On topological degree and Poincaré duality

Franco Cardin

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1995)

  • Volume: 6, Issue: 1, page 73-78
  • ISSN: 1120-6330

Abstract

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In this Note we investigate about some relations between Poincaré dual and other topological objects, such as intersection index, topological degree, and Maslov index of Lagrangian submanifolds. A simple proof of the Poincaré-Hopf theorem is recalled. The Lagrangian submanifolds are the geometrical, multi-valued, solutions of physical problems of evolution governed by Hamilton-Jacobi equations: the computation of the algebraic number of the branches is showed to be performed by using Poincaré dual.

How to cite

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Cardin, Franco. "On topological degree and Poincaré duality." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 6.1 (1995): 73-78. <http://eudml.org/doc/244072>.

@article{Cardin1995,
abstract = {In this Note we investigate about some relations between Poincaré dual and other topological objects, such as intersection index, topological degree, and Maslov index of Lagrangian submanifolds. A simple proof of the Poincaré-Hopf theorem is recalled. The Lagrangian submanifolds are the geometrical, multi-valued, solutions of physical problems of evolution governed by Hamilton-Jacobi equations: the computation of the algebraic number of the branches is showed to be performed by using Poincaré dual.},
author = {Cardin, Franco},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Topological degree; Poincaré duality; Maslov index; Lagrangian manifolds; Solutions of Hamilton-Jacobi equation; Poincaré-Hopf theorem; intersection index; topological degree},
language = {eng},
month = {3},
number = {1},
pages = {73-78},
publisher = {Accademia Nazionale dei Lincei},
title = {On topological degree and Poincaré duality},
url = {http://eudml.org/doc/244072},
volume = {6},
year = {1995},
}

TY - JOUR
AU - Cardin, Franco
TI - On topological degree and Poincaré duality
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1995/3//
PB - Accademia Nazionale dei Lincei
VL - 6
IS - 1
SP - 73
EP - 78
AB - In this Note we investigate about some relations between Poincaré dual and other topological objects, such as intersection index, topological degree, and Maslov index of Lagrangian submanifolds. A simple proof of the Poincaré-Hopf theorem is recalled. The Lagrangian submanifolds are the geometrical, multi-valued, solutions of physical problems of evolution governed by Hamilton-Jacobi equations: the computation of the algebraic number of the branches is showed to be performed by using Poincaré dual.
LA - eng
KW - Topological degree; Poincaré duality; Maslov index; Lagrangian manifolds; Solutions of Hamilton-Jacobi equation; Poincaré-Hopf theorem; intersection index; topological degree
UR - http://eudml.org/doc/244072
ER -

References

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  1. ARNOLD, V., Mathematical Methods of Classical Mechanics. Springer, Berlin-New York1978. Zbl0386.70001MR690288
  2. BORISOVICH, YU. - BLIZNYAKOV, N. - IZRAILEVICH, YA. - FOMENKO, T., Introduction to Topology. Mir Publishers, Moskow1985. Zbl0478.57001MR824983
  3. BOTT, R. - TU, L. W., Differential Forms in Algebraic Topology. Springer GTM 82, Berlin-New York1982. Zbl0496.55001MR658304
  4. CARDIN, F., On the geometrical Cauchy problem for the Hamilton-Jacobi equation. Il Nuovo Cimento, vol. 104, sect. B, 1989, 525-544. MR1043810DOI10.1007/BF02726162
  5. CARDIN, F., On viscosity and geometrical solutions of Hamilton-Jacobi equations. Nonlinear Analysis, T.M.A., vol. 20, n. 6, 1993, 713-719. Zbl0771.35069MR1214737DOI10.1016/0362-546X(93)90029-R
  6. CARDIN, F. - SPERA, M., Some topological properties of generalized hyperelastic materials. Preprint Dipartimento di Matematica Pura e Applicata, Università di Padova, 1994. 
  7. DUBROVIN, B. A. - FOMENKO, A. T. - NOVIKOV, S. P., Modern Geometry. Methods and Applications, Springer GTM, 3 vols., Berlin-New York1985. Zbl0751.53001MR822729
  8. GUELLEMIN, V. - STERNBERG, S., Geometrie Asymptotics. American Mathematical Society, Providence, Rhode Island1977. Zbl0364.53011
  9. HIRSCH, M. W., Differential Topology. Springer GTM 33, Berlin-New York1976. Zbl0356.57001MR448362
  10. LIBERMANN, P. - MARLE, C. M., Symplectic Geometry and Analytical Mechanics. D. Reidel Publishing Co., 1987. Zbl0643.53002MR882548DOI10.1007/978-94-009-3807-6
  11. MILNOR, J. W., Topology from the Differential Viewpoint. The University Press of Virginia, Charlottesville1965. Zbl0136.20402MR226651
  12. WEINSTEIN, A., Lectures on Symplectic Manifolds. CBMS Conf. Series, AMS29, 1977. Zbl0406.53031MR464312

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