Constructive algebraic topology

Rubio, Julio; Sergeraert, Francis

doi:10.1016/s0007-4497(02)01119-3

Cited by 63 publications

(86 citation statements)

References 10 publications

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“…For example, the first one is represented through: There exist special homotopy operators called contracting homotopies which express algorithmically that the chain complex is acyclic [24]. Definition 8.…”

Section: Computing With Infinite Data Structures In Coqmentioning

confidence: 99%

“…Under good conditions, its homology groups are, however, of finite type. Computing these homology groups was one of the first challenges solved by Kenzo (see [24]), and working with them in Coq would be an interesting issue.…”

Section: Conclusion and Further Workmentioning

confidence: 99%

“…In this work, we undertake another via to manage the infinity, working with algebraic structures of infinite type (that is to say, generated by infinite sets) [24]. We report in this paper on an experiment of this nature, in the area of homological algebra.…”

Section: Introductionmentioning

confidence: 99%

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Computing in Coq with Infinite Algebraic Data Structures

Domínguez

Rubio

2010

Lecture Notes in Computer Science

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Section: Computing With Infinite Data Structures In Coqmentioning

confidence: 99%

Section: Conclusion and Further Workmentioning

confidence: 99%

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Computing in Coq with Infinite Algebraic Data Structures

Domínguez

Rubio

2010

Lecture Notes in Computer Science

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“…In the 1990s, three independent groups of researchers proposed general frameworks to make various more advanced methods of algebraic topology effective (algorithmic): Schön [45], Smith [50], and Sergeraert, Rubio, Dousson, and Romero (e.g., [48,42,41,43]; also see [44] for an exposition). These frameworks yielded general computability results for homotopy-theoretic questions (including new algorithms for the computation of higher homotopy groups [40]), and in the case of Sergeraert and co-workers, a practical implementation as well.…”

Section: Introductionmentioning

confidence: 99%

Computing All Maps into a Sphere

Čadek

Krčál

Matoušek

et al. 2014

J. ACM

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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.

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“…In the 1990s, three independent collections of works appeared with the goal of making various more advanced methods of algebraic topology effective (algorithmic): by Schön [Sch91], by Smith [Smi98], and by Sergeraert, Rubio, Dousson, and Romero (e.g., [Ser94,RS02,RRS06,RS05]; also see [RS12] for an exposition). New algorithms for computing higher homotopy groups follow from these methods; see Real [Rea96] for an algorithm based on Sergeraert et al…”

Section: Introductionmentioning

confidence: 99%

Extendability of Continuous Maps Is Undecidable

Čadek

Krčál

Matoušek

et al. 2013

Discrete Comput Geom

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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 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.

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Constructive algebraic topology

Cited by 63 publications

References 10 publications

Computing in Coq with Infinite Algebraic Data Structures

Computing in Coq with Infinite Algebraic Data Structures

Computing All Maps into a Sphere

Extendability of Continuous Maps Is Undecidable

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