Eulerian 2-Strata Spaces

Chen, Beifang; Yan, Min

doi:10.1006/jcta.1998.2891

Journal of Combinatorial Theory, Series A

1999

DOI: 10.1006/jcta.1998.2891

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Eulerian 2-Strata Spaces

Beifang Chen

Min Yan

Abstract: In (B. F. Chen and M. Yan, 1998, Eulerian stratification of polyhedra, Adv. in Appl. Math. 21, 22 57), we introduced the notion of Eulerian stratified spaces. In this paper, we study in detail the topological and combinatorial properties of Eulerian 2-strata spaces, as the first step toward a deeper understanding of more strata cases. We show that Eulerian 2-strata spaces have similar topological structure as topological 2-strata spaces. Moreover, we find all the linear conditions on f-vectors over arbitrary a… Show more

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“…Following Klee's work, the authors 7,8 showed that the notion of Eulerian manifolds is actually independent of triangulations. Furthermore, Ž we studied the more general two-strata spaces including manifolds with .…”

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Eulerian Stratification of Polyhedra

Chen

Yan

1998

Advances in Applied Mathematics

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In this paper we introduce Eulerian stratifications, weight functions, and boundary weight functions to study linear conditions on f-vectors of stratified triangulations on arbitrary polyhedra. The Eulerian stratified spaces are characterized by the Euler characteristics of the links between strata. With the new concepts, the classical Dehn᎐Sommerville equations are generalized to weighted f-vectors of arbitrary polyhedra; and the linear conditions on weighted f-vectors of all stratified triangulations are classified. Moreover, the necessary and sufficient conditions are obtained for given numbers to be the Euler characteristics of the links between strata, and a procedure of constructing Eulerian stratifications from the given numbers on posets is provided.Our study of Eulerian stratifications and weight functions suggests that the underlying combinatorial information should become a rather general setup that includes many classical linear combinatorial theories. It also points out a possible approach toward the study on f-vectors of triangulations of more general spaces.

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Eulerian Stratification of Polyhedra

Chen

Yan

1998

Advances in Applied Mathematics

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The Angle Defect for Odd-Dimensional Simplicial Manifolds

Bloch¹

2005

Discrete Comput Geom

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Abstract. In a 1967 paper, Banchoff stated that a certain type of polyhedral curvature, that applies to all finite polyhedra, was zero at all vertices of an odd-dimensional polyhedral manifold; one then obtains an elementary proof that odd-dimensional manifolds have zero Euler characteristic. In a previous paper, the author defined a different approach to curvature for arbitrary simplicial complexes, based upon a direct generalization of the angle defect. The generalized angle defect is not zero at the simplices of every odd-dimensional manifold. In this paper we use a sequence based upon the Bernoulli numbers to define a variant of the angle defect for finite simplicial complexes that still satisfies a Gauss-Bonnet type theorem, but is also zero at any simplex of an odddimensional simplicial complex K (of dimension at least 3), such that χ(link(η i , K)) = 2 for all i-simplices η i of K, where i is an even integer such that 0 ≤ i ≤ n − 1. As a corollary, an elementary proof is given that any such simplicial complex has Euler characteristic zero.

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On semi-Eulerian partially ordered sets with boundary

Chen

Lau

2003

European Journal of Combinatorics

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Eulerian 2-Strata Spaces

Cited by 3 publications

References 32 publications

Eulerian Stratification of Polyhedra

Eulerian Stratification of Polyhedra

The Angle Defect for Odd-Dimensional Simplicial Manifolds

On semi-Eulerian partially ordered sets with boundary

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