August  2011, 30(3): 623-639. doi: 10.3934/dcds.2011.30.623

Discrete gradient fields on infinite complexes

1. 

Dpto. de Geometría y Topología, Universidad de Sevilla, 41080, Sevilla, Spain

2. 

Dpto. de Geometrĺa y Topologĺa, Universidad de Sevilla, 41080, Sevilla, Spain

3. 

The Faculty of Computer and Information Science, University of Ljubljana, Tržaška 25, 1000 Ljubljana, Slovenia, Slovenia

Received  December 2009 Revised  November 2010 Published  March 2011

The aim of this work is to characterize the discrete gradient vector fields on infinite and locally finite simplicial complexes which are induced by a proper discrete Morse function. This characterization is essentially given by the non-existence of closed trajectories and the absence of a certain kind of incidence between monotonous rays in the given field.
Citation: Rafael Ayala, Jose Antonio Vilches, Gregor Jerše, Neža Mramor Kosta. Discrete gradient fields on infinite complexes. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 623-639. doi: 10.3934/dcds.2011.30.623
References:
[1]

R. Ayala, L. M. Fernández, D. Fernández-Ternero and J. A. Vilches, Discrete morse theory on graphs, Topology and its Applications, 156 (2009), 3091-3100. doi: 10.1016/j.topol.2009.01.022.

[2]

R. Ayala, L. M. Fernández and J. A. Vilches, Morse inequalities on certain infinite 2-complexes, Glasgow Mathematical Journal, 49 (2007), 155-165. doi: 10.1017/S0017089507003643.

[3]

Anders Björner and Michelle L. Wachs, Shellable nonpure complexes and posets. II, Trans. Amer. Math. Soc., 349 (1997), 3945-3975. doi: 10.1090/S0002-9947-97-01838-2.

[4]

Manoj K. Chari, On discrete morse functions and combinatorial decompositions, Discrete Mathematics, 217 (2000), 101-113. doi: 10.1016/S0012-365X(99)00258-7.

[5]

Robin Forman, Combinatorial vector fields and dynamical systems, Mathematische Zeitschrift, 228 (1998), 629-681. doi: 10.1007/PL00004638.

[6]

Robin Forman, Morse theory for cell complexes, Advances in Mathematics, 184 (1998), 90-145. doi: 10.1006/aima.1997.1650.

[7]

Etienne Gallais, Combinatorial realization of the thom-smale complex via discrete morse theory, (2008), preprint, arXiv:0803.2616v1.

[8]

J. F. P. Hudson, "Piecewise Linear Topology," University of Chicago, Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees, W. A. Benjamin, Inc., New York-Amsterdam, 1969.

[9]

Gregor Jerse and Neza Mramor Kosta, Ascending and descending regions of a discrete morse function, Computational Geometry, 42 (2009), 639-651. doi: 10.1016/j.comgeo.2008.11.001.

[10]

Jakob Jonsson, "Simplicial Complexes of Graphs," Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-75859-4.

[11]

Michael Joswig and Marc E. Pfetsch, Computing optimal morse matchings, SIAM Journal on Discrete Mathematics, 20 (2006), 11-25. doi: 10.1137/S0895480104445885.

[12]

Thomas Lewiner, Helio Lopes and Geovan Tavares, Visualizing forman's discrete vector field, in "Mathematical Visualization III," Polthier (Eds.), Springer, Berlin, (2003), 95-112.

[13]

John Milnor, "Morse Theory," Princeton University Press, New Jersey, USA, 1963.

[14]

James R. Munkres, "Elements of Algebraic Topology," Addison-Wesley Publishing Company, Menlo Park, CA, 1984.

[15]

A. Quintero R. Ayala, L. M. Fernández and J. A. Vilches, A note on the pure Morse complex of a graph, Topology Appl., 155 (2008), 2084-2089. doi: 10.1016/j.topol.2007.04.023.

[16]

Stephen Smale, On dynamical systems, Bol. Soc. Mat. Mexicana (2), 5 (1960), 195-198.

[17]

M. van Gemmeren, Total absolute curvature and tightness of noncompact manifolds, Transactions of the American Mathematical Society, 348 (1996), 2413-2426. doi: 10.1090/S0002-9947-96-01632-7.

show all references

References:
[1]

R. Ayala, L. M. Fernández, D. Fernández-Ternero and J. A. Vilches, Discrete morse theory on graphs, Topology and its Applications, 156 (2009), 3091-3100. doi: 10.1016/j.topol.2009.01.022.

[2]

R. Ayala, L. M. Fernández and J. A. Vilches, Morse inequalities on certain infinite 2-complexes, Glasgow Mathematical Journal, 49 (2007), 155-165. doi: 10.1017/S0017089507003643.

[3]

Anders Björner and Michelle L. Wachs, Shellable nonpure complexes and posets. II, Trans. Amer. Math. Soc., 349 (1997), 3945-3975. doi: 10.1090/S0002-9947-97-01838-2.

[4]

Manoj K. Chari, On discrete morse functions and combinatorial decompositions, Discrete Mathematics, 217 (2000), 101-113. doi: 10.1016/S0012-365X(99)00258-7.

[5]

Robin Forman, Combinatorial vector fields and dynamical systems, Mathematische Zeitschrift, 228 (1998), 629-681. doi: 10.1007/PL00004638.

[6]

Robin Forman, Morse theory for cell complexes, Advances in Mathematics, 184 (1998), 90-145. doi: 10.1006/aima.1997.1650.

[7]

Etienne Gallais, Combinatorial realization of the thom-smale complex via discrete morse theory, (2008), preprint, arXiv:0803.2616v1.

[8]

J. F. P. Hudson, "Piecewise Linear Topology," University of Chicago, Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees, W. A. Benjamin, Inc., New York-Amsterdam, 1969.

[9]

Gregor Jerse and Neza Mramor Kosta, Ascending and descending regions of a discrete morse function, Computational Geometry, 42 (2009), 639-651. doi: 10.1016/j.comgeo.2008.11.001.

[10]

Jakob Jonsson, "Simplicial Complexes of Graphs," Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-75859-4.

[11]

Michael Joswig and Marc E. Pfetsch, Computing optimal morse matchings, SIAM Journal on Discrete Mathematics, 20 (2006), 11-25. doi: 10.1137/S0895480104445885.

[12]

Thomas Lewiner, Helio Lopes and Geovan Tavares, Visualizing forman's discrete vector field, in "Mathematical Visualization III," Polthier (Eds.), Springer, Berlin, (2003), 95-112.

[13]

John Milnor, "Morse Theory," Princeton University Press, New Jersey, USA, 1963.

[14]

James R. Munkres, "Elements of Algebraic Topology," Addison-Wesley Publishing Company, Menlo Park, CA, 1984.

[15]

A. Quintero R. Ayala, L. M. Fernández and J. A. Vilches, A note on the pure Morse complex of a graph, Topology Appl., 155 (2008), 2084-2089. doi: 10.1016/j.topol.2007.04.023.

[16]

Stephen Smale, On dynamical systems, Bol. Soc. Mat. Mexicana (2), 5 (1960), 195-198.

[17]

M. van Gemmeren, Total absolute curvature and tightness of noncompact manifolds, Transactions of the American Mathematical Society, 348 (1996), 2413-2426. doi: 10.1090/S0002-9947-96-01632-7.

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