1,316 research outputs found

    Combinatorial models for topology-based geometric modeling

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    Many combinatorial (topological) models have been proposed in geometric modeling, computational geometry, image processing or analysis, for representing subdivided geometric objects, i.e. partitionned into cells of different dimensions: vertices, edges, faces, volumes, etc. We can distinguish among models according to the type of cells (regular or not regular ones), the type of assembly ("manifold" or "non manifold"), the type of representation (incidence graphs or ordered models), etc

    A Boundary Operator for Computing the Homology of Cellular Structures

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    71 pagesThe paper focuses on homology computation over cellular structures through the computation of incidence numbers. Roughly speaking, if two cells are incident, then their incidence number characterizes how they are attached. Having these numbers naturally leads to the definition of a boundary operator, which induces a cellular homology. More precisely, the two main families of cellular structures (incidence graphs and ordered models) are studied through various models. A boundary operator is then proposed for the most general structure, and is optimized for the other structures. It is proved that, under specific conditions, the cellular boundary operator proposed in this paper defines a cellular homology equivalent to the simplicial one

    Homology of Cellular Structures Allowing Multi-incidence

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    International audienceThis paper focuses on homology computation over ‘cellular’ structures that allow multi-incidence between cells. We deal here with combinatorial maps, more precisely chains of maps and subclasses such as maps and generalized maps. Homology computation on such structures is usually achieved by computing simplicial homology on a simplicial analog. But such an approach is computationally expensive because it requires computing this simplicial analog and performing the homology computation on a structure containing many more cells (simplices) than the initial one. Our work aims at providing a way to compute homologies directly on a cellular structure. This is done through the computation of incidence numbers. Roughly speaking, if two cells are incident, then their incidence number characterizes how they are attached. Having these numbers naturally leads to the definition of a boundary operator, which induces a homology. Hence, we propose a boundary operator for chains of maps and provide optimization for maps and generalized maps. It is proved that, under specific conditions, the homology of a combinatorial map as defined in the paper is equivalent to the homology of its simplicial analogue

    Moment-angle complexes, monomial ideals, and Massey products

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    Associated to every finite simplicial complex K there is a "moment-angle" finite CW-complex, Z_K; if K is a triangulation of a sphere, Z_K is a smooth, compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study the cohomology ring, the homotopy groups, and the triple Massey products of a moment-angle complex, relating these topological invariants to the algebraic combinatorics of the underlying simplicial complex. Applications to the study of non-formal manifolds and subspace arrangements are given.Comment: 30 pages. Published versio

    Density of isoperimetric spectra

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    We show that the set of k-dimensional isoperimetric exponents of finitely presented groups is dense in the interval [1, \infty) for k > 1. Hence there is no higher-dimensional analogue of Gromov's gap (1,2) in the isoperimetric spectrum.Comment: 34 pages, 3 figure

    Incremental Computation of the Homology of Generalized Maps: An Application of Effective Homology Results

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    This paper deals with the incremental computation of the homology of " cellular " combinatorial structures, namely combinatorial maps and incidence graphs. " Incremental " is related to the operations which are applied to construct such structures: basic operations, i.e. the creation of cells and the identification of cells, are considered in the paper. Such incremental computation is done by applying results of effective homology [RS06]: a correspondence between the chain complex associated with a given combinatorial structure is maintained with a " smaller " chain complex , from which the homology groups and homology generators can be more efficiently computed
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