Searcharxiv⌕ Search

arXiv subjects

Uli Wagner

Publications and source records attributed to Uli Wagner.

30 records · Page 2Linked to original sources
arXiv2015

Eliminating Higher-Multiplicity Intersections, I. A Whitney Trick for Tverberg-Type Problems

Motivated by topological Tverberg-type problems and by classical results about embeddings (maps without double points), we study the question whether a finite simplicial complex K can be mapped into R^d without triple, quadruple, or, more generally, r-fold points. Specifically, we are interested in maps f from K to R^d that have no r-Tverberg points, i.e., no r-fold points with preimages in r pairwise disjoint simplices of K, and we seek necessary and sufficient conditions for the existence of such maps. We present a higher-multiplicity analogue of the completeness of the Van Kampen obstruction for embeddability in twice the dimension. Specifically, we show that under suitable restrictions on the dimensions, a well-known Deleted Product Criterion (DPC) is not only necessary but also sufficient for the existence of maps without r-Tverberg points. Our main technical tool is a higher-multiplicity version of the classical Whitney trick. An important guiding idea for our work was that sufficiency of the DPC, together with an old result of Ozaydin on the existence of equivariant maps, might yield an approach to disproving the remaining open cases of the long-standing topological Tverberg conjecture. Unfortunately, our proof of the sufficiency of the DPC requires a "codimension 3" proviso, which is not satisfied for when K is the N-simplex. Recently, Frick found an extremely elegant way to overcome this last "codimension 3" obstacle and to construct counterexamples to the topological Tverberg conjecture for d at least 3r+1 (r not a prime power). Here, we present a different construction that yields counterexamples for d at least 3r (r not a prime power).

math.GT↗
arXiv2015

On Topological Minors in Random Simplicial Complexes

For random graphs, the containment problem considers the probability that a binomial random graph $G(n,p)$ contains a given graph as a substructure. When asking for the graph as a topological minor, i.e., for a copy of a subdivision of the given graph, it is well-known that the (sharp) threshold is at $p=1/n$. We consider a natural analogue of this question for higher-dimensional random complexes $X^k(n,p)$, first studied by Cohen, Costa, Farber and Kappeler for $k=2$. Improving previous results, we show that $p=Θ(1/\sqrt{n})$ is the (coarse) threshold for containing a subdivision of any fixed complete $2$-complex. For higher dimensions $k>2$, we get that $p=O(n^{-1/k})$ is an upper bound for the threshold probability of containing a subdivision of a fixed $k$-dimensional complex.

math.CO↗
arXiv2014

Collection of abstracts of the Workshop on Triangulations in Geometry and Topology at CG Week 2014 in Kyoto

This workshop about triangulations of manifolds in computational geometry and topology was held at the 2014 CG-Week in Kyoto, Japan. It focussed on computational and combinatorial questions regarding triangulations, with the goal of bringing together researchers working on various aspects of triangulations and of fostering a closer collaboration within the computational geometry and topology community. Triangulations are highly suitable for computations due to their clear combinatorial structure. As a consequence, they have been successfully employed in discrete algorithms to solve purely theoretical problems in a broad variety of mathematical research areas (knot theory, polytope theory, 2- and 3-manifold topology, geometry, and others). However, due to the large variety of applications, requirements vary from field to field and thus different types of triangulations, different tools, and different frameworks are used in different areas of research. This is why today closely related research areas are sometimes largely disjoint leaving potential reciprocal benefits unused. To address these potentials a workshop on Triangulations was held at Oberwolfach Research Institute in 2012. Since then many new collaborations between researchers of different mathematical communities have been established. Regarding the computational geometry community, the theory of manifolds continues to contribute to advances in more applied areas of the field. Many researchers are interested in fundamental mathematical research about triangulations and thus will benefit from a broad set of knowledge about different research areas using different techniques. We hope that this workshop brought together researchers from many different fields of computational geometry to have fruitful discussions which will lead to new interdisciplinary collaborations and solutions.

cs.CG↗
arXiv2014

Polynomial-time computation of homotopy groups and Postnikov systems in fixed dimension

For several computational problems in homotopy theory, we obtain algorithms with running time polynomial in the input size. In particular, for every fixed k>1, there is a polynomial-time algorithm that, for a 1-connected topological space X given as a finite simplicial complex, or more generally, as a simplicial set with polynomial-time homology, computes the k-th homotopy group π_k(X), as well as the first k stages of a Postnikov system of X. Combined with results of an earlier paper, this yields a polynomial-time computation of [X,Y], i.e., all homotopy classes of continuous mappings X -> Y, under the assumption that Y is (k-1)-connected and dim X < 2k-1. We also obtain a polynomial-time solution of the extension problem, where the input consists of finite simplicial complexes X,Y, where Y is (k-1)-connected and dim X < 2k, plus a subspace A\subseteq X and a (simplicial) map f:A -> Y, and the question is the extendability of f to all of X. The algorithms are based on the notion of a simplicial set with polynomial-time homology, which is an enhancement of the notion of a simplicial set with effective homology developed earlier by Sergeraert and his co-workers. Our polynomial-time algorithms are obtained by showing that simplicial sets with polynomial-time homology are closed under various operations, most notably, Cartesian products, twisted Cartesian products, and classifying space. One of the key components is also polynomial-time homology for the Eilenberg--MacLane space K(Z,1), provided in another recent paper by Krcal, Matousek, and Sergeraert.

cs.CG↗
arXiv2014

Untangling two systems of noncrossing curves

We consider two systems of curves $(α_1,...,α_m)$ and $(β_1,...,β_n)$ drawn on a compact two-dimensional surface $M$ with boundary. Each $α_i$ and each $β_j$ is either an arc meeting the boundary of $M$ at its two endpoints, or a closed curve. The $α_i$ are pairwise disjoint except for possibly sharing endpoints, and similarly for the $β_j$. We want to "untangle" the $β_j$ from the $α_i$ by a self-homeomorphism of $M$; more precisely, we seek a homeomorphism $ϕ:M\rightarrow M$ fixing the boundary of $M$ pointwise such that the total number of crossings of the $α_i$ with the $ϕ(β_j)$ is as small as possible. This problem is motivated by an application in the algorithmic theory of embeddings and 3-manifolds. We prove that if $M$ is planar, i.e., a sphere with $h\geq 0$ boundary components ("holes"), then $O(mn)$ crossings can be achieved (independently of $h$), which is asymptotically tight, as an easy lower bound shows. In general, for an arbitrary (orientable or nonorientable) surface $M$ with $h$ holes and of (orientable or nonorientable) genus $g$, we obtain an $O((m+n)^4)$ upper bound, again independent of $h$ and $g$. The proofs rely, among others, on a result concerning simultaneous planar drawings of graphs by Erten and Kobourov.

math.CO↗
arXiv2014

Embeddability in the 3-sphere is decidable

We show that the following algorithmic problem is decidable: given a $2$-dimensional simplicial complex, can it be embedded (topologically, or equivalently, piecewise linearly) in $\mathbf{R}^3$? By a known reduction, it suffices to decide the embeddability of a given triangulated 3-manifold $X$ into the 3-sphere $S^3$. The main step, which allows us to simplify $X$ and recurse, is in proving that if $X$ can be embedded in $S^3$, then there is also an embedding in which $X$ has a short meridian, i.e., an essential curve in the boundary of $X$ bounding a disk in $S^3\setminus X$ with length bounded by a computable function of the number of tetrahedra of $X$.

math.GT↗
arXiv2014

Computing all maps into a sphere

Given topological spaces X and Y, a fundamental problem of algebraic topology is understanding the structure of all continuous maps X -> Y . We consider a computational version, where X, Y are given as finite simplicial complexes, and the goal is to compute [X,Y], i.e., all homotopy classes of such maps. We solve this problem in the stable range, where for some d >= 2, we have dim X <= 2d - 2 and Y is (d - 1)-connected; in particular, Y can be the d-dimensional sphere S^d. The algorithm combines classical tools and ideas from homotopy theory (obstruction theory, Postnikov systems, and simplicial sets) with algorithmic tools from effective algebraic topology (locally effective simplicial sets and objects with effective homology). In contrast, [X,Y] is known to be uncomputable for general X,Y, since for X = S^1 it includes a well known undecidable problem: testing triviality of the fundamental group of Y. In follow-up papers, the algorithm is shown to run in polynomial time for d fixed, and extended to other problems, such as the extension problem, where we are given a subspace A of X and a map A -> Y and ask whether it extends to a map X -> Y, or computing the Z_2-index---everything in the stable range. Outside the stable range, the extension problem is undecidable.

cs.CG↗
arXiv2013

Extendability of continuous maps is undecidable

We consider two basic problems of algebraic topology, the extension problem and the computation of higher homotopy groups, from the point of view of computability and computational complexity. The extension problem is the following: Given topological spaces X and Y, a subspace A\subseteq X, and a (continuous) map f:A->Y, decide whether f can be extended to a continuous map \bar{f}:X->Y. All spaces are given as finite simplicial complexes and the map f is simplicial. Recent positive algorithmic results, proved in a series of companion papers, show that for (k-1)-connected Y, k>=2, the extension problem is algorithmically solvable if the dimension of X is at most 2k-1, and even in polynomial time when k is fixed. Here we show that the condition \dim X<=2k-1 cannot be relaxed: for \dim X=2k, the extension problem with (k-1)-connected Y becomes undecidable. Moreover, either the target space Y or the pair (X,A) can be fixed in such a way that the problem remains undecidable. Our second result, a strengthening of a result of Anick, says that the computation of π_k(Y) of a 1-connected simplicial complex Y is #P-hard when k is considered as a part of the input.

cs.CG↗
arXiv2011

A geometric proof of the colored Tverberg theorem

The colored Tverberg theorem asserts that for every d and r there exists t=t(d,r) such that for every set C in R^d of cardinality (d+1)t, partitioned into t-point subsets C_1,C_2,...,C_{d+1} (which we think of as color classes; e.g., the points of C_1 are red, the points of C_2 blue, etc.), there exist r disjoint sets R_1,R_2,...,R_r \subseteq C that are "rainbow", meaning that |R_i \cap C_j| < 2 for every i,j, and whose convex hulls all have a common point. All known proofs of this theorem are topological. We present a geometric version of a recent beautiful proof by Blagojević, Matschke, and Ziegler, avoiding a direct use of topological methods. The purpose of this de-topologization is to make the proof more concrete and intuitive, and accessible to a wider audience.

math.CO↗
arXiv2011

On Gromov's Method of Selecting Heavily Covered Points

A result of Boros and Füredi ($d=2$) and of Bárány (arbitrary $d$) asserts that for every $d$ there exists $c_d>0$ such that for every $n$-point set $P\subset \R^d$, some point of $\R^d$ is covered by at least $c_d{n\choose d+1}$ of the $d$-simplices spanned by the points of $P$. The largest possible value of $c_d$ has been the subject of ongoing research. Recently Gromov improved the existing lower bounds considerably by introducing a new, topological proof method. We provide an exposition of the combinatorial component of Gromov's approach, in terms accessible to combinatorialists and discrete geometers, and we investigate the limits of his method. In particular, we give tighter bounds on the \emph{cofilling profiles} for the $(n-1)$-simplex. These bounds yield a minor improvement over Gromov's lower bounds on $c_d$ for large $d$, but they also show that the room for further improvement through the {\cofilling} profiles alone is quite small. We also prove a slightly better lower bound for $c_3$ by an approach using an additional structure besides the {\cofilling} profiles. We formulate a combinatorial extremal problem whose solution might perhaps lead to a tight lower bound for $c_d$.

math.CO↗
arXiv2009

Hardness of embedding simplicial complexes in $\R^d$

Let EMBED(k,d) be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into R^d? Known results easily imply polynomiality of EMBED(k,2) (k=1,2; the case k=1, d=2 is graph planarity) and of EMBED(k,2k) for all k>2 (even if k is not considered fixed). We observe that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that EMBED(d,d) and EMBED(d-1,d) are undecidable for each d>4. Our main result is NP-hardness of EMBED(2,4) and, more generally, of EMBED(k,d) for all k,d with d>3 and d\geq k \geq (2d-2)/3. These dimensions fall outside the so-called metastable range of a theorem of Haefliger and Weber, which characterizes embeddability using the deleted product obstruction. Our reductions are based on examples, due to Segal, Spież, Freedman, Krushkal, Teichner, and Skopenkov, showing that outside the metastable range the deleted product obstruction is not sufficient to characterize embeddability.

cs.CG↗