Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the rr-Metastable Range

Motivated by Tverberg-type problems in topological combinatorics 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 higher-multiplicity intersections. We focus on conditions for the existence of almost r-embeddings, i.e., maps from K to R^d without r-intersection points among any set of r pairwise disjoint simplices of K. Generalizing the classical Haefliger-Weber embeddability criterion, we show that a well-known necessary deleted product condition for the existence of almost r-embeddings is sufficient in a suitable r-metastable range of dimensions (r d > (r+1) dim K +2). This significantly extends one of the main results of our previous paper (which treated the special case where d=rk and dim K=(r-1)k, for some k> 3).Comment: 35 pages, 10 figures (v2: reference for the algorithmic aspects updated & appendix on Block Bundles added

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 $p=\Theta(1/\sqrt{n})$ 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.Comment: 15 page

On Expansion and Topological Overlap

We give a detailed and easily accessible proof of Gromov's Topological Overlap Theorem. Let $X$ be a finite simplicial complex or, more generally, a finite polyhedral cell complex of dimension $d$. Informally, the theorem states that if $X$ has sufficiently strong higher-dimensional expansion properties (which generalize edge expansion of graphs and are defined in terms of cellular cochains of $X$) then $X$ has the following topological overlap property: for every continuous map $X\rightarrow \mathbf{R}^d$ there exists a point $p\in \mathbf{R}^d$ that is contained in the images of a positive fraction $\mu>0$ of the $d$-cells of $X$. More generally, the conclusion holds if $\mathbf{R}^d$ is replaced by any $d$-dimensional piecewise-linear (PL) manifold $M$, with a constant $\mu$ that depends only on $d$ and on the expansion properties of $X$, but not on $M$.Comment: Minor revision, updated reference

Computing simplicial representatives of homotopy group elements

A central problem of algebraic topology is to understand the homotopy groups $\pi_d(X)$ of a topological space $X$. For the computational version of the problem, it is well known that there is no algorithm to decide whether the fundamental group $\pi_1(X)$ of a given finite simplicial complex $X$ is trivial. On the other hand, there are several algorithms that, given a finite simplicial complex $X$ that is simply connected (i.e., with $\pi_1(X)$ trivial), compute the higher homotopy group $\pi_d(X)$ for any given $d\geq 2$. %The first such algorithm was given by Brown, and more recently, \v{C}adek et al. However, these algorithms come with a caveat: They compute the isomorphism type of $\pi_d(X)$, $d\geq 2$ as an \emph{abstract} finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of $\pi_d(X)$. Converting elements of this abstract group into explicit geometric maps from the $d$-dimensional sphere $S^d$ to $X$ has been one of the main unsolved problems in the emerging field of computational homotopy theory. Here we present an algorithm that, given a~simply connected space $X$, computes $\pi_d(X)$ and represents its elements as simplicial maps from a suitable triangulation of the $d$-sphere $S^d$ to $X$. For fixed $d$, the algorithm runs in time exponential in $size(X)$, the number of simplices of $X$. Moreover, we prove that this is optimal: For every fixed $d\geq 2$, we construct a family of simply connected spaces $X$ such that for any simplicial map representing a generator of $\pi_d(X)$, the size of the triangulation of $S^d$ on which the map is defined, is exponential in $size(X)$

Shape Dimension and Intrinsic Metric from Samples of Manifolds

We introduce the adaptive neighborhood graph as a data structure for modeling a smooth manifold M embedded in some Euclidean space Rd. We assume that M is known to us only through a finite sample P \subset M, as is often the case in applications. The adaptive neighborhood graph is a geometric graph on P. Its complexity is at most \min{2^{O(k)n, n2}, where n = |P| and k = dim M, as opposed to the n\lceil d/2 \rceil complexity of the Delaunay triangulation, which is often used to model manifolds. We prove that we can correctly infer the connected components and the dimension of M from the adaptive neighborhood graph provided a certain standard sampling condition is fulfilled. The running time of the dimension detection algorithm is d2O(k^{7} log k) for each connected component of M. If the dimension is considered constant, this is a constant-time operation, and the adaptive neighborhood graph is of linear size. Moreover, the exponential dependence of the constants is only on the intrinsic dimension k, not on the ambient dimension d. This is of particular interest if the co-dimension is high, i.e., if k is much smaller than d, as is the case in many applications. The adaptive neighborhood graph also allows us to approximate the geodesic distances between the points in

The Clique Problem in Intersection Graphs of Ellipses and Triangles

Intersection graphs of disks and of line segments, respectively, have been well studied, because of both practical applications and theoretically interesting properties of these graphs. Despite partial results, the complexity status of the Clique problem for these two graph classes is still open. Here, we consider the Clique problem for intersection graphs of ellipses, which, in a sense, interpolate between disks and line segments, and show that the problem is APX-hard in that case. Moreover, this holds even if for all ellipses, the ratio of the larger over the smaller radius is some prescribed number. Furthermore, the reduction immediately carries over to intersection graphs of triangles. To our knowledge, this is the first hardness result for the Clique problem in intersection graphs of convex objects with finite description complexity. We also describe a simple approximation algorithm for the case of ellipses for which the ratio of radii is bounde