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Discrete & Computational Geometry

, Volume 54, Issue 1, pp 42–77 | Cite as

Homology of Cellular Structures Allowing Multi-incidence

  • Sylvie Alayrangues
  • Guillaume Damiand
  • Pascal Lienhardt
  • Samuel Peltier
Article
  • 123 Downloads

Abstract

This 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.

Keywords

Homology computation Boundary operator Combinatorial maps 

Mathematics Subject Classification

68U05 

Notes

Acknowledgments

The authors wish to thank the reviewers for their many useful remarks and suggestions and in particular Pol Vanhaecke and Francis Sergeraert for many informative discussions.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Sylvie Alayrangues
    • 1
  • Guillaume Damiand
    • 2
  • Pascal Lienhardt
    • 1
  • Samuel Peltier
    • 1
  1. 1.Université de Poitiers, Laboratoire XLIM, Département SIC, CNRS 7252, Bâtiment SP2MI - Téléport 2Futuroscope-Chasseneuil CedexFrance
  2. 2.Université de Lyon, CNRS, LIRIS, UMR5205VilleurbanneFrance

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