Effective chain complexes for twisted products

Filakovský, Marek

doi:10.48550/arxiv.1209.1240

Cited by 2 publications

(5 citation statements)

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“…The effective-homology analogs of this result are due to Rubio and Sergeraert [45,Theorem 132] when B is 1-reduced and due to Filakovský [14,Corollary 12] when G is 0-reduced. (…”

Section: Twisted Productmentioning

confidence: 90%

“…This follows from the obvious fact that such a chain homotopy never increases the filtration degree 22 by more than 1, plus a result showing that if G is 0-reduced or B is 1-reduced, then δT decreases the filtration degree at least by 2. We refer to [14,Corollary 9 and 11] for a proof of the latter result (also see the proof of Lemma 3.14 below, where a very similar situation is discussed). Then a constant nilpotency bound with N k ≤ k + 1 follows, and the proposition is proved.…”

Section: Twisted Productmentioning

confidence: 97%

“…Then we can define a new parameterized simplicial set ( X(I) : I ∈ I) by X(I) := X(F (I)); if X is locally polynomial-time, then so is X. 14 Chain complexes with a distinguished basis for each chain group are sometimes called cellular.…”

Section: Locally Polynomial-time Simplicial Sets and Chain Complexesmentioning

confidence: 99%

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Polynomial-time computation of homotopy groups and Postnikov systems in fixed dimension

Čadek¹,

Krčál²,

Matoušek³

et al. 2012

Preprint

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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 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.

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“…The effective-homology analogs of this result are due to Rubio and Sergeraert [45,Theorem 132] when B is 1-reduced and due to Filakovský [14,Corollary 12] when G is 0-reduced. (…”

Section: Twisted Productmentioning

confidence: 90%

Section: Twisted Productmentioning

confidence: 97%

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Polynomial-time computation of homotopy groups and Postnikov systems in fixed dimension

Čadek¹,

Krčál²,

Matoušek³

et al. 2012

Preprint

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“…13 For t = j k j σ j , we may choose h = j σ k j j (choosing any order of the simplices). 14 The connectivity assumption on F was exploited in the existence of the contraction c A j on the Abelian part.…”

Section: Arrowmentioning

confidence: 99%

“…We remark that the paper[9] uses a different formalization of twised cartesian product than the one employed by us. However, the paper[14], on which the Corollary 3.18 of[9] is based, can be reformulated in context of the definition used here. We do not provide full details, only remark that one has to make a choice of Eilenberg-Zilber reduction data that corresponds to the definition of twisted cartesian product.…”

mentioning

confidence: 99%

Computing simplicial representatives of homotopy group elements

Filakovský

Franek

Wagner

et al. 2018

J Appl. and Comput. Topology

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A central problem of algebraic topology is to understand the homotopy groups of a topological space X. For the computational version of the problem, it is well known that there is no algorithm to decide whether the fundamental group of a given finite simplicial complex X is trivial. On the other hand, there are several algorithms that, given a finite simplicial complex X that is simply connected (i.e., with trivial), compute the higher homotopy group for any given . However, these algorithms come with a caveat: They compute the isomorphism type of , as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of . Converting elements of this abstract group into explicit geometric maps from the d-dimensional sphere to X has been one of the main unsolved problems in the emerging field of computational homotopy theory. Here we present an algorithm that, given a simply connected space X, computes and represents its elements as simplicial maps from a suitable triangulation of the d-sphere to X. For fixed d, the algorithm runs in time exponential in , the number of simplices of X. Moreover, we prove that this is optimal: For every fixed , we construct a family of simply connected spaces X such that for any simplicial map representing a generator of , the size of the triangulation of on which the map is defined, is exponential in .

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Effective chain complexes for twisted products

Cited by 2 publications

References 0 publications

Polynomial-time computation of homotopy groups and Postnikov systems in fixed dimension

Polynomial-time computation of homotopy groups and Postnikov systems in fixed dimension

Computing simplicial representatives of homotopy group elements

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