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

Abstract

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 π k (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 dim X ≤ 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 dim X ≤ 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.

Extending continuous maps

“…This is a simple extension of Corollary 1.4, presented in [5]; for dim X ≤ 2k − 2, the algorithm even finds a certain description of the set of all possible extensions up to homotopy.…”

“…Some of the tools and methods from the proof are sketched in Section 3, and for a full proof we refer to [5].…”

“…(In [5], a slightly different but simplicially isomorphic representation is used, using pullbacks which we do not want to introduce here.) Twisted product is a simplicial analog of the topological notion of fiber bundle (this is a generalization of a vector bundle), which we will now outline to convey some intuition.…”

“…The way the bundle is "twisted" is determined by the Postnikov class k k−1 , which can be regarded as a simplicial map P k−1 → K(π k (Y ), k + 1), or alternatively, using the correspondence mentioned earlier, as a cocycle in Z k+1 (P k−1 , π k (Y )). Technically, the twisting is achieved by forming the ordinary simplicial product K(π k (Y ), k)× P k−1 , and then modifying the 0th face operator in it using k k−1 -we refer to [5,15] for the details.…”

“…This survey summarizes the results of the three papers [12,5,4]. Together these comprise well over 100 pages, and they deal with numerous moderately advanced topological concepts, as well as many algorithmic notions and results.…”