223 research outputs found
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
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
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