are $k$-stacked balls. We show that for $d\geq 2k + 2$, there is a natural bijection $M \mapsto \bar{M}$ from ${\cal K}_k(d)$ onto $\bar{{\cal K}}_k(d + 1)$ which is the inverse to the boundary map $\partial \colon \bar{{\cal K}}_k(d + 1) \to {\cal K}_k(d)$.Comment: Revised Version. Theorem 2.24 is new. 18 pages. arXiv admin note: substantial text overlap with arXiv:1102.085

Non-existence of 6-dimensional pseudomanifolds with complementarity

In a previous paper the second author showed that if $M$ is a pseudomanifold with complementarity other than the 6-vertex real projective plane and the 9-vertex complex projective plane, then $M$ must have dimension $\geq 6$, and - in case of equality - $M$ must have exactly 12 vertices. In this paper we prove that such a 6-dimensional pseudomanifold does not exist. On the way to proving our main result we also prove that all combinatorial triangulations of the 4-sphere with at most 10 vertices are combinatorial 4-spheres.Comment: 11 pages. To appear in Advances in Geometr

Minimal triangulations of sphere bundles over the circle

For integers $d \geq 2$ and $\epsilon = 0$ or 1, let $S^{1, d - 1}(\epsilon)$ denote the sphere product $S^{1} \times S^{d - 1}$ if $\epsilon = 0$ and the twisted $S^{d - 1}$ bundle over $S^{1}$ if $\epsilon = 1$. The main results of this paper are: (a) if $d \equiv \epsilon$ (mod 2) then $S^{1, d - 1}(\epsilon)$ has a unique minimal triangulation using $2d + 3$ vertices, and (b) if $d \equiv 1 - \epsilon$ (mod 2) then $S^{1, d - 1}(\epsilon)$ has minimal triangulations (not unique) using $2d + 4$ vertices. The second result confirms a recent conjecture of Lutz. The first result provides the first known infinite family of closed manifolds (other than spheres) for which the minimal triangulation is unique. Actually, we show that while $S^{1, d - 1}(\epsilon)$ has at most one $(2d + 3)$-vertex triangulation (one if $d \equiv \epsilon$ (mod 2), zero otherwise), in sharp contrast, the number of non-isomorphic $(2d + 4)$-vertex triangulations of these $d$-manifolds grows exponentially with $d$ for either choice of $\epsilon$. The result in (a), as well as the minimality part in (b), is a consequence of the following result: (c) for $d \geq 3$, there is a unique $(2d + 3)$-vertex simplicial complex which triangulates a non-simply connected closed manifold of dimension $d$. This amazing simplicial complex was first constructed by K\"{u}hnel in 1986. Generalizing a 1987 result of Brehm and K\"{u}hnel, we prove that (d) any triangulation of a non-simply connected closed $d$-manifold requires at least $2d + 3$ vertices. The result (c) completely describes the case of equality in (d). The proofs rest on the Lower Bound Theorem for normal pseudomanifolds and on a combinatorial version of Alexander duality.Comment: 15 pages, Revised, To appear in `Journal of Combinatorial Theory, Ser. A

On polytopal upper bound spheres

Generalizing a result (the case $k = 1$) due to M. A. Perles, we show that any polytopal upper bound sphere of odd dimension $2k + 1$ belongs to the generalized Walkup class ${\cal K}_k(2k + 1)$, i.e., all its vertex links are $k$-stacked spheres. This is surprising since the $k$-stacked spheres minimize the face-vector (among all polytopal spheres with given $f_0,..., f_{k - 1}$) while the upper bound spheres maximize the face vector (among spheres with a given $f_0$). It has been conjectured that for $d\neq 2k + 1$, all $(k + 1)$-neighborly members of the class ${\cal K}_k(d)$ are tight. The result of this paper shows that, for every $k$, the case $d = 2k +1$ is a true exception to this conjecture.Comment: 4 page