233 research outputs found

    Computing Invariants of Simplicial Manifolds

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    This is a survey of known algorithms in algebraic topology with a focus on finite simplicial complexes and, in particular, simplicial manifolds. Wherever possible an elementary approach is chosen. This way the text may also serve as a condensed but very basic introduction to the algebraic topology of simplicial manifolds. This text will appear as a chapter in the forthcoming book "Triangulated Manifolds with Few Vertices" by Frank H. Lutz.Comment: 13 pages, 3 figure

    Recent progress on the combinatorial diameter of polytopes and simplicial complexes

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    The Hirsch conjecture, posed in 1957, stated that the graph of a dd-dimensional polytope or polyhedron with nn facets cannot have diameter greater than ndn - d. The conjecture itself has been disproved, but what we know about the underlying question is quite scarce. Most notably, no polynomial upper bound is known for the diameters that were conjectured to be linear. In contrast, no polyhedron violating the conjecture by more than 25% is known. This paper reviews several recent attempts and progress on the question. Some work in the world of polyhedra or (more often) bounded polytopes, but some try to shed light on the question by generalizing it to simplicial complexes. In particular, we include here our recent and previously unpublished proof that the maximum diameter of arbitrary simplicial complexes is in nTheta(d)n^{Theta(d)} and we summarize the main ideas in the polymath 3 project, a web-based collective effort trying to prove an upper bound of type nd for the diameters of polyhedra and of more general objects (including, e. g., simplicial manifolds).Comment: 34 pages. This paper supersedes one cited as "On the maximum diameter of simplicial complexes and abstractions of them, in preparation

    A Discrete Multivariate Mean Value Theorem with Applications

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    AMS classifications: 47H10; 54H25; 55M20; 90C33; 91B50Discrete set;mean value theorem;fixed point;algorithm;equilibrium;complementarity
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