6,123 research outputs found

    Universality theorems for inscribed polytopes and Delaunay triangulations

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    We prove that every primary basic semialgebraic set is homotopy equivalent to the set of inscribed realizations (up to M\"obius transformation) of a polytope. If the semialgebraic set is moreover open, then, in addition, we prove that (up to homotopy) it is a retract of the realization space of some inscribed neighborly (and simplicial) polytope. We also show that all algebraic extensions of Q\mathbb{Q} are needed to coordinatize inscribed polytopes. These statements show that inscribed polytopes exhibit the Mn\"ev universality phenomenon. Via stereographic projections, these theorems have a direct translation to universality theorems for Delaunay subdivisions. In particular, our results imply that the realizability problem for Delaunay triangulations is polynomially equivalent to the existential theory of the reals.Comment: 15 pages, 2 figure

    Topology of random simplicial complexes: a survey

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    This expository article is based on a lecture from the Stanford Symposium on Algebraic Topology: Application and New Directions, held in honor of Gunnar Carlsson, Ralph Cohen, and Ib Madsen.Comment: After revisions, now 21 pages, 5 figure

    Bounding Helly numbers via Betti numbers

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    We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers bb and dd there exists an integer h(b,d)h(b,d) such that the following holds. If F\mathcal F is a finite family of subsets of Rd\mathbb R^d such that β~i(⋂G)≤b\tilde\beta_i\left(\bigcap\mathcal G\right) \le b for any G⊊F\mathcal G \subsetneq \mathcal F and every 0≤i≤⌈d/2⌉−10 \le i \le \lceil d/2 \rceil-1 then F\mathcal F has Helly number at most h(b,d)h(b,d). Here β~i\tilde\beta_i denotes the reduced Z2\mathbb Z_2-Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these ⌈d/2⌉\lceil d/2 \rceil first Betti numbers allow for families with unbounded Helly number. Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex KK, some well-behaved chain map C∗(K)→C∗(Rd)C_*(K) \to C_*(\mathbb R^d).Comment: 29 pages, 8 figure

    Good covers are algorithmically unrecognizable

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    A good cover in R^d is a collection of open contractible sets in R^d such that the intersection of any subcollection is either contractible or empty. Motivated by an analogy with convex sets, intersection patterns of good covers were studied intensively. Our main result is that intersection patterns of good covers are algorithmically unrecognizable. More precisely, the intersection pattern of a good cover can be stored in a simplicial complex called nerve which records which subfamilies of the good cover intersect. A simplicial complex is topologically d-representable if it is isomorphic to the nerve of a good cover in R^d. We prove that it is algorithmically undecidable whether a given simplicial complex is topologically d-representable for any fixed d \geq 5. The result remains also valid if we replace good covers with acyclic covers or with covers by open d-balls. As an auxiliary result we prove that if a simplicial complex is PL embeddable into R^d, then it is topologically d-representable. We also supply this result with showing that if a "sufficiently fine" subdivision of a k-dimensional complex is d-representable and k \leq (2d-3)/3, then the complex is PL embeddable into R^d.Comment: 22 pages, 5 figures; result extended also to acyclic covers in version
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