A note on the existence of {k, k}-equivelar polyhedral maps

A polyhedral map is called $\{p, q\}$-equivelar if each face has $p$ edges and each vertex belongs to $q$ faces. In 1983, it was shown that there exist infinitely many geometrically realizable $\{p, q\}$-equivelar polyhedral maps if $q > p = 4$, $p > q = 4$ or $q - 3 > p = 3$. It was shown in 2001 that there exist infinitely many $\{4, 4\}$- and $\{3, 6\}$-equivelar polyhedral maps. In 1990, it was shown that $\{5, 5\}$- and $\{6, 6\}$-equivelar polyhedral maps exist. In this note, examples are constructed, to show that infinitely many self dual $\{k, k\}$-equivelar polyhedral maps exist for each $k \geq 5$. Also vertex-minimal non-singular $\{p, p\}$-pattern are constructed for all odd primes $p$.Comment: 7 pages. To appear in `Contributions to Algebra and Geometry

Minimal Triangulations of Manifolds

In this survey article, we are interested on minimal triangulations of closed pl manifolds. We present a brief survey on the works done in last 25 years on the following: (i) Finding the minimal number of vertices required to triangulate a given pl manifold. (ii) Given positive integers $n$ and $d$, construction of $n$-vertex triangulations of different $d$-dimensional pl manifolds. (iii) Classifications of all the triangulations of a given pl manifold with same number of vertices. In Section 1, we have given all the definitions which are required for the remaining part of this article. In Section 2, we have presented a very brief history of triangulations of manifolds. In Section 3, we have presented examples of several vertex-minimal triangulations. In Section 4, we have presented some interesting results on triangulations of manifolds. In particular, we have stated the Lower Bound Theorem and the Upper Bound Theorem. In Section 5, we have stated several results on minimal triangulations without proofs. Proofs are available in the references mentioned there.Comment: Survey article, 29 page

On stacked triangulated manifolds

We prove two results on stacked triangulated manifolds in this paper: (a) every stacked triangulation of a connected manifold with or without boundary is obtained from a simplex or the boundary of a simplex by certain combinatorial operations; (b) in dimension $d \geq 4$, if $\Delta$ is a tight connected closed homology $d$-manifold whose $i$th homology vanishes for $1 < i < d-1$, then $\Delta$ is a stacked triangulation of a manifold.These results give affirmative answers to questions posed by Novik and Swartz and by Effenberger.Comment: 11 pages, minor changes in the organization of the paper, add information about recent result

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

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