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**1 - 4**of**4**### Polynomial-time computation of homotopy groups and Postnikov systems in fixed dimension∗

, 2014

"... 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, ..."

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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 pik(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, un-der 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 sim-plicial 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 ho-mology, which is an enhancement of the notion of a simplicial set with effective homology

### Extendability of continuous maps is undecidable∗

, 2013

"... 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 (continuo ..."

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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 map f ̄ : 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 dimX ≤ 2k − 1 cannot be relaxed: for dimX = 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 pik(Y) of a 1-connected simplicial complex Y is #P-hard when k is considered as a part of the input. 1