72 research outputs found

### 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 $k$-stellated and $k$-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

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