X. [. Alayrangues, J. Daragon, P. Lachaud, and . Lienhardt, Equivalence between Closed Connected n-G-Maps without??Multi-Incidence and n-Surfaces, Journal of Mathematical Imaging and Vision, vol.14, issue.3, pp.1-22, 2008.
DOI : 10.1007/s10851-008-0084-3
URL : https://hal.archives-ouvertes.fr/hal-00340945

G. [. Alayrangues, P. Damiand, S. Lienhardt, and . Peltier, Homology of Cellular Structures Allowing Multi-incidence, Discrete & Computational Geometry, vol.41, issue.3, 2015.
DOI : 10.1007/s00454-015-9662-5
URL : https://hal.archives-ouvertes.fr/hal-01189215

]. M. Ago76 and . Agoston, Algebraic Topology, a first course. Pure and applied mathematics, 1976.

[. Alayrangues, S. Lienhardt, and . Peltier, Conversion between chains of maps and chains of surfaces; application to the computation of incidence graphs homology, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01130543

]. T. Bas10 and . Basak, Combinatorial cell complexes and poincaré duality, Geometriae Dedicata, vol.147, pp.357-387, 2010.

]. D. Bcma-+-11, D. Boltcheva, S. Canino, J. Merino-aceituno, L. Léon et al., An iterative algorithm for homology computation on simplical shapes New notions for discrete topology, Proc. of 8th Disc. Geom. for Computer Imagery, pp.1457-1467, 1999.

S. [. Boltcheva, J. Merino, F. Léon, and . Hétroy, Constructive Mayer-Vietoris Algorithm: Computing the Homology of Unions of Simplicial Complexes, 2010.
URL : https://hal.archives-ouvertes.fr/inria-00542717

]. N. Bou89 and . Bourbaki, Algebra I: Chapters 1-3. Elements of mathematics, 1989.

. Bpa-+-10-]-t, M. Bellet, A. Poudret, P. L. Arnould, L. Gall et al., Designing a Topological Modeler Kernel: A Rule-Based Approach, Shape Modeling International (SMI'10), 2010.

]. E. Bri93 and . Brisson, Representing geometric structures in d dimensions: Topology and order, Discrete & Computational Geometry, vol.9, pp.387-426, 1993.

S. Brandel, S. Schneider, M. Perrin, N. Guiard, J. Rainaud et al., Automatic Building of Structured Geological Models, Journal of Computing and Information Science in Engineering, vol.5, issue.2, 2005.
DOI : 10.1115/1.1884145

M. [. Daragon, G. Couprie, and . Bertrand, Discrete Surfaces and Frontier Orders, Journal of Mathematical Imaging and Vision, vol.147, issue.2???3, pp.379-399, 2005.
DOI : 10.1007/s10851-005-2029-4
URL : https://hal.archives-ouvertes.fr/hal-00622397

G. [. Dupas and . Damiand, Region merging with topological control, Discrete Applied Mathematics, vol.157, issue.16, pp.3435-3446, 2009.
DOI : 10.1016/j.dam.2009.04.005
URL : https://hal.archives-ouvertes.fr/hal-00422688

[. Dumas, F. Heckenbach, B. D. Saunders, and V. Welker, Computing Simplicial Homology Based on Efficient Smith Normal Form Algorithms, Algebra, Geometry, and Software Systems, pp.177-206, 2003.
DOI : 10.1007/978-3-662-05148-1_10

T. [. Dlotko, M. Kaczynski, T. Mrozek, and . Wanner, Coreduction Homology Algorithm for Regular CW-Complexes, Discrete & Computational Geometry, vol.58, issue.1, pp.361-388, 2011.
DOI : 10.1007/s00454-010-9303-y

P. [. Damiand and . Lienhardt, Removal and Contraction for n-Dimensional Generalized Maps, Proc. of 11th International Conference on Discrete Geometry for Computer Imagery (DGCI) REFERENCES [DL14] G. Damiand and P. Lienhardt. Combinatorial Maps: Efficient Data Structures for Computer Graphics and Image Processing. A K Peters, pp.408-419, 2003.
DOI : 10.1007/978-3-540-39966-7_39

B. [. Dumas, G. Saunders, and . Villard, On Efficient Sparse Integer Matrix Smith Normal Form Computations, Journal of Symbolic Computation, vol.32, issue.1-2, 2001.
DOI : 10.1006/jsco.2001.0451

H. Edelsbrunner and J. Harer, Computational Topology an Introduction, 2010.

P. [. Elter and . Lienhardt, CELLULAR COMPLEXES AS STRUCTURED SEMI-SIMPLICIAL SETS, International Journal of Shape Modeling, vol.01, issue.02, pp.191-217, 1995.
DOI : 10.1142/S021865439400013X

M. [. González-díaz, B. Jiménez, and P. Medrano, Chain homotopies for object topological representations, Discrete Applied Mathematics, vol.157, issue.3, pp.490-499, 2009.
DOI : 10.1016/j.dam.2008.05.029

]. M. Gie96 and . Giesbrecht, Probabilistic computation of the smith normal form of a sparse integer matrix, Proceedings of the Second Int. Symp. on Algorithmic Number Theory, pp.173-186, 1996.

]. P. Lie94 and . Lienhardt, N-dimensional generalized combinatorial maps and cellular quasi-manifolds, Int. J. on Comput. Geom. & App, vol.4, issue.3, pp.275-324, 1994.

P. [. Lang and . Lienhardt, Geometric modeling with simplicial sets, Computer Graphics and Applications, Pacific Graphics, pp.475-494, 1995.

]. J. May67 and . May, Simplicial Objects in Algebraic Topology, 1967.

B. [. Mrozek and . Batko, Coreduction Homology Algorithm, Discrete & Computational Geometry, vol.33, issue.1, pp.96-118, 2009.
DOI : 10.1007/s00454-008-9073-y

S. [. Peltier, L. Alayrangues, J. Fuchs, and . Lachaud, Computation of homology groups and generators, Computers & Graphics, vol.30, issue.1, pp.62-69, 2006.
DOI : 10.1016/j.cag.2005.10.011
URL : https://hal.archives-ouvertes.fr/hal-00308012

L. [. Peltier, P. Fuchs, ]. J. Lienhardt, F. Rubio, and . Sergeraert, Simploidals sets: Definitions , operations and comparison with simplicial sets. Discrete App Constructive homological algebra and applications, Math. Genova Summer School on Mathematics, vol.157, pp.542-557, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00366069

F. [. Romero and . Sergeraert, Discrete vector fields and fundamental algebraic topology. CoRR, abs/1005, 2010.

]. C. Sddj15, G. Solnon, C. Damiand, J. De-la-higuera, and . Janodet, On the complexity of submap isomorphism and maximum common submap problems, Pattern Recognition (PR), vol.48, issue.2, pp.302-316, 2015.

]. A. Sto96 and . Storjohann, Near optimal algorithms for computing smith normal forms of integer matrices REFERENCES The properties of any reduction (C, C , f, g, h) can be explained on an elementary reduction example, for instance, Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation, pp.267-274, 1996.

@. ??-=, ?h? 0 ? 0 h? = ? + ? ? id + ?h? 0 ? 0 h? ? ?f = f ? ?? 0 (hf ) = f ? g? = g ? (gh)? 0 ? = g ? h? = ?h = h ? g ? f ? = g??f = g(?f ) + (g?)f ? (gf )

@. and ?. S1, is such that: ? C S1 contains only 0?dimentional generators, each one corresponding to a complete simplex of S 1