arXiv subjects

Günter M. Ziegler

Publications and source records attributed to Günter M. Ziegler.

75 records · Page 5Linked to original sources

arXiv1999

Lectures on 0/1-polytopes

These lectures on the combinatorics and geometry of 0/1-polytopes are meant as an \emph{introduction} and \emph{invitation}. Rather than heading for an extensive survey on 0/1-polytopes I present some interesting aspects of these objects; all of them are related to some quite recent work and progress. 0/1-polytopes have a very simple definition and explicit descriptions; we can enumerate and analyze small examples explicitly in the computer (e.g. using {\tt polymake}). However, any intuition that is derived from the analysis of examples in ``low dimensions'' will miss the true complexity of 0/1-polytopes. Thus, in the following we will study several aspects of the complexity of higher-dimensional 0/1-polytopes: the doubly-exponential number of combinatorial types, the number of facets which can be huge, and the coefficients of defining inequalities which sometimes turn out to be extremely large. Some of the effects and results will be backed by proofs in the course of these lectures; we will also be able to verify some of them on explicit examples, which are accessible as a {\tt polymake} database.

math.CO↗

arXiv1995

Realization spaces of 4-polytopes are universal

Let $P\subset\R^d$ be a $d$-dimensional polytope. The {\em realization space} of~$P$ is the space of all polytopes $P'\subset\R^d$ that are combinatorially equivalent to~$P$, modulo affine transformations. We report on work by the first author, which shows that realization spaces of \mbox{4-dimensional} polytopes can be ``arbitrarily bad'': namely, for every primary semialgebraic set~$V$ defined over~$\Z$, there is a $4$-polytope $P(V)$ whose realization space is ``stably equivalent'' to~$V$. This implies that the realization space of a $4$-polytope can have the homotopy type of an arbitrary finite simplicial complex, and that all algebraic numbers are needed to realize all $4$- polytopes. The proof is constructive. These results sharply contrast the $3$-dimensional case, where realization spaces are contractible and all polytopes are realizable with integral coordinates (Steinitz's Theorem). No similar universality result was previously known in any fixed dimension.

math.MG↗

arXiv1992

On the difference between real and complex arrangements

Let $B$ be an arrangement of linear complex hyperplanes in $C^d$. Then a classical result by Orlik \& Solomon asserts that the cohomology algebra of the complement can be constructed from the combinatorial data that are given by the intersection lattice. If $B'$ is, more generally, a $2$-arrangement in $R^{2d}$ (an arrangement of real subspaces of codimension $2$ with even-dimensional intersections), then the intersection lattice still determines the cohomology {\it groups} of the complement, as was shown by Goresky \& MacPherson. We prove, however, that for $2$-arrangements the cohomology {\it algebra} is not determined by the intersection lattice. It encodes extra information on sign patterns, which can be computed from determinants of linear relations or, equivalently, from linking coefficients in the sense of knot theory. This also allows us (in the case $d=2$) to identify arrangements with the same lattice but different fundamental groups.

alg-geom↗