arXiv subjects
Publications and source records attributed to Günter M. Ziegler.
We survey Jean-Louis Loday's vertex description of the associahedron, and its far reaching influence in combinatorics, discrete geometry and algebra. We present in particular four topics were it plays a central role: lattice congruences of the weak order and their quotientopes, cluster algebras and their generalized associahedra, nested complexes and their nestohedra, and operads and the associahedron diagonal.
In his first mathematical paper, published in 1895 when he was 18, Edmund Landau suggested a new way to determine the winner of a chess tournament by not simply adding for each player the fixed number of points they would get for each win or draw, but rather by considering the performance of all players in the tournament relative to each other: each player would get more credit for games won or drawn against stronger players. Landau called this "relative Wertbemessung", which translates to relative score. The basic idea from linear algebra behind this scoring system was rediscovered and reused in many contexts since 1895; in particular, it is a central ingredient in Google's PageRank.
The equivariant cohomology of the classical configuration space $F(\mathbb{R}^d,n)$ has been been of great interest and has been studied intensively starting with the classical papers by Artin (1925/1947) on the theory of braids, by Fox and Neuwirth (1962), Fadell and Neuwirth (1962), and Arnol'd (1969). We give a brief treatment of the subject from the beginnings to recent developments. However, we focus on the mod 2 equivariant cohomology algebras of the classical configuration space $F(\mathbb{R}^d,n)$, as described in an influential paper by Hung (1990). We show with a new, detailed proof that his main result is correct, but that the arguments that were given by Hung on the way to his result are not, as are some of the intermediate results in his paper. This invalidates a paper by three of the present authors, Blagojević, Lück \& Ziegler (2016), who used a claimed intermediate result from Hung (1990) in order to derive lower bounds for the existence of $k$-regular and $\ell$-skew embeddings. Using our new proof for Hung's main result, we get new lower bounds for existence of highly regular embeddings: Some of them agree with the previously claimed bounds, some are weaker.
Robertson (1988) suggested a model for the realization space of a convex d-dimensional polytope and an approach via the implicit function theorem, which -- in the case of a full rank Jacobian -- proves that the realization space is a manifold of dimension NG(P):=d(f_0+f_{d-1})-f_{0,d-1}, which is the natural guess for the dimension given by the number of variables minus the number of quadratic equations that are used in the definition of the realization space. While this indeed holds for many natural classes of polytopes (including simple and simplicial polytopes, as well as all polytopes of dimension at most 3),and Robertson claimed this to be true for all polytopes, Mnev's (1986/1988) Universality Theorem implies that it is not true in general: Indeed, (1) the centered realization space is not a smoothly embedded manifold in general, and (2) it does not have the dimension NG(P) in general. In this paper we develop Jacobian criteria for the analysis of realization spaces. From these we get easily that for various large and natural classes of polytopes the realization spaces are indeed manifolds, whose dimensions are given by NG(P). However, we also identify the smallest polytopes where the dimension count (2) and thus Robertson's claim fails, among them the bipyramid over a triangular prism. For the property (1), we analyze the classical 24-cell: We show that the realization space has at least the dimension 48, and it has points where it is a manifold of this dimension, but it is not smoothly embedded as a manifold everywhere.
We provide lower bounds on the number of periodic Finsler billiard trajectories inside a quadratically convex smooth closed hypersurface $M$ in a $d$-dimensional Finsler space with possibly irreversible Finsler metric. An example of such a system is a billiard in a sufficiently weak magnetic field. The $r$-periodic Finsler billiard trajectories correspond to $r$-gons inscribed in $M$ and having extremal Finsler length. The cyclic group ${\mathbb Z}_r$ acts on these extremal polygons, and one counts the ${\mathbb Z}_r$-orbits. Using Morse and Lusternik-Schnirelmann theories, we prove that if $r\ge 3$ is prime, then the number of $r$-periodic Finsler billiard trajectories is not less than $(r-1)(d-2)+1$. We also give stronger lower bounds when $M$ is in general position. The problem of estimating the number of periodic billiard trajectories from below goes back to Birkhoff. Our work extends to the Finsler setting the results previously obtained for Euclidean billiards by Babenko, Farber, Tabachnikov, and Karasev.
For $3$-dimensional convex polytopes, inscribability is a classical property that is relatively well-understood due to its relation with Delaunay subdivisions of the plane and hyperbolic geometry. In particular, inscribability can be tested in polynomial time, and for every $f$-vector of $3$-polytopes, there exists an inscribable polytope with that $f$-vector. For higher-dimensional polytopes, much less is known. Of course, for any inscribable polytope, all of its lower-dimensional faces need to be inscribable, but this condition does not appear to be very strong. We observe non-trivial new obstructions to the inscribability of polytopes that arise when imposing that a certain inscribable face be inscribed. Using this obstruction, we show that the duals of the $4$-dimensional cyclic polytopes with at least $8$ vertices---all of whose faces are inscribable---are not inscribable. This result is optimal in the following sense: We prove that the duals of the cyclic $4$-polytopes with up to $7$ vertices are, in fact, inscribable. Moreover, we interpret this obstruction combinatorially as a forbidden subposet of the face lattice of a polytope, show that $d$-dimensional cyclic polytopes with at least $d+4$ vertices are not circumscribable, and that no dual of a neighborly $4$-polytope with $8$ vertices, that is, no polytope with $f$-vector $(20,40,28,8)$, is inscribable.
The classical 1966 theorem of Tverberg with its numerous variations was and still is a motivating force behind many important developments in convex and computational geometry as well as the testing ground for methods from equivariant algebraic topology. In 2018, Bárány and Soberón presented a new variation, the "Tverberg plus minus theorem." In this paper, we give a new proof of the Tverberg plus minus theorem, by using a projective transformation. The same tool allows us to derive plus minus analogues of all known affine Tverberg type results. In particular, we prove a plus minus analogue of the optimal colored Tverberg theorem.
Holmsen, Kynčl and Valculescu recently conjectured that if a finite set $X$ with $\ell n$ points in $\mathbb{R}^d$ that is colored by $m$ different colors can be partitioned into $n$ subsets of $\ell$ points each, such that each subset contains points of at least $d$ different colors, then there exists such a partition of $X$ with the additional property that the convex hulls of the $n$ subsets are pairwise disjoint. We prove a continuous analogue of this conjecture, generalized so that each subset contains points of at least $c$ different colors, where we also allow $c$ to be greater than $d$. Furthermore, we give lower bounds on the fraction of the points each of the subsets contains from $c$ different colors. For example, when $n\geq 2$, $d\geq 2$, $c\geq d$ with $m\geq n(c-d)+d$ are integers, and $μ_1, \dots, μ_m$ are $m$ positive finite absolutely continuous measures on $\mathbb{R}^d$, we prove that there exists a partition of $\mathbb{R}^d$ into $n$ convex pieces which equiparts the measures $μ_1, \dots, μ_{d-1}$, and in addition every piece of the partition has positive measure with respect to at least $c$ of the measures $μ_1, \dots, μ_m$.
Using the authors' 2014 "constraints method," we give a short proof for a 2015 result of Dobbins on representations of a point in a polytope as the barycenter of points in a skeleton, and show that the "r-fold Whitney trick" of Mabillard and Wagner (2014/2015) implies that the Topological Tverberg Conjecture for r-fold intersections fails dramatically for all r that are not prime powers.
The amplituhedra arise as images of the totally nonnegative Grassmannians by projections that are induced by linear maps. They were introduced in Physics by Arkani-Hamed \& Trnka (Journal of High Energy Physics, 2014) as model spaces that should provide a better understanding of the scattering amplitudes of quantum field theories. The topology of the amplituhedra has been known only in a few special cases, where they turned out to be homeomorphic to balls. The amplituhedra are special cases of Grassmann polytopes introduced by Lam (Current Developments in Mathematics 2014, Int.\ Press). In this paper we show that that some further amplituhedra are homeomorphic to balls, and that some more Grassmann polytopes and amplituhedra are contractible.
Grünbaum, Barnette, and Reay in 1974 completed the characterization of the pairs $(f_i,f_j)$ of face numbers of $4$-dimensional polytopes. Here we obtain a complete characterization of the pairs of flag numbers $(f_0,f_{03})$ for $4$-polytopes. Furthermore, we describe the pairs of face numbers $(f_0,f_{d-1})$ for $d$-polytopes; this description is complete for even $d\ge6$ except for finitely many exceptional pairs that are "small" in a well-defined sense, while for odd $d$ we show that there are also "large" exceptional pairs. Our proofs rely on the insight that "small" pairs need to be defined and to be treated separately; in the $4$-dimensional case, these may be characterized with the help of the characterizations of the $4$-polytopes with at most $8$ vertices by Altshuler and Steinberg (1984).
Polytope theory has produced a great number of remarkably simple and complete characterization results for face-number sets or f-vector sets of classes of polytopes. We observe that in most cases these sets can be described as the intersection of a semi-algebraic set with an integer lattice. Such "semi-algebraic sets of lattice points" have not received much attention, which is surprising in view of a close connection to Hilbert's Tenth problem, which deals with their projections. We develop proof techniques in order to show that, despite the observations above, some f-vector sets are NOT semi-algebraic sets of lattice points. This is then proved for the set of all pairs $(f_1,f_2)$ of 4-dimensional polytopes, the set of all f-vectors of simplicial $d$-polytopes for $d\ge6$, and the set of all f-vectors of general $d$-polytopes for $d\ge6$. For the f-vector set of all 4-polytopes this remains open.
In 1960 Grünbaum asked whether for any finite mass in $\mathbb{R}^d$ there are $d$ hyperplanes that cut it into $2^d$ equal parts. This was proved by Hadwiger (1966) for $d\le3$, but disproved by Avis (1984) for $d\ge5$, while the case $d=4$ remained open. More generally, Ramos (1996) asked for the smallest dimension $Δ(j,k)$ in which for any $j$ masses there are $k$ affine hyperplanes that simultaneously cut each of the masses into $2^k$ equal parts. At present the best lower bounds on $Δ(j,k)$ are provided by Avis (1984) and Ramos (1996), the best upper bounds by Mani-Levitska, Vrećica \& Živaljević (2006). The problem has been an active testing ground for advanced machinery from equivariant topology. We give a critical review of the work on the Grünbaum--Hadwiger--Ramos problem, which includes the documentation of essential gaps in the proofs for some previous claims. Furthermore, we establish that $Δ(j,2)= \frac12(3j+1)$ in the cases when $j-1$ is a power of $2$, $j\ge5$.
Monsky's theorem from 1970 states that a square cannot be dissected into an odd number of triangles of the same area, but it does not give a lower bound for the area differences that must occur. We extend Monsky's theorem to "constrained framed maps"; based on this we can apply a gap theorem from semi-algebraic geometry to a polynomial area difference measure and thus get a lower bound for the area differences that decreases doubly-exponentially with the number of triangles. On the other hand, we obtain the first superpolynomial upper bounds for this problem, derived from an explicit construction that uses the Thue-Morse sequence.
We show that any $d$-colored set of points in general position in $\mathbb{R}^d$ can be partitioned into $n$ subsets with disjoint convex hulls such that the set of points and all color classes are partitioned as evenly as possible. This extends results by Holmsen, Kynčl & Valculescu (2017) and establishes a special case of their general conjecture. Our proof utilizes a result obtained independently by Soberón and by Karasev in 2010, on simultaneous equipartitions of $d$ continuous measures in $\mathbb{R}^d$ by $n$ convex regions. This gives a convex partition of $\mathbb{R}^d$ with the desired properties, except that points may lie on the boundaries of the regions. In order to resolve the ambiguous assignment of these points, we set up a network flow problem. The equipartition of the continuous measures gives a fractional flow. The existence of an integer flow then yields the desired partition of the point set.
We show that the $f$-vector sets of $d$-polytopes have non-trivial additive structure: They span affine lattices and are embedded in monoids that we describe explicitly. Moreover, for many large subclasses, such as the simple polytopes, or the simplicial polytopes, there are monoid structures on the set of $f$-vectors by themselves: "addition of $f$-vectors minus the $f$-vector of the $d$-simplex" always yields a new $f$-vector. For general $4$-polytopes, we show that the modified addition operation does not always produce an $f$-vector, but that the result is always close to an $f$-vector. In this sense, the set of $f$-vectors of \emph{all} $4$-polytopes forms an "approximate affine semigroup." The proof relies on the fact for $d=4$ every $d$-polytope, or its dual, has a "small facet." This fails for $d>4$. We also describe a two further modified addition operations on $f$-vectors that can be geometrically realized by glueing corresponding polytopes. The second one of these may yield a semigroup structure on the $f$-vector set of all $4$-polytopes.
We show that there are simple 4-dimensional polytopes with n vertices such that all separators of the graph have size at least $Ω(n/\log n)$. This establishes a strong form of a claim by Thurston, for which the construction and proof had been lost. We construct the polytopes by cutting off the vertices and then the edges of a particular type of neighborly cubical polytopes. The graphs of simple polytopes thus obtained are 4-regular; they contain 3-regular "cube-connected cycle graphs" as minors of spanning subgraphs.
Bárány's "topological Tverberg conjecture" from 1976 states that any continuous map of an $N$-simplex $Δ_N$ to $\mathbb{R}^d$, for $N\ge(d+1)(r-1)$, maps points from $r$ disjoint faces in $Δ_N$ to the same point in $\mathbb{R}^d$. The proof of this result for the case when $r$ is a prime, as well as some colored version of the same result, using the results of Borsuk-Ulam and Dold on the non-existence of equivariant maps between spaces with a free group action, were main topics of Matoušek's 2003 book "Using the Borsuk-Ulam theorem." In this paper we show how advanced equivariant topology methods allow one to go beyond the prime case of the topological Tverberg conjecture. First we explain in detail how equivariant cohomology tools (employing the Borel construction, comparison of Serre spectral sequences, Fadell-Husseini index, etc.) can be used to prove the topological Tverberg conjecture whenever $r$ is a prime power. Our presentation includes a number of improved proofs as well as new results, such as a complete determination of the Fadell-Husseini index of chessboard complexes in the prime case. Then we introduce the "constraint method," which applied to suitable "unavoidable complexes" yields a great variety of variations and corollaries to the topological Tverberg theorem, such as the "colored" and the "dimension-restricted" (Van Kampen-Flores type) versions. Both parts have provided crucial components to the recent spectacular counter-examples in high dimensions for the case when $r$ is not a prime power.