Reeb spaces of piecewise linear mappings

ABSTRACT

Generalizing the concept of a Reeb graph, the Reeb space of a multivariate continuous mapping identifies points of the domain that belong to a common component of the preimage of a point in the range. We study the local and global structure of this space for generic, piecewise linear mappings on a combinatorial manifold.

REFERENCES

• 1: Marc van Kreveld , René van Oostrum , Chandrajit Bajaj , Valerio Pascucci , Dan Schikore, Contour trees and small seed sets for isosurface traversal, Proceedings of the thirteenth annual symposium on Computational geometry, p.212-220, June 04-06, 1997, Nice, France [doi>10.1145/262839.269238]
• 2: O. BURLET AND G. DE RHAM. Sur certaines applications génériques d'une variété close à 3 dimensions dans le plan. L'Enseign. Math. 20 (1974), 275--292.
• 3: Hamish Carr , Jack Snoeyink , Michiel van de Panne, Simplifying Flexible Isosurfaces Using Local Geometric Measures, Proceedings of the conference on Visualization '04, p.497-504, October 10-15, 2004 [doi>10.1109/VIS.2004.96]
• 4: Herbert Edelsbrunner, Algorithms in combinatorial geometry, Springer-Verlag New York, Inc., New York, NY, 1987
• 5: H. EDELSBRUNNER AND J. HARER. Jacobi sets of multiple Morse functions. In F. Cucker, R. Devore and P. Olver (eds.), Foundations of Computational Mathematics, Minneapolis 2002, 35--57, Cambridge University Press, 2004.
• 6: Herbert Edelsbrunner , John Harer , Ajith Mascarenhas , Valerio Pascucci, Time-varying reeb graphs for continuous space-time data, Proceedings of the twentieth annual symposium on Computational geometry, June 08-11, 2004, Brooklyn, New York, USA [doi>10.1145/997817.997872]
• 7: H. EDELSBRUNNER, D. LETSCHER AND A. ZOMORODIAN. Topological persistence and simplification. Discrete Comput. Geom. 28 (2002), 511--533.
• 8: R. D. EDWARDS. Approximating certain cell-like maps by homeomorphisms. Notices Amer. Math. Soc. 24 (1977), A649.
• 9: Y. K. S. FURUYA. Sobre applicações genéricas M4 -- R2. Tese, Departamento de Matemática, ICMSC-USP, São Carlos, Brazil, 1986.
• 10: M. GOLUBITSKY AND V. GUILLEMIN. Stable Mappings and Their Singularities. Springer-Verlag, New York, 1973.
• 11: M. KOBAYASHI AND O. SAEKI. Simplifying stable mappings into the plane from a global viewpoint. Trans. Amer. Math. Soc. 348 (1996), 2607--2636.
• 12: L. KUSHNER, H. LEVINE AND J. P. PORTO. Mapping three-manifolds into the plane I. Bol. Soc. Mat. Mexicana 29 (1984), 11--33.
• 13: H. LEVINE. Classifying Immersions into R4 over Stable Maps of 3-Manifolds in R2. Springer-Verlag, Berlin, Germany, Lecture Notes in Mathematics 1157, 1985.
• 14: A. A. MARKOV. Insolubility of the problem of homeomorphy. In Proc. Int. Congr. Math., 1958, 14--21.
• 15: J. R. MUNKRES. Topology. Second edition, Prentice Hall, Englewood Cliffs, New Jersey, 2000.
• 16: J. R. MUNKRES. Elements of Algebraic Topology. Addison-Wesley, Redwood City, California, 1984.
• 17: J. P. PORTO AND Y. K. S. FURUYA. On special generic maps from a closed manifold into the plane. Topology Appl. 35 (1990), 41--52.
• 18: G. REEB. Sur les points singuliers d'une forme de Pfaff complèment intégrable ou d'une fonction numérique. Comptes Rendus de L'Académie ses Séances, Paris 222 (1946), 847--849.
• 19: D. P. SULLIVAN. On the Hauptvermutung for manifolds. Bull. Amer. Math. Soc. 73 (1967), 598--600.

INDEX TERMS

Primary Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Computations on discrete structures

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Computations on discrete structures; Geometrical problems and computations

General Terms:
Algorithms, Theory

Keywords:
algorithms, combinatorial manifolds, cone neighborhoods, reeb spaces, smooth and pl topology, stratifications, triangulations