Abstract
We recall that the calculation of homology with integer coefficients of a simplicial complex reduces to the calculation of the Smith Normal Form of the boundary matrices which in general are sparse. We provide a review of several algorithms for the calculation of Smith Normal Form of sparse matrices and compare their running times for actual boundary matrices. Then we describe alternative approaches to the calculation of simplicial homology. The last section then describes motivating examples and actual experiments with the GAP package that was implemented by the authors. These examples also include as an example of other homology theories some calculations of Lie algebra homology.
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References
Eric Babson, Anders Björner, Svante Linusson, John Shareshian and Volkmar Welker Complexes of not i-connected graphs. Topology 38 (1999) 271–299.
Achim Bachem and Ravindran Kannan. Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix. SIAM J. Comput. 8 499–507 (1979).
William W. Boone. Certain simple unsolvable problems in group theory. Indig. Math. 16 231–237 (1955).
Alfred Brauer. Limits for the characteristic roots of a matrix. I. Duke Math. J. 13 387–395 (1946).
Alfred Brauer. Limits for the characteristic roots of a matrix. II. Duke Math. J. 14 21–26 (1947).
Richard A. Brualdi and Stephen Mellendorf. Regions in the complex plane containing the eigenvalues of a matrix. American Mathematical Monthly 101 975–985 (1994).
Antonio Capani, Gianfranco Niesi and Lorenzo Robbiano, CoCoA, a system for doing Computations in Commutative Algebra. Available via anonymous ftp from: cocoa.dima.unige.it
David Eisenbud. Commutative Algebra with a View Toward Algebraic Geometry,Springer, 1993.
Jean-Guillaume Dumas, Frank Heckenbach, B. David Saunders and Volkmar Welker. Simplicial Homology, a (proposed) share package for GAP, March 2000. Manual(http://www.cis.udel.edu/~dumas/Homology).
Jean-Guillaume Dumas, B. David Saunders and Gilles Villard. On efficient sparse integer matrix Smith normal form computations. J. Symb. Comp. 32 71–99 (2001).
Pierre Dusart. Autour de la fonction qui compte le nombre de nombres premiers. PhD thesis, Université de Limoges, 1998.
William G. Dwyer. Homology of integral upper-triangular matrices. Proc. Amer. Math. Soc. 94 523–528 (1985).
Wayne Eberly and Erich Kaltofen. On randomized Lanczos algorithms. In Wolfgang W. Küchlin, editor, Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, Maui, Hawaii, pages 176–183. ACM Press, New York, July 1997. Computing Simplicial Homology 205
Michael L. Fredman and Leonid Khachiyan. On the complexity of dualization of monotone disjunctive normal forms, J. Algorithms 21 (1996) 618–628.
The GAP Group, GAP Groups, Algorithms, and Programming, Version 4.2; 2000. (http://www.gap-system.org)
Joachim von zur Gathen and Jürgen Gerhard. Modern Computer Algebra. Cambridge University Press, New York, NY, USA, 1999.
Mark W. Giesbrecht. Probabilistic Computation of the Smith Normal Form of a Sparse Integer Matrix. Lecture Notes in Comp. Sci. 1122, 173–186. Springer, 1996.
Gene H. Golub and Charles F. Van Loan. Matrix computations. Johns Hopkins Studies in the Mathematical Sciences. The Johns Hopkins University Press, Baltimore, MD, USA, third edition, 1996.
Daniel R. Grayson and Michael E. Stillman, Macaulay 2, a software system for research in algebraic geometry,Available at http://www.math.uiuc.edu/ Macaulay2/
Melvin Hochster. Cohen-Macaulay rings, combinatorics, and simplicial complexes. Ring Theory II, Proc. 2nd Okla. Conf. 1975. Lecture Notes in Pure and Applied Mathematics. Vol. 26. New York. Marcel Dekker. 171–223 (1977).
Phil Hanlon. A survey of combinatorial problems in Lie algebra homology. Billera, Louis J. (ed.) et al., Formal power series and algebraic combinatorics. Providence, RI: American Mathematical Society. DIMACS, Ser. Discrete Math. Theor. Comput. Sci. 24, 89–113, 1996.
Iztok Hozo, Inclusion of poset homology into Lie algebra homology. J. Pure Appl. Algebra 111 169–180 (1996).
Costas S. Iliopoulos. Worst case bounds an algorithms for computing the canonical structure of finite Abelian groups and the Hermite and Smith normal forms of an integer matrix. SIAM J. Comput. 18 (1989) 658–669.
Costas S. Iliopoulos. Worst case bounds an algorithms for computing the canonical structure of infinite Abelian groups and solving systems of linear diophantine equations. SIAM J. Comput. 18 (1989) 670–678.
Gil Kalai. Algebraic Shfiting. Computational Commutative Algebra and Combinatorics Advanced Studies in Pure Math., Vol. 33. Tokyo. Math. Soc. of Japan. 121–165 (2002).
Erich Kaltofen, Wen-Shin Lee and Austin A. Lobo. Early termination in BenOr/Tiwari sparse interpolation and a hybrid of Zippel’s algorithm. In Carlo Traverso, editor, Proceedings of the 2000 International Symposium on Symbolic and Algebraic Computation, Saint Andrews, Scotland, pages 192–201. ACM Press, New York, 2000.
Volker Kaibel and Marc E. Pfetsch. Some algorithmic problems in polytope theory, this volume, pages 23–47.
Dimitris Kavvadias and Elias Stavroploulos. Evaluation of an algorithm for the transversal hypergraph problem. in: Proc. 3rd Workshop on Algorithm Engineering (WAE’99), Lecture Notes in Comp. Sci. 1668, Springer, 1999.
Anthony W. Knapp. Lie Groups, Lie Algebras, and Cohomology, Mathematical Notes 34, Princeton University Press, 1988.
Bertram Kostant. Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. Math. (2) 74 (1961) 329–387.
Robert H. Lewis. Fermat, a computer algebra system for polynomial and matrix computations,1997. http://www.bway.net/~lewis.
James R. Munkres, Elements of Algebraic Topology, Addison-Wesley, Menlo Park, 1984.
James R. Munkres, Topology, 2nd Edition, Prentice Hall, 2000.
Sergey P. Novikov. On the algorithmic unsolvability of the word problem in group theory, Trudy Mat. Inst 44 (1955) 143.
Sarah Rees and Leonard H. Soicher An algorithmic approach to fundamental groups and covers of combinatorial cell complexes J. of Symb. Comp. 29 (2000) 59–77.
Ernest Sibert, Harold F. Mattson and Paul Jackson. Finite Field Arithmetic Using the Connection Machine. In Richard Zippel, editor, Proceedings of the second International Workshop on Parallel Algebraic Computation, Ithaca, USA, volume 584 of Lecture Notes in Computer Science, 51–61. Springer, 1990.
Arne Storjohann, Near Optimal Algorithms for Computing Smith Normal Forms of Integer Matrices, Lakshman, Y. N. (ed.), Proceedings of the 1996 international symposium, on symbolic and algebraic computation, ISSAC 96, New York, NY: ACM Press. 267–274 (1996).
Olga Taussky. Bounds for characteristic roots of matrices. Duke Math. J. 15 1043–1044 (1948).
Vladimir Tuchin. Homologies of complexes of doubly connected graphs. Russian Math. Surveys 52 426–427 (1997).
Richard S. Varga. Matrix iterative analysis. Number 27 in Springer series in Computational Mathematics. Springer, second edition, 2000.
Michelle Wachs. Topology of matching, chessboard and general bounded degree graph complexes. Preprint 2001.
Douglas H. Wiedemann. Solving sparse linear equations over finite fields. IEEE Trans. Inf. Theory 32 54–62 (1986).
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Dumas, JG., Heckenbach, F., Saunders, D., Welker, V. (2003). Computing Simplicial Homology Based on Efficient Smith Normal Form Algorithms. In: Joswig, M., Takayama, N. (eds) Algebra, Geometry and Software Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05148-1_10
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DOI: https://doi.org/10.1007/978-3-662-05148-1_10
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