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In this page, a selection of literature more or less strictly related to the subject of computational algebraic topology is gathered.
1. Max K. Agoston. Algebraic Topology, A First Course. Vol. 32 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, Inc., New York and Basel, 1976.
2. Fusun; Ionescu Akman.
A Survey of Huebschmann and Stasheff's Paper: Formal Solution of the Master Equation via HPT and Deformation Theory.
(2007)
[direct link]
3. Madjid Allili, Tomasz Kaczynski.
An algorithmic approach to the construction of homomorphisms induced by maps in homology.
Trans. Amer. Math. Soc. Vol. 352 No. 5 (2000) 2261-2281.
DOI: 10.1090/S0002-9947-99-02527-1
4. Madjid Allili, Konstantin Mischaikow, Allen Tannenbaum.
Cubical homology and the topological classification of 2D and 3D imagery.
[In:] Proceedings of the 2001 International Conference on Image Processing, ICIP 2001, 2001, 173-176, vol. 2.
DOI: 10.1109/ICIP.2001.958452
5. Victor Álvarez, Jose Andres Armario, María Dolores Frau, Pedro Real.
Algebra Structures on the Twisted Eilenberg-Zilber Theorem.
Communications in Algebra. Vol. 35 (2007) 3273-3291.
DOI: 10.1080/00914030701410369
[direct link]
6. Victor Álvarez, Jose Andres Armario, María Dolores Frau, Pedro Real.
A system of equations for describing cocyclic Hadamard matrices.
Journal of Combinatorial Designs. Vol. 16 No. 4 (2008) 276-290.
DOI: 10.1002/jcd.20191
7. Victor Alvarez, Jose Andres Armario, María Dolores Frau, Pedro Real.
The homological reduction method for computing cocyclic Hadamard matrices.
Journal of Symbolic Computation. Vol. 44 No. 5 (2009) 558-570.
DOI: 10.1016/j.jsc.2007.06.009
[direct link]
8. Zin Arai, William Kalies, Hiroshi Kokubu, Konstantin Mischaikow, Hiroe Oka, Paweł Pilarczyk.
A Database Schema for the Analysis of Global Dynamics of Multiparameter Systems.
SIAM Journal on Applied Dynamical Systems. Vol. 8 No. 3 (2009) 757-789.
DOI: 10.1137/080734935
[direct link]
9. A. Berciano, H. Molina-Abril, P. Real.
Searching high order invariants in computer imagery.
Applicable Algebra in Engineering, Communication and Computing. Vol. 23 No. 1-2 (2012) 17-28.
DOI: 10.1007/s00200-012-0169-5
10. Ainhoa Berciano, María Jiménez, Pedro Real.
On the Computation of $A_\infty$-Maps.
[In:] Computer Algebra in Scientific Computing, Lecture Notes in Computer Science, Vol. 4770, 2007, 45-57.
DOI: 10.1007/978-3-540-75187-8_5
11. Ainhoa Berciano, Helena Molina-Abril, Ana Pacheco, Paweł Pilarczyk, Pedro Real.
Decomposing Cavities in Digital Volumes into Products of Cycles.
[In:] Discrete Geometry for Computer Imagery, Lecture Notes in Computer Science, Vol. 5810, 2009, 263-274.
DOI: 10.1007/978-3-642-04397-0_23
12. Gilles Bertrand, Michel Couprie.
On Parallel Thinning Algorithms: Minimal Non-simple Sets, P-simple Points and Critical Kernels.
Journal of Mathematical Imaging and Vision. Vol. 35 (2009) 23-35.
DOI: 10.1007/s10851-009-0152-3
13. Hepzibah A. Christinal, Daniel Díaz-Pernil, Pedro Real Jurado.
Using Membrane Computing for Obtaining Homology Groups of Binary 2D Digital Images.
[In:] Combinatorial Image Analysis, Lecture Notes in Computer Science, Vol. 5852, 2009, 383-396.
DOI: 10.1007/978-3-642-10210-3_30
14. Hepzibah Christinal, Daniel Díaz-Pernil, Pedro Real Jurado.
Segmentation in 2D and 3D Image Using Tissue-Like P System.
[In:] Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications, Lecture Notes in Computer Science, Vol. 5856, 2009, 169-176.
DOI: 10.1007/978-3-642-10268-4_20
15. Cecil Jose A. Delfinado, Herbert Edelsbrunner.
An incremental algorithm for Betti numbers of simplicial complexes.
[In:] Proceedings of the ninth annual symposium on Computational geometry, SCG '93, 1993, 232-239.
DOI: 10.1145/160985.161140
[direct link]
16. Cecil Jose A. Delfinado, Herbert Edelsbrunner.
An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere.
Computer Aided Geometric Design. Vol. 12 No. 7 (1995) 771-784.
DOI: 10.1016/0167-8396(95)00016-Y
17. Mathieu Desbrun, Eva Kanso, Yiying Tong. Discrete Differential Forms for Computational Sciences. [In:] Discrete Differential Geometry, Course Notes, 2006,
18. Jean-Guillaume Dumas, B. David Saunders, Gilles Villard.
On Efficient Sparse Integer Matrix Smith Normal Form Computations.
Journal of Symbolic Computation. Vol. 32 No. 1-2 (2001) 71-99.
DOI: 10.1006/jsco.2001.0451
[direct link]
19. Eduardo Liz, Paweł Pilarczyk.
Global dynamics in a stage-structured discrete-time population model with harvesting.
Journal of Theoretical Biology. Vol. 297 (2012) 148-165.
DOI: 10.1016/j.jtbi.2011.12.012
[direct link]
20. Samuel Eilenberg, Saunders Mac Lane.
On the Groups H(Π,n), I.
The Annals of Mathematics. Vol. 58 (1953) 55-106.
[direct link]
21. Samuel Eilenberg, Saunders Mac Lane.
On the Groups H(Π,n), II: Methods of Computation.
The Annals of Mathematics. Vol. 60 (1954) 49-139.
[direct link]
22. Samuel Eilenberg, Saunders Mac Lane.
On the Groups H(Π,n), III: Operations and Obstructions.
The Annals of Mathematics. Vol. 60 (1954) 513-557.
[direct link]
23. Samuel Eilenberg, J.A. Zilber.
On Products of Complexes.
American Journal of Mathematics. Vol. 75 (1953) 200-204.
[direct link]
24. Robin Forman. A Discrete Morse Theory for Cell Complexes. [In:] Geometry, Topology, and Physics for Raoul Bott (Conference Proceedings and Lecture Notes in Geometry and Topology), 1995, 112-125.
25. Robin Forman.
Morse Theory for Cell Complexes.
Advances in Mathematics. Vol. 134 No. 1 (1998) 90-145.
DOI: 10.1006/aima.1997.1650
[direct link]
26. Robin Forman.
Witten-Morse theory for cell complexes.
Topology. Vol. 37 No. 5 (1998) 945-979.
DOI: 10.1016/S0040-9383(97)00071-2
[direct link]
27. Robin Forman.
Discrete Morse Theory and the Cohomology Ring.
Trans. Amer. Math. Soc. Vol. 354 No. 12 (2002) 5063-5085.
DOI: 10.1090/S0002-9947-02-03041-6
28. Marcio Gameiro, Konstantin Mischaikow, William Kalies. Topological Characterization of Spatial-Temporal Chaos. Physical Review E. Vol. 70 (2004) 035203.
29. Rocío González-Díaz, Adrian Ion, Mabel Iglesias-Ham, Walter Kropatsch.
Irregular Graph Pyramids and Representative Cocycles of Cohomology Generators.
[In:] Graph-Based Representations in Pattern Recognition, Lecture Notes in Computer Science, Vol. 5534, 2009, 263-272.
DOI: 10.1007/978-3-642-02124-4_27
30. Rocío González-Díaz, María J. Jiménez, Belén Medrano, Pedro Real.
A tool for integer homology computation: λ-AT-model.
Image and Vision Computing. Vol. 27 No. 7 (2009) 837-845.
DOI: 10.1016/j.imavis.2008.10.001
[direct link]
31. Rocío González-Díaz, María Jiménez, Belén Medrano, Pedro Real.
Extending the Notion of AT-Model for Integer Homology Computation.
[In:] Graph-Based Representations in Pattern Recognition, Lecture Notes in Computer Science, Vol. 4538, 2007, 330-339.
DOI: 10.1007/978-3-540-72903-7_30
32. Rocío González-Díaz, María Jiménez, Belén Medrano, Pedro Real.
Chain homotopies for object topological representations.
Discrete Applied Mathematics. Vol. 157 No. 3 (2009) 490-499.
DOI: 10.1016/j.dam.2008.05.029
[direct link]
33. Rocío González-Díaz, María Jiménez, Belen Medrano, Pedro Real.
A Graph-with-Loop Structure for a Topological Representation of 3D Objects.
[In:] Computer Analysis of Images and Patterns, Lecture Notes in Computer Science, Vol. 4673, 2007, 506-513.
DOI: 10.1007/978-3-540-74272-2_63
34. Rocío González-Díaz, Maria Jimenez, Belen Medrano, Helena Molina-Abril, Pedro Real.
Integral Operators for Computing Homology Generators at Any Dimension.
[In:] Progress in Pattern Recognition, Image Analysis and Applications, Lecture Notes in Computer Science, Vol. 5197, 2008, 356-363.
DOI: 10.1007/978-3-540-85920-8_44
35. Rocío González-Díaz, Belén Medrano, Javier Sánchez-Peláez, Pedro Real.
Simplicial Perturbation Techniques and Effective Homology.
[In:] Computer Algebra in Scientific Computing, Lecture Notes in Computer Science, Vol. 4194, 2006, 166-177.
DOI: 10.1007/11870814_14
36. Rocío González-Díaz, Belén Medrano, Javier Sánchez-Peláez, Pedro Real.
Reusing Integer Homology Information of Binary Digital Images.
[In:] Discrete Geometry for Computer Imagery, Lecture Notes in Computer Science, Vol. 4245, 2006, 199-210.
DOI: 10.1007/11907350_17
37. Rocío González-Díaz, Belen Medrano, Pedro Real, Javier Sánchez-Peláez.
Algebraic Topological Analysis of Time-Sequence of Digital Images.
[In:] Computer Algebra in Scientific Computing, Lecture Notes in Computer Science, Vol. 3718, 2005, 208-219.
DOI: 10.1007/11555964_18
38. Rocío González-Díaz, Pedro Real.
A combinatorial method for computing Steenrod squares.
Journal of Pure and Applied Algebra. Vol. 139 No. 1-3 (1999) 89-108.
DOI: 10.1016/S0022-4049(99)00006-7
[direct link]
39. Rocío González-Díaz, Pedro Real.
Computation of Cohomology Operations on Finite Simplicial Complexes.
Homology, Homotopy and Applications. Vol. 5 No. 2 (2003) 83-93.
[direct link]
40. Rocío González-Díaz, Pedro Real.
Towards Digital Cohomology.
[In:] Discrete Geometry for Computer Imagery, Lecture Notes in Computer Science, Vol. 2886, 2003, 92-101.
DOI: 10.1007/978-3-540-39966-7_8
41. Rocío González-Díaz, Pedro Real.
HPT and Cocyclic Operations.
Homology, Homotopy and Applications. Vol. 7 No. 2 (2005) 95-108.
[direct link]
42. Rocío González-Díaz, Pedro Real.
Simplification techniques for maps in simplicial topology.
Journal of Symbolic Computation. Vol. 40 No. 4-5 (2005) 1208-1224.
DOI: 10.1016/j.jsc.2004.04.008
[direct link]
43. Rocío González-Díaz, Pedro Real.
On the cohomology of 3D digital images.
Discrete Applied Mathematics. Vol. 147 No. 2-3 (2005) 245-263.
DOI: 10.1016/j.dam.2004.09.014
[direct link]
44. Allen Hatcher.
Algebraic Topology.
Cambridge University Press, Cambridge, U.K., 2002.
[direct link]
45. Johannes Huebschmann, Jim Stasheff.
Formal solution of the master equation via HPT and deformation theory.
Forum Mathematicum. Vol. 14 847-–868.
DOI: 10.1515/form.2002.037
46. P. R. Hurado, V. Álvarez, J. A. Armario, Rocío González-Díaz.
Algorithms in Algebraic Topology and Homological Algebra: Problem of Complexity.
Journal of Mathematical Sciences. Vol. 108 (2002) 1015-1033.
DOI: 10.1023/A:1013544506151
47. Costas S. Iliopoulos.
Worst-Case Complexity Bounds on Algorithms for Computing the Canonical Structure of Finite Abelian Groups and the Hermite and Smith Normal Forms of an Integer Matrix.
SIAM Journal on Computing. Vol. 18 No. 4 (1989) 658-669.
DOI: 10.1137/0218045
[direct link]
48. Gerold Jäger, Clemens Wagner.
Efficient parallelizations of Hermite and Smith normal form algorithms.
Parallel Computing. Vol. 35 No. 6 (2009) 345-357.
DOI: 10.1016/j.parco.2009.01.003
[direct link]
49. Nathan Jacobson. Basic Algebra I. 2nd ed. Dover Publications, Mineola, NY, 2009.
50. Nathan Jacobson. Basic Algebra II. 2nd ed. Dover Publications, Mineola, NY, 2009.
51. María J. Jiménez, Pedro Real.
Rectifications of A∞-Algebras.
Communications in Algebra. Vol. 35 (2007) 2731-2743.
DOI: 10.1080/00927870701353514
[direct link]
52. Justin Bush, Marcio Gameiro, Shaun Harker, Hiroshi Kokubu, Konstantin Mischaikow, Ippei Obayashi, Paweł Pilarczyk.
Combinatorial-topological framework for the analysis of global dynamics.
Chaos: An Interdisciplinary Journal of Nonlinear Science. Vol. 22 No. 4 (2012) 047508.
DOI: 10.1063/1.4767672
[direct link]
53. Tomasz Kaczyński, Marian Mrozek, Maciej Ślusarek.
Homology computation by reduction of chain complexes.
Computers & Mathematics with Applications. Vol. 35 No. 4 (1998) 59-70.
DOI: 10.1016/S0898-1221(97)00289-7
[direct link]
54. Tomasz Kaczynski, Paweł Dłotko, Marian Mrozek. Computing the Cubical Cohomology Ring. [In:] Proceedings of the 3rd International Workshop on Computational Topology in Image Context (CTIC), Image A, Vol. 3, 2010, 137-142.
55. Tomasz Kaczynski, Konstantin Mischaikow, Marian Mrozek. Computational homology. Vol. 157 of Applied Mathematical Sciences. Springer-Verlag, New York, 2004.
56. Ravindran Kannan, Achim Bachem.
Polynomial Algorithms for Computing the Smith and Hermite Normal Forms of an Integer Matrix.
SIAM Journal on Computing. Vol. 8 No. 4 (1979) 499-507.
DOI: 10.1137/0208040
[direct link]
57. K. Krishan, M. Gameiro, Konstantin Mischaikow, M.F. Schatz, H. Kurtuldu, S. Madruga. Homology and symmetry breaking in Rayleigh-Bénard convection: Experiments and simulations. Physics of Fluids. Vol. 19 (2007) 117105.
58. Saunders MacLane. Homology. Classics in Mathematics. Springer-Verlag, Berlin and Heidelberg, 1995.
59. William S. Massey. A Basic Course in Algebraic Topology. Addison-Wesley, Menlo Park, CA, 1991.
60. J. Peter May. Simplicial objects in algebraic topology. The University of Chicago Press, Chicago and London, 1967.
61. Marcin Mazur, Jacek Szybowski. The implementation of the Allili-Kaczyński algorithm for the construction of a chain homomorphism induced by a multivalued map. [In:] Proceedings of the International Conference on Differential Equations, Equadiff '99, 2000, 225-227.
62. Konstantin Mischaikow, Marian Mrozek, Paweł Pilarczyk.
Graph Approach to the Computation of the Homology of Continuous Maps.
Foundations of Computational Mathematics. Vol. 5 (2005) 199-229.
DOI: 10.1007/s10208-004-0125-2
63. Helena Molina-Abril, Pedro Real.
Advanced Homology Computation of Digital Volumes Via Cell Complexes.
[In:] Structural, Syntactic, and Statistical Pattern Recognition, Lecture Notes in Computer Science, Vol. 5342, 2008, 361-371.
DOI: 10.1007/978-3-540-89689-0_40
64. Helena Molina-Abril, Pedro Real.
Homological Computation Using Spanning Trees.
[In:] Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications, Lecture Notes in Computer Science, Vol. 5856, 2009, 272-278.
DOI: 10.1007/978-3-642-10268-4_32
65. Helena Molina-Abril, Pedro Real.
Homological spanning forest framework for 2D image analysis.
Annals of Mathematics and Artificial Intelligence. Vol. 64 No. 4 (2012) 385-409.
DOI: 10.1007/s10472-012-9297-7
66. Marian Mrozek, Paweł Pilarczyk, Natalia Żelazna.
Homology algorithm based on acyclic subspace.
Computers & Mathematics with Applications. Vol. 55 No. 11 (2008) 2395-2412.
DOI: 10.1016/j.camwa.2007.08.044
[direct link]
67. James R. Munkres. Elements of Algebraic Topology. Addison-Wesley, Reading, MA, 1984.
68. M. Niethammer, W.D. Kalies, K. Mischaikow, A. Tannenbaum.
On the detection of simple points in higher dimensions using cubical homology.
Image Processing, IEEE Transactions on. Vol. 15 No. 8 (2006) 2462-2469.
DOI: 10.1109/TIP.2006.877309
69. Marc Niethammer, A.N. Stein, William D. Kalies, Paweł Pilarczyk, Konstantin Mischaikow, Allen Tannenbaum.
Analysis of blood vessel topology by cubical homology.
[In:] Proceedings of the International Conference on Image Processing, ICIP 2002, Vol. 2, 2002, 969-972.
DOI: 10.1109/ICIP.2002.1040114
70. Ana Pacheco, Pedro Real.
Getting Topological Information for a 80-Adjacency Doxel-Based 4\it D Volume through a Polytopal Cell Complex.
[In:] Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications, Lecture Notes in Computer Science, Vol. 5856, 2009, 279-286.
DOI: 10.1007/978-3-642-10268-4_33
71. Paweł Pilarczyk, Luis García, Benjamin A. Carreras, Irene Llerena.
A dynamical model for plasma confinement transitions.
Journal of Physics A: Mathematical and Theoretical. Vol. 45 No. 12 (2012) 125502.
DOI: 10.1088/1751-8113/45/12/125502
[direct link]
72. Paweł Pilarczyk.
Computer assisted method for proving existence of periodic orbits.
Topol. Methods Nonlinear Anal. Vol. 13 No. 2 (1999) 365-377.
[direct link]
73. Paweł Pilarczyk, Kinga Stolot.
Excision-preserving cubical approach to the algorithmic computation of the discrete Conley index.
Topology and its Applications. Vol. 155 No. 10 (2008) 1149-1162.
DOI: 10.1016/j.topol.2008.02.003
[direct link]
74. Pedro Real.
Sur le calcul des groupes d'homotopie.
C.R. Acad. Sci., Paris, Ser-I. Vol. 319 (1994) 475-478.
[direct link]
75. Pedro Real.
On the computability of the Steenrod squares.
Ann. Univ. Ferrara, Nuova Ser., Sez. VII, Sc. Mat. Vol. 42 (1996) 57-63.
DOI: 10.1007/BF02955020
76. Pedro Real.
An algorithm computing homotopy groups.
Mathematics and Computers in Simulation. Vol. 42 No. 4-6 (1996) 461-465.
DOI: 10.1016/S0378-4754(96)00021-3
[direct link]
77. Pedro Real.
Homological Perturbation Theory and Associativity.
Homology, Homotopy and Applications. Vol. 2 No. 5 (2000) 51-88.
[direct link]
78. Pedro Real.
Connectivity Forests for Homological Analysis of Digital Volumes.
[In:] Bio-Inspired Systems: Computational and Ambient Intelligence, Lecture Notes in Computer Science, Vol. 5517, 2009, 415-423.
DOI: 10.1007/978-3-642-02478-8_52
79. Pedro Real, Helena Molina-Abril.
Cell AT-Models for Digital Volumes.
[In:] Graph-Based Representations in Pattern Recognition, Lecture Notes in Computer Science, Vol. 5534, 2009, 314-323.
DOI: 10.1007/978-3-642-02124-4_32
80. Pedro Real, Helena Molina-Abril, Walter Kropatsch.
Homological Tree-Based Strategies for Image Analysis.
[In:] Computer Analysis of Images and Patterns, Lecture Notes in Computer Science, Vol. 5702, 2009, 326-333.
DOI: 10.1007/978-3-642-03767-2_40
81. Francis Sergeraert.
The Computability Problem in Algebraic Topology.
Advances in Mathematics. Vol. 104 No. 1 (1994) 1-29.
DOI: 10.1006/aima.1994.1018
[direct link]
82. Jean-Pierre Serre.
Homologie Singulière des Espaces Fibrés. Applications.
Annals of Math. Vol. 54 No. 3 (1951) 425-505.
[direct link]
83. Weishu Shih.
Homologie des espaces fibrés.
Publ. Math. I.H.E.S. Vol. 13 (1962) 5-87.
[direct link]
84. Henry J. Stephen Smith.
On Systems of Linear Indeterminate Equations and Congruences.
Philos. Trans. Vol. 151 (1861) 293-326.
[direct link]
85. Arne Storjohann.
Near optimal algorithms for computing Smith normal forms of integer matrices.
[In:] Proceedings of the 1996 international symposium on Symbolic and algebraic computation, ISSAC '96, 1996, 267-274.
DOI: 10.1145/236869.237084
[direct link]
86. John Tate.
Homology of Noetherian rings and local rings.
Illinois J. Math. Vol. 1 No. 1 (1957) 14-27.
[direct link]
87. Tomasz Kaczynski, Marian Mrozek.
The Cubical Cohomology Ring: An Algorithmic Approach.
Foundations of Computational Mathematics. (2012)
DOI: 10.1007/s10208-012-9138-4
88. Oswald Veblen. Analysis situs, 2nd ed. Vol. 5, Part 2 of Amer. Math. Soc. Colloq. Publ. Amer. Math. Soc., New York, 1931.
89. Djemel Ziou, Madjid Allili.
Generating cubical complexes from image data and computation of the Euler number.
Pattern Recognition. Vol. 35 No. 12 (2002) 2833-2839.
DOI: 10.1016/S0031-3203(01)00238-2
[direct link]
Note: This page contains a possibly comprehensive list of selected literature related to the subject, with links to digital version of documents whenever available. However, if there are some positions which you would like to suggest for adding to this list, please, contact the webmaster by email, and provide the information, preferably a BibTeX entry, including a DOI and/or a direct link to the publication. Please, note that addresses of websites related to the subject are gathered in the Links page.
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