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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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2. Y. Félix , S. Halperin and J.-C. Thomas , Rational homotopy theory, Graduate Texts in Mathematics, Volume 205 (Springer, 2001).

4. A. Garvín and L. Lechuga , The computation of the Betti numbers of an elliptic space is a NP-hard problem. Topol. Applic. 131(3) (2003), 235238.

6. L. Lechuga and A. Murillo , Complexity in rational homotopy. Topology 39(1) (2000), 8994.

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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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