@article{13974, abstract = {The Tverberg theorem is one of the cornerstones of discrete geometry. It states that, given a set X of at least (d+1)(r−1)+1 points in Rd, one can find a partition X=X1∪⋯∪Xr of X, such that the convex hulls of the Xi, i=1,…,r, all share a common point. In this paper, we prove a trengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concrete application, we show that any n points in the plane in general position span ⌊n/3⌋ vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Álvarez-Rebollar et al. guarantees ⌊n/6⌋pairwise crossing triangles. Our result generalizes to a result about simplices in Rd, d≥2.}, author = {Fulek, Radoslav and Gärtner, Bernd and Kupavskii, Andrey and Valtr, Pavel and Wagner, Uli}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, publisher = {Springer Nature}, title = {{The crossing Tverberg theorem}}, doi = {10.1007/s00454-023-00532-x}, year = {2023}, } @article{14445, abstract = {We prove the following quantitative Borsuk–Ulam-type result (an equivariant analogue of Gromov’s Topological Overlap Theorem): Let X be a free ℤ/2-complex of dimension d with coboundary expansion at least ηk in dimension 0 ≤ k < d. Then for every equivariant map F: X →ℤ/2 ℝd, the fraction of d-simplices σ of X with 0 ∈ F (σ) is at least 2−d Π d−1k=0ηk. As an application, we show that for every sufficiently thick d-dimensional spherical building Y and every map f: Y → ℝ2d, we have f(σ) ∩ f(τ) ≠ ∅ for a constant fraction μd > 0 of pairs {σ, τ} of d-simplices of Y. In particular, such complexes are non-embeddable into ℝ2d, which proves a conjecture of Tancer and Vorwerk for sufficiently thick spherical buildings. We complement these results by upper bounds on the coboundary expansion of two families of simplicial complexes; this indicates some limitations to the bounds one can obtain by straighforward applications of the quantitative Borsuk–Ulam theorem. Specifically, we prove • an upper bound of (d + 1)/2d on the normalized (d − 1)-th coboundary expansion constant of complete (d + 1)-partite d-dimensional complexes (under a mild divisibility assumption on the sizes of the parts); and • an upper bound of (d + 1)/2d + ε on the normalized (d − 1)-th coboundary expansion of the d-dimensional spherical building associated with GLd+2(Fq) for any ε > 0 and sufficiently large q. This disproves, in a rather strong sense, a conjecture of Lubotzky, Meshulam and Mozes.}, author = {Wagner, Uli and Wild, Pascal}, issn = {1565-8511}, journal = {Israel Journal of Mathematics}, number = {2}, pages = {675--717}, publisher = {Springer Nature}, title = {{Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes}}, doi = {10.1007/s11856-023-2521-9}, volume = {256}, year = {2023}, } @article{10776, abstract = {Let K be a convex body in Rn (i.e., a compact convex set with nonempty interior). Given a point p in the interior of K, a hyperplane h passing through p is called barycentric if p is the barycenter of K∩h. In 1961, Grünbaum raised the question whether, for every K, there exists an interior point p through which there are at least n+1 distinct barycentric hyperplanes. Two years later, this was seemingly resolved affirmatively by showing that this is the case if p=p0 is the point of maximal depth in K. However, while working on a related question, we noticed that one of the auxiliary claims in the proof is incorrect. Here, we provide a counterexample; this re-opens Grünbaum’s question. It follows from known results that for n≥2, there are always at least three distinct barycentric cuts through the point p0∈K of maximal depth. Using tools related to Morse theory we are able to improve this bound: four distinct barycentric cuts through p0 are guaranteed if n≥3.}, author = {Patakova, Zuzana and Tancer, Martin and Wagner, Uli}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, pages = {1133--1154}, publisher = {Springer Nature}, title = {{Barycentric cuts through a convex body}}, doi = {10.1007/s00454-021-00364-7}, volume = {68}, year = {2022}, } @article{12129, abstract = {Given a finite point set P in general position in the plane, a full triangulation of P is a maximal straight-line embedded plane graph on P. A partial triangulation of P is a full triangulation of some subset P′ of P containing all extreme points in P. A bistellar flip on a partial triangulation either flips an edge (called edge flip), removes a non-extreme point of degree 3, or adds a point in P∖P′ as vertex of degree 3. The bistellar flip graph has all partial triangulations as vertices, and a pair of partial triangulations is adjacent if they can be obtained from one another by a bistellar flip. The edge flip graph is defined with full triangulations as vertices, and edge flips determining the adjacencies. Lawson showed in the early seventies that these graphs are connected. The goal of this paper is to investigate the structure of these graphs, with emphasis on their vertex connectivity. For sets P of n points in the plane in general position, we show that the edge flip graph is ⌈n/2−2⌉-vertex connected, and the bistellar flip graph is (n−3)-vertex connected; both results are tight. The latter bound matches the situation for the subfamily of regular triangulations (i.e., partial triangulations obtained by lifting the points to 3-space and projecting back the lower convex hull), where (n−3)-vertex connectivity has been known since the late eighties through the secondary polytope due to Gelfand, Kapranov, & Zelevinsky and Balinski’s Theorem. For the edge flip-graph, we additionally show that the vertex connectivity is at least as large as (and hence equal to) the minimum degree (i.e., the minimum number of flippable edges in any full triangulation), provided that n is large enough. Our methods also yield several other results: (i) The edge flip graph can be covered by graphs of polytopes of dimension ⌈n/2−2⌉ (products of associahedra) and the bistellar flip graph can be covered by graphs of polytopes of dimension n−3 (products of secondary polytopes). (ii) A partial triangulation is regular, if it has distance n−3 in the Hasse diagram of the partial order of partial subdivisions from the trivial subdivision. (iii) All partial triangulations of a point set are regular iff the partial order of partial subdivisions has height n−3. (iv) There are arbitrarily large sets P with non-regular partial triangulations and such that every proper subset has only regular triangulations, i.e., there are no small certificates for the existence of non-regular triangulations.}, author = {Wagner, Uli and Welzl, Emo}, issn = {1432-0444}, journal = {Discrete & Computational Geometry}, keywords = {Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology, Theoretical Computer Science}, number = {4}, pages = {1227--1284}, publisher = {Springer Nature}, title = {{Connectivity of triangulation flip graphs in the plane}}, doi = {10.1007/s00454-022-00436-2}, volume = {68}, year = {2022}, } @article{14381, abstract = {Expander graphs (sparse but highly connected graphs) have, since their inception, been the source of deep links between Mathematics and Computer Science as well as applications to other areas. In recent years, a fascinating theory of high-dimensional expanders has begun to emerge, which is still in a formative stage but has nonetheless already lead to a number of striking results. Unlike for graphs, in higher dimensions there is a rich array of non-equivalent notions of expansion (coboundary expansion, cosystolic expansion, topological expansion, spectral expansion, etc.), with differents strengths and applications. In this talk, we will survey this landscape of high-dimensional expansion, with a focus on two main results. First, we will present Gromov’s Topological Overlap Theorem, which asserts that coboundary expansion (a quantitative version of vanishing mod 2 cohomology) implies topological expansion (roughly, the property that for every map from a simplicial complex to a manifold of the same dimension, the images of a positive fraction of the simplices have a point in common). Second, we will outline a construction of bounded degree 2-dimensional topological expanders, due to Kaufman, Kazhdan, and Lubotzky.}, author = {Wagner, Uli}, issn = {2102-622X}, journal = {Bulletin de la Societe Mathematique de France}, pages = {281--294}, publisher = {Societe Mathematique de France}, title = {{High-dimensional expanders (after Gromov, Kaufman, Kazhdan, Lubotzky, and others)}}, doi = {10.24033/ast.1188}, volume = {438}, year = {2022}, } @article{10220, abstract = {We study conditions under which a finite simplicial complex K can be mapped to ℝd without higher-multiplicity intersections. An almost r-embedding is a map f: K → ℝd such that the images of any r pairwise disjoint simplices of K do not have a common point. We show that if r is not a prime power and d ≥ 2r + 1, then there is a counterexample to the topological Tverberg conjecture, i.e., there is an almost r-embedding of the (d +1)(r − 1)-simplex in ℝd. This improves on previous constructions of counterexamples (for d ≥ 3r) based on a series of papers by M. Özaydin, M. Gromov, P. Blagojević, F. Frick, G. Ziegler, and the second and fourth present authors. The counterexamples are obtained by proving the following algebraic criterion in codimension 2: If r ≥ 3 and if K is a finite 2(r − 1)-complex, then there exists an almost r-embedding K → ℝ2r if and only if there exists a general position PL map f: K → ℝ2r such that the algebraic intersection number of the f-images of any r pairwise disjoint simplices of K is zero. This result can be restated in terms of a cohomological obstruction and extends an analogous codimension 3 criterion by the second and fourth authors. As another application, we classify ornaments f: S3 ⊔ S3 ⊔ S3 → ℝ5 up to ornament concordance. It follows from work of M. Freedman, V. Krushkal and P. Teichner that the analogous criterion for r = 2 is false. We prove a lemma on singular higher-dimensional Borromean rings, yielding an elementary proof of the counterexample.}, author = {Avvakumov, Sergey and Mabillard, Isaac and Skopenkov, Arkadiy B. and Wagner, Uli}, issn = {1565-8511}, journal = {Israel Journal of Mathematics}, pages = {501–534 }, publisher = {Springer Nature}, title = {{Eliminating higher-multiplicity intersections. III. Codimension 2}}, doi = {10.1007/s11856-021-2216-z}, volume = {245}, year = {2021}, } @inproceedings{7806, abstract = {We consider the following decision problem EMBEDk→d in computational topology (where k ≤ d are fixed positive integers): Given a finite simplicial complex K of dimension k, does there exist a (piecewise-linear) embedding of K into ℝd? The special case EMBED1→2 is graph planarity, which is decidable in linear time, as shown by Hopcroft and Tarjan. In higher dimensions, EMBED2→3 and EMBED3→3 are known to be decidable (as well as NP-hard), and recent results of Čadek et al. in computational homotopy theory, in combination with the classical Haefliger–Weber theorem in geometric topology, imply that EMBEDk→d can be solved in polynomial time for any fixed pair (k, d) of dimensions in the so-called metastable range . Here, by contrast, we prove that EMBEDk→d is algorithmically undecidable for almost all pairs of dimensions outside the metastable range, namely for . This almost completely resolves the decidability vs. undecidability of EMBEDk→d in higher dimensions and establishes a sharp dichotomy between polynomial-time solvability and undecidability. Our result complements (and in a wide range of dimensions strengthens) earlier results of Matoušek, Tancer, and the second author, who showed that EMBEDk→d is undecidable for 4 ≤ k ϵ {d – 1, d}, and NP-hard for all remaining pairs (k, d) outside the metastable range and satisfying d ≥ 4.}, author = {Filakovský, Marek and Wagner, Uli and Zhechev, Stephan Y}, booktitle = {Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms}, isbn = {9781611975994}, location = {Salt Lake City, UT, United States}, pages = {767--785}, publisher = {SIAM}, title = {{Embeddability of simplicial complexes is undecidable}}, doi = {10.1137/1.9781611975994.47}, volume = {2020-January}, year = {2020}, } @inproceedings{7807, abstract = {In a straight-line embedded triangulation of a point set P in the plane, removing an inner edge and—provided the resulting quadrilateral is convex—adding the other diagonal is called an edge flip. The (edge) flip graph has all triangulations as vertices, and a pair of triangulations is adjacent if they can be obtained from each other by an edge flip. The goal of this paper is to contribute to a better understanding of the flip graph, with an emphasis on its connectivity. For sets in general position, it is known that every triangulation allows at least edge flips (a tight bound) which gives the minimum degree of any flip graph for n points. We show that for every point set P in general position, the flip graph is at least -vertex connected. Somewhat more strongly, we show that the vertex connectivity equals the minimum degree occurring in the flip graph, i.e. the minimum number of flippable edges in any triangulation of P, provided P is large enough. Finally, we exhibit some of the geometry of the flip graph by showing that the flip graph can be covered by 1-skeletons of polytopes of dimension (products of associahedra). A corresponding result ((n – 3)-vertex connectedness) can be shown for the bistellar flip graph of partial triangulations, i.e. the set of all triangulations of subsets of P which contain all extreme points of P. This will be treated separately in a second part.}, author = {Wagner, Uli and Welzl, Emo}, booktitle = {Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms}, isbn = {9781611975994}, location = {Salt Lake City, UT, United States}, pages = {2823--2841}, publisher = {SIAM}, title = {{Connectivity of triangulation flip graphs in the plane (Part I: Edge flips)}}, doi = {10.1137/1.9781611975994.172}, volume = {2020-January}, year = {2020}, } @inproceedings{7990, abstract = {Given a finite point set P in general position in the plane, a full triangulation is a maximal straight-line embedded plane graph on P. A partial triangulation on P is a full triangulation of some subset P' of P containing all extreme points in P. A bistellar flip on a partial triangulation either flips an edge, removes a non-extreme point of degree 3, or adds a point in P ⧵ P' as vertex of degree 3. The bistellar flip graph has all partial triangulations as vertices, and a pair of partial triangulations is adjacent if they can be obtained from one another by a bistellar flip. The goal of this paper is to investigate the structure of this graph, with emphasis on its connectivity. For sets P of n points in general position, we show that the bistellar flip graph is (n-3)-connected, thereby answering, for sets in general position, an open questions raised in a book (by De Loera, Rambau, and Santos) and a survey (by Lee and Santos) on triangulations. This matches the situation for the subfamily of regular triangulations (i.e., partial triangulations obtained by lifting the points and projecting the lower convex hull), where (n-3)-connectivity has been known since the late 1980s through the secondary polytope (Gelfand, Kapranov, Zelevinsky) and Balinski’s Theorem. Our methods also yield the following results (see the full version [Wagner and Welzl, 2020]): (i) The bistellar flip graph can be covered by graphs of polytopes of dimension n-3 (products of secondary polytopes). (ii) A partial triangulation is regular, if it has distance n-3 in the Hasse diagram of the partial order of partial subdivisions from the trivial subdivision. (iii) All partial triangulations are regular iff the trivial subdivision has height n-3 in the partial order of partial subdivisions. (iv) There are arbitrarily large sets P with non-regular partial triangulations, while every proper subset has only regular triangulations, i.e., there are no small certificates for the existence of non-regular partial triangulations (answering a question by F. Santos in the unexpected direction).}, author = {Wagner, Uli and Welzl, Emo}, booktitle = {36th International Symposium on Computational Geometry}, isbn = {9783959771436}, issn = {18688969}, location = {Zürich, Switzerland}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Connectivity of triangulation flip graphs in the plane (Part II: Bistellar flips)}}, doi = {10.4230/LIPIcs.SoCG.2020.67}, volume = {164}, year = {2020}, } @inproceedings{7992, abstract = {Let K be a convex body in ℝⁿ (i.e., a compact convex set with nonempty interior). Given a point p in the interior of K, a hyperplane h passing through p is called barycentric if p is the barycenter of K ∩ h. In 1961, Grünbaum raised the question whether, for every K, there exists an interior point p through which there are at least n+1 distinct barycentric hyperplanes. Two years later, this was seemingly resolved affirmatively by showing that this is the case if p=p₀ is the point of maximal depth in K. However, while working on a related question, we noticed that one of the auxiliary claims in the proof is incorrect. Here, we provide a counterexample; this re-opens Grünbaum’s question. It follows from known results that for n ≥ 2, there are always at least three distinct barycentric cuts through the point p₀ ∈ K of maximal depth. Using tools related to Morse theory we are able to improve this bound: four distinct barycentric cuts through p₀ are guaranteed if n ≥ 3.}, author = {Patakova, Zuzana and Tancer, Martin and Wagner, Uli}, booktitle = {36th International Symposium on Computational Geometry}, isbn = {9783959771436}, issn = {18688969}, location = {Zürich, Switzerland}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Barycentric cuts through a convex body}}, doi = {10.4230/LIPIcs.SoCG.2020.62}, volume = {164}, year = {2020}, } @article{9308, author = {Avvakumov, Sergey and Wagner, Uli and Mabillard, Isaac and Skopenkov, A. B.}, issn = {0036-0279}, journal = {Russian Mathematical Surveys}, number = {6}, pages = {1156--1158}, publisher = {IOP Publishing}, title = {{Eliminating higher-multiplicity intersections, III. Codimension 2}}, doi = {10.1070/RM9943}, volume = {75}, year = {2020}, } @article{5986, abstract = {Given a triangulation of a point set in the plane, a flip deletes an edge e whose removal leaves a convex quadrilateral, and replaces e by the opposite diagonal of the quadrilateral. It is well known that any triangulation of a point set can be reconfigured to any other triangulation by some sequence of flips. We explore this question in the setting where each edge of a triangulation has a label, and a flip transfers the label of the removed edge to the new edge. It is not true that every labelled triangulation of a point set can be reconfigured to every other labelled triangulation via a sequence of flips, but we characterize when this is possible. There is an obvious necessary condition: for each label l, if edge e has label l in the first triangulation and edge f has label l in the second triangulation, then there must be some sequence of flips that moves label l from e to f, ignoring all other labels. Bose, Lubiw, Pathak and Verdonschot formulated the Orbit Conjecture, which states that this necessary condition is also sufficient, i.e. that all labels can be simultaneously mapped to their destination if and only if each label individually can be mapped to its destination. We prove this conjecture. Furthermore, we give a polynomial-time algorithm (with 𝑂(𝑛8) being a crude bound on the run-time) to find a sequence of flips to reconfigure one labelled triangulation to another, if such a sequence exists, and we prove an upper bound of 𝑂(𝑛7) on the length of the flip sequence. Our proof uses the topological result that the sets of pairwise non-crossing edges on a planar point set form a simplicial complex that is homeomorphic to a high-dimensional ball (this follows from a result of Orden and Santos; we give a different proof based on a shelling argument). The dual cell complex of this simplicial ball, called the flip complex, has the usual flip graph as its 1-skeleton. We use properties of the 2-skeleton of the flip complex to prove the Orbit Conjecture.}, author = {Lubiw, Anna and Masárová, Zuzana and Wagner, Uli}, issn = {1432-0444}, journal = {Discrete & Computational Geometry}, number = {4}, pages = {880--898}, publisher = {Springer Nature}, title = {{A proof of the orbit conjecture for flipping edge-labelled triangulations}}, doi = {10.1007/s00454-018-0035-8}, volume = {61}, year = {2019}, } @inproceedings{6647, abstract = {The Tverberg theorem is one of the cornerstones of discrete geometry. It states that, given a set X of at least (d+1)(r-1)+1 points in R^d, one can find a partition X=X_1 cup ... cup X_r of X, such that the convex hulls of the X_i, i=1,...,r, all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concrete application, we show that any n points in the plane in general position span floor[n/3] vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Alvarez-Rebollar et al. guarantees floor[n/6] pairwise crossing triangles. Our result generalizes to a result about simplices in R^d,d >=2.}, author = {Fulek, Radoslav and Gärtner, Bernd and Kupavskii, Andrey and Valtr, Pavel and Wagner, Uli}, booktitle = {35th International Symposium on Computational Geometry}, isbn = {9783959771047}, issn = {1868-8969}, location = {Portland, OR, United States}, pages = {38:1--38:13}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{The crossing Tverberg theorem}}, doi = {10.4230/LIPICS.SOCG.2019.38}, volume = {129}, year = {2019}, } @article{7093, abstract = {In graph theory, as well as in 3-manifold topology, there exist several width-type parameters to describe how "simple" or "thin" a given graph or 3-manifold is. These parameters, such as pathwidth or treewidth for graphs, or the concept of thin position for 3-manifolds, play an important role when studying algorithmic problems; in particular, there is a variety of problems in computational 3-manifold topology - some of them known to be computationally hard in general - that become solvable in polynomial time as soon as the dual graph of the input triangulation has bounded treewidth. In view of these algorithmic results, it is natural to ask whether every 3-manifold admits a triangulation of bounded treewidth. We show that this is not the case, i.e., that there exists an infinite family of closed 3-manifolds not admitting triangulations of bounded pathwidth or treewidth (the latter implies the former, but we present two separate proofs). We derive these results from work of Agol, of Scharlemann and Thompson, and of Scharlemann, Schultens and Saito by exhibiting explicit connections between the topology of a 3-manifold M on the one hand and width-type parameters of the dual graphs of triangulations of M on the other hand, answering a question that had been raised repeatedly by researchers in computational 3-manifold topology. In particular, we show that if a closed, orientable, irreducible, non-Haken 3-manifold M has a triangulation of treewidth (resp. pathwidth) k then the Heegaard genus of M is at most 18(k+1) (resp. 4(3k+1)).}, author = {Huszár, Kristóf and Spreer, Jonathan and Wagner, Uli}, issn = {1920-180X}, journal = {Journal of Computational Geometry}, number = {2}, pages = {70–98}, publisher = {Computational Geometry Laborartoy}, title = {{On the treewidth of triangulated 3-manifolds}}, doi = {10.20382/JOGC.V10I2A5}, volume = {10}, year = {2019}, } @article{7108, abstract = {We prove that for every d ≥ 2, deciding if a pure, d-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every d ≥ 2 and k ≥ 0, deciding if a pure, d-dimensional, simplicial complex is k-decomposable is NP-hard. For d ≥ 3, both problems remain NP-hard when restricted to contractible pure d-dimensional complexes. Another simple corollary of our result is that it is NP-hard to decide whether a given poset is CL-shellable.}, author = {Goaoc, Xavier and Patak, Pavel and Patakova, Zuzana and Tancer, Martin and Wagner, Uli}, issn = {0004-5411}, journal = {Journal of the ACM}, number = {3}, publisher = {ACM}, title = {{Shellability is NP-complete}}, doi = {10.1145/3314024}, volume = {66}, year = {2019}, } @inproceedings{184, abstract = {We prove that for every d ≥ 2, deciding if a pure, d-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every d ≥ 2 and k ≥ 0, deciding if a pure, d-dimensional, simplicial complex is k-decomposable is NP-hard. For d ≥ 3, both problems remain NP-hard when restricted to contractible pure d-dimensional complexes.}, author = {Goaoc, Xavier and Paták, Pavel and Patakova, Zuzana and Tancer, Martin and Wagner, Uli}, location = {Budapest, Hungary}, pages = {41:1 -- 41:16}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Shellability is NP-complete}}, doi = {10.4230/LIPIcs.SoCG.2018.41}, volume = {99}, year = {2018}, } @inproceedings{285, abstract = {In graph theory, as well as in 3-manifold topology, there exist several width-type parameters to describe how "simple" or "thin" a given graph or 3-manifold is. These parameters, such as pathwidth or treewidth for graphs, or the concept of thin position for 3-manifolds, play an important role when studying algorithmic problems; in particular, there is a variety of problems in computational 3-manifold topology - some of them known to be computationally hard in general - that become solvable in polynomial time as soon as the dual graph of the input triangulation has bounded treewidth. In view of these algorithmic results, it is natural to ask whether every 3-manifold admits a triangulation of bounded treewidth. We show that this is not the case, i.e., that there exists an infinite family of closed 3-manifolds not admitting triangulations of bounded pathwidth or treewidth (the latter implies the former, but we present two separate proofs). We derive these results from work of Agol and of Scharlemann and Thompson, by exhibiting explicit connections between the topology of a 3-manifold M on the one hand and width-type parameters of the dual graphs of triangulations of M on the other hand, answering a question that had been raised repeatedly by researchers in computational 3-manifold topology. In particular, we show that if a closed, orientable, irreducible, non-Haken 3-manifold M has a triangulation of treewidth (resp. pathwidth) k then the Heegaard genus of M is at most 48(k+1) (resp. 4(3k+1)).}, author = {Huszár, Kristóf and Spreer, Jonathan and Wagner, Uli}, issn = {18688969}, location = {Budapest, Hungary}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{On the treewidth of triangulated 3-manifolds}}, doi = {10.4230/LIPIcs.SoCG.2018.46}, volume = {99}, year = {2018}, } @article{425, abstract = {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 R3? By a known reduction, it suffices to decide the embeddability of a given triangulated 3-manifold X into the 3-sphere S3. The main step, which allows us to simplify X and recurse, is in proving that if X can be embedded in S3, then there is also an embedding in which X has a short meridian, that is, an essential curve in the boundary of X bounding a disk in S3 \ X with length bounded by a computable function of the number of tetrahedra of X.}, author = {Matoušek, Jiří and Sedgwick, Eric and Tancer, Martin and Wagner, Uli}, journal = {Journal of the ACM}, number = {1}, publisher = {ACM}, title = {{Embeddability in the 3-Sphere is decidable}}, doi = {10.1145/3078632}, volume = {65}, year = {2018}, } @article{6774, abstract = {A central problem of algebraic topology is to understand the homotopy groups 𝜋𝑑(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 𝜋1(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 𝜋1(𝑋) trivial), compute the higher homotopy group 𝜋𝑑(𝑋) for any given 𝑑≥2 . However, these algorithms come with a caveat: They compute the isomorphism type of 𝜋𝑑(𝑋) , 𝑑≥2 as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of 𝜋𝑑(𝑋) . Converting elements of this abstract group into explicit geometric maps from the d-dimensional sphere 𝑆𝑑 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 𝜋𝑑(𝑋) and represents its elements as simplicial maps from a suitable triangulation of the d-sphere 𝑆𝑑 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≥2 , we construct a family of simply connected spaces X such that for any simplicial map representing a generator of 𝜋𝑑(𝑋) , the size of the triangulation of 𝑆𝑑 on which the map is defined, is exponential in size(𝑋) .}, author = {Filakovský, Marek and Franek, Peter and Wagner, Uli and Zhechev, Stephan Y}, issn = {2367-1734}, journal = {Journal of Applied and Computational Topology}, number = {3-4}, pages = {177--231}, publisher = {Springer}, title = {{Computing simplicial representatives of homotopy group elements}}, doi = {10.1007/s41468-018-0021-5}, volume = {2}, year = {2018}, } @article{742, abstract = {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 (Formula presented.) there exists a point (Formula presented.) that is contained in the images of a positive fraction (Formula presented.) of the d-cells of X. More generally, the conclusion holds if (Formula presented.) is replaced by any d-dimensional piecewise-linear manifold M, with a constant (Formula presented.) that depends only on d and on the expansion properties of X, but not on M.}, author = {Dotterrer, Dominic and Kaufman, Tali and Wagner, Uli}, journal = {Geometriae Dedicata}, number = {1}, pages = {307–317}, publisher = {Springer}, title = {{On expansion and topological overlap}}, doi = {10.1007/s10711-017-0291-4}, volume = {195}, year = {2018}, }