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.

No abstract

Let X and Y be finite simplicial sets (e.g. finite simplicial complexes), both equipped with a free simplicial action of a finite group G. Assuming that Y is dconnected and dim X ≤ 2d, for some d ≥ 1, we provide an algorithm that computes the set of all equivariant homotopy classes of equivariant continuous maps |X| → |Y |; the existence of such a map can be decided even for dim X ≤ 2d + 1. For fixed G and d, the algorithm runs in polynomial time. This yields the first algorithm for deciding topological embeddability of a k-dimensional finite simplicial complex into R n under the condition k ≤ 2 3 n − 1. More generally, we present an algorithm that, given a lifting-extension problem satisfying an appropriate stability assumption, computes the set of all homotopy classes of solutions. This result is new even in the non-equivariant situation. * 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. was supported by the Center of Excellence -Inst. for Theor. Comput. Sci., Prague (project P202/12/G061 of GA ČR) and by the Project LL1201 ERCCZ CORES. The research of L. V. was supported by the Center of Excellence -Eduard Čech Institute (project P201/12/G028 of GA ČR).

We consider two basic problems of algebraic topology, the extension problem and the computation of higher homotopy groups, from the point of view of computability and computational complexity.The extension problem is the following: Given topological spaces X and Y , a subspace A ⊆ X, and a (continuous) map f : A → Y , decide whether f can be extended to a continuous mapf : X → Y . All spaces are given as finite simplicial complexes and the map f is simplicial.Recent positive algorithmic results, proved in a series of companion papers, show that for (k − 1)-connected Y , k ≥ 2, the extension problem is algorithmically solvable if the dimension of X is at most 2k − 1, and even in polynomial time when k is fixed.Here we show that the condition dim X ≤ 2k − 1 cannot be relaxed: for dim X = 2k, the extension problem with (k − 1)-connected Y becomes undecidable. Moreover, either the target space Y or the pair (X, A) can be fixed in such a way that the problem remains undecidable.Our second result, a strengthening of a result of Anick, says that the computation of π k (Y ) of a 1-connected simplicial complex Y is #P-hard when k is considered as a part of the input.

Given topological spaces X, Y , a fundamental problem of algebraic topology is understanding the structure of all continuous maps X → Y . We consider a computational version, where X, Y are given as finite simplicial complexes, and the goal is to compute [X, Y ], i.e., all homotopy classes of such maps.We solve this problem in the stable range, where for some d ≥ 2, we have dimThe algorithm combines classical tools and ideas from homotopy theory (obstruction theory, Postnikov systems, and simplicial sets) with algorithmic tools from effective algebraic topology (locally effective simplicial sets and objects with effective homology).In contrast, [X, Y ] is known to be uncomputable for general X, Y , since for X = S 1 it includes a well known undecidable problem: testing triviality of the fundamental group of Y . In follow-up papers, the algorithm is shown to run in polynomial time for d fixed, and extended to other problems, such as the extension problem, where we are given a subspace A ⊂ X and a map A → Y and ask whether it extends to a map X → Y , or computing the Z 2 -index -everything in the stable range. Outside the stable range, the extension problem is undecidable.