Polynomial-Time Computation of Homotopy Groups and Postnikov Systems in Fixed Dimension

For several computational problems in homotopy theory, we obtain algorithms with running time polynomial in the input size. In particular, for every fixed k ≥ 2, there is a polynomial-time algorithm that, for a 1-connected topological space X given as a finite simplicial complex, or more generally, as a simplicial set with polynomial-time homology, computes the kth homotopy group πk(X), as well as the first k stages of a Postnikov system of X. Combined with results of an earlier paper, this yields a polynomial-time computation of [X,Y ], i.e., all homotopy classes of continuous mappings X → Y , under the assumption that Y is (k−1)-connected and dimX ≤ 2k − 2. We also obtain a polynomial-time solution of the extension problem, where the input consists of finite simplicial complexes X,Y , where Y is (k−1)-connected and dimX ≤ 2k− 1, plus a subspace A ⊆ X and a (simplicial) map f : A → Y , and the question is the extendability of f to all of X. The algorithms are based on the notion of a simplicial set with polynomial-time homology, which is an enhancement of the notion of a simplicial set with effective homology developed earlier by Sergeraert and his co-workers. Our polynomial-time algorithms are obtained by showing that simplicial sets with polynomial-time homology are closed under various operations, most notably, Cartesian products, twisted Cartesian products, and classifying space. One of the key components is also polynomial-time homology for the Eilenberg–MacLane space K(Z, 1), provided in another recent paper by Krčál, Matoušek, and Sergeraert. ∗This research was supported by the ERC Advanced Grant No. 267165. The research of M. Č. was supported by the project CZ.1.07/2.3.00/20.0003 of the Operational Programme Education for Competitiveness of the Ministry of Education, Youth and Sports of the Czech Republic. The research by M. K. and J. M. was supported by the Center of Excellence – Inst. for Theor. Comput. Sci., Prague (project P202/12/G061 of GA ČR). The research of L. V. was supported by the Center of Excellence – Eduard Čech Institute (project P201/12/G028 of GA ČR). The research by U. W. was supported by the Swiss National Science Foundation (grants SNSF200020-138230 and SNSF-PP00P2-138948). Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic Department of Computer Science, ETH Zurich, 8092 Zurich, Switzerland Institut de Mathématiques de Géométrie et Applications, École Polytechnique Fédérale de Lausanne, EPFL SB MATHGEOM, MA C1 553, Station 8, 1015 Lausanne, Switzerland 1 ar X iv :1 21 1. 30 93 v2 [ cs .C G ] 2 8 M ay 2 01 4