arXiv subjects

Martin Tancer

Publications and source records attributed to Martin Tancer.

44 records · Page 3Linked to original sources

arXiv2011

d-Representability of simplicial complexes of fixed dimension

Let K be a simplicial complex with vertex set V = {v_1,..., v_n}. The complex K is d-representable if there is a collection {C_1,...,C_n} of convex sets in R^d such that a subcollection {C_{i_1},...,C_{i_j}} has a nonempty intersection if and only if {v_{i_1},...,v_{i_j}} is a face of K. In 1967 Wegner proved that every simplicial complex of dimension d is (2d+1)-representable. He also suggested that his bound is the best possible, i.e., that there are $d$-dimensional simplicial complexes which are not 2d-representable. However, he was not able to prove his suggestion. We prove that his suggestion was indeed right. Thus we add another piece to the puzzle of intersection patterns of convex sets in Euclidean space.

math.CO↗

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↗

arXiv2010

A counterexample to Wegner's conjecture on good covers

In 1975 Wegner conjectured that the nerve of every finite good cover in R^d is d-collapsible. We disprove this conjecture. A good cover is a collection of open sets in R^d such that the intersection of every subcollection is either empty or homeomorphic to an open d-ball. A simplicial complex is d-collapsible if it can be reduced to an empty complex by repeatedly removing a face of dimension at most d-1 which is contained in a unique maximal face.

math.CO↗

arXiv2010

d-collapsibility is NP-complete for d greater or equal to 4

A simplicial complex is d-collapsible if it can be reduced to an empty complex by repeatedly removing (collapsing) a face of dimension at most d-1 that is contained in a unique maximal face. We prove that the algorithmic question whether a given simplicial complex is d-collapsible is NP-complete for d greater or equal to 4 and polynomial time solvable for d at most 2. As an intermediate step, we prove that d-collapsibility can be recognized by the greedy algorithm for d at most 2, but the greedy algorithm does not work for d greater or equal 3. A simplicial complex is d-representable if it is the nerve of a collection of convex sets in R^d. The main motivation for studying d-collapsible complexes is that every d-representable complex is d-collapsible. We also observe that known results imply that analogical algorithmic question for d-representable complexes is NP-hard for d greater or equal to 2.

math.CO↗

arXiv2009

Non-representability of finite projective planes by convex sets

We prove that there is no d such that all finite projective planes can be represented by convex sets in R^d, answering a question of Alon, Kalai, Matousek, and Meshulam. Here, if P is a projective plane with lines l_1,...,l_n, a representation of P by convex sets in R^d is a collection of convex sets C_1,...,C_n in R^d such that C_{i_1},...,C_{i_k} have a common point if and only if the corresponding lines l_{i_1},...,l_{i_k} have a common point in P. The proof combines a positive-fraction selection lemma of Pach with a result of Alon on "expansion" of finite projective planes. As a corollary, we show that for every $d$ there are 2-collapsible simplicial complexes that are not d-representable, strengthening a result of Matousek and the author.

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↗

arXiv2008

On the gap between representability and collapsibility

A simplicial complex K is called d-representable if it is the nerve of a collection of convex sets in R^d; K is d-collapsible if it can be reduced to an empty complex by repeatedly removing a face of dimension at most d-1 that is contained in a unique maximal face; and K is d-Leray if every induced subcomplex of K has vanishing homology of dimension d and larger. It is known that d-representable implies d-collapsible implies d-Leray, and no two of these notions coincide for d greater or equal to 2. The famous Helly theorem and other important results in discrete geometry can be regarded as results about d-representable complexes, and in many of these results "d-representable" in the assumption can be replaced by "d-collapsible" or even "d-Leray". We investigate "dimension gaps" among these notions, and we construct, for all positive integers d, a 2d-Leray complex that is not (3d-1)-collapsible and a d-collapsible complex that is not (2d-2)-representable. In the proofs we obtain two results of independent interest: (i) The nerve of every finite family of sets, each of size at most d, is d-collapsible. (ii) If the nerve of a simplicial complex K is d-representable, then K embeds in R^d.

math.CO↗

arXiv2008

Combinatorial Alexander Duality -- a Short and Elementary Proof

Let X be a simplicial complex with the ground set V. Define its Alexander dual as a simplicial complex X* = {A \subset V: V \setminus A \notin X}. The combinatorial Alexander duality states that the i-th reduced homology group of X is isomorphic to the (|V|-i-3)-th reduced cohomology group of X* (over a given commutative ring R). We give a self-contained proof.

math.CO↗