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Computational Complexity of Topological Invariants

  • Manuel Amann (a1)

We answer the following question posed by Lechuga: given a simply connected space X with both H* (X; ℚ) and π*(X) ⊗ ℚ being finite dimensional, what is the computational complexity of an algorithm computing the cup length and the rational Lusternik—Schnirelmann category of X?

Basically, by a reduction from the decision problem of whether a given graph is k-colourable for k ≥ 3, we show that even stricter versions of the problems above are NP-hard.

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1.Anick, D. J., The computation of rational homotopy groups is #P-hard, in Computers in geometry and topology, Lecture Notes in Pure and Applied Mathematics, Volume 114, pp. 156 (Dekker, New York, 1989).
2.Félix, Y., Halperin, S. and Thomas, J.-C., Rational homotopy theory, Graduate Texts in Mathematics, Volume 205 (Springer, 2001).
3.Garey, M. R. and Johnson, D. S., Computers and intractability (Freeman, San Francisco, CA, 1979).
4.Garvín, A. and Lechuga, L., The computation of the Betti numbers of an elliptic space is a NP-hard problem. Topol. Applic. 131(3) (2003), 235238.
5.Lechuga, L., A Groebner basis algorithm for computing the rational L.-S. category of elliptic pure spaces. Bull. Belg. Math. Soc. Simon Stevin 9(4) (2002), 533544.
6.Lechuga, L. and Murillo, A., Complexity in rational homotopy. Topology 39(1) (2000), 8994.
7.Sipser, M., Introduction to the theory of computation, 2nd edn (Thomson, 2006).
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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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