12 research outputs found
Hamiltonian submanifolds of regular polytopes
We investigate polyhedral -manifolds as subcomplexes of the boundary
complex of a regular polytope. We call such a subcomplex {\it -Hamiltonian}
if it contains the full -skeleton of the polytope. Since the case of the
cube is well known and since the case of a simplex was also previously studied
(these are so-called {\it super-neighborly triangulations}) we focus on the
case of the cross polytope and the sporadic regular 4-polytopes. By our results
the existence of 1-Hamiltonian surfaces is now decided for all regular
polytopes.
Furthermore we investigate 2-Hamiltonian 4-manifolds in the -dimensional
cross polytope. These are the "regular cases" satisfying equality in Sparla's
inequality. In particular, we present a new example with 16 vertices which is
highly symmetric with an automorphism group of order 128. Topologically it is
homeomorphic to a connected sum of 7 copies of . By this
example all regular cases of vertices with or, equivalently, all
cases of regular -polytopes with are now decided.Comment: 26 pages, 4 figure
Combinatorial properties of the K3 surface: Simplicial blowups and slicings
The 4-dimensional abstract Kummer variety K^4 with 16 nodes leads to the K3
surface by resolving the 16 singularities. Here we present a simplicial
realization of this minimal resolution. Starting with a minimal 16-vertex
triangulation of K^4 we resolve its 16 isolated singularities - step by step -
by simplicial blowups. As a result we obtain a 17-vertex triangulation of the
standard PL K3 surface. A key step is the construction of a triangulated
version of the mapping cylinder of the Hopf map from the real projective
3-space onto the 2-sphere with the minimum number of vertices. Moreover we
study simplicial Morse functions and the changes of their levels between the
critical points. In this way we obtain slicings through the K3 surface of
various topological types.Comment: 31 pages, 3 figure
Simple crystallizations of 4-manifolds
Minimal crystallizations of simply connected PL 4-manifolds are very natural
objects. Many of their topological features are reflected in their
combinatorial structure which, in addition, is preserved under the connected
sum operation. We present a minimal crystallization of the standard PL K3
surface. In combination with known results this yields minimal crystallizations
of all simply connected PL 4-manifolds of "standard" type, that is, all
connected sums of , , and the K3 surface. In
particular, we obtain minimal crystallizations of a pair of homeomorphic but
non-PL-homeomorphic 4-manifolds. In addition, we give an elementary proof that
the minimal 8-vertex crystallization of is unique and its
associated pseudotriangulation is related to the 9-vertex combinatorial
triangulation of by the minimum of four edge contractions.Comment: 23 pages, 7 figures. Minor update, replacement of Figure 7. To appear
in Advances in Geometr
Stacked polytopes and tight triangulations of manifolds
Tightness of a triangulated manifold is a topological condition, roughly
meaning that any simplexwise linear embedding of the triangulation into
euclidean space is "as convex as possible". It can thus be understood as a
generalization of the concept of convexity. In even dimensions,
super-neighborliness is known to be a purely combinatorial condition which
implies the tightness of a triangulation.
Here we present other sufficient and purely combinatorial conditions which
can be applied to the odd-dimensional case as well. One of the conditions is
that all vertex links are stacked spheres, which implies that the triangulation
is in Walkup's class . We show that in any dimension
\emph{tight-neighborly} triangulations as defined by Lutz, Sulanke and Swartz
are tight.
Furthermore, triangulations with -stacked vertex links and the centrally
symmetric case are discussed.Comment: 28 pages, 2 figure
On possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane
In 1987 Brehm and K\"uhnel showed that any triangulation of a combinatorial
-manifold (without boundary) that is not homeomorphic to the sphere has at
least vertices. Moreover, triangulations with exactly
vertices may exist only for `manifolds like projective planes', which can have
dimensions , , , and only. There is a -vertex triangulation of
, a -vertex triangulation of , and -vertex
triangulations of . Recently, the author has constructed first
examples of -vertex triangulations of manifolds like the octonionic
projective plane . The four most symmetrical have symmetry group
of order . These triangulations
were constructed using a computer program after the symmetry group was guessed.
However, it remained unclear why exactly this group is realized as the symmetry
group and whether -vertex triangulations of manifolds like
exist with other (possibly larger) symmetry groups. In this paper we find
strong restrictions on symmetry groups of such -vertex triangulations.
Namely, we present a list of subgroups of containing all
possible symmetry groups of -vertex triangulations of manifolds like the
octonionic projective plane. (We do not know whether all these subgroups can be
realized as symmetry groups.) The group
is the largest group in this list, and the orders of all other groups do not
exceed . A key role in our approach is played by the use of Smith and
Bredon's results on the cohomology of fixed point sets of finite transformation
groups.Comment: 38 page