Algorithmic Solvability of the Lifting-Extension Problem

Čadek, Martin; Krăźál, Marek; Vokřínek, Lukáš

doi:10.1007/s00454-016-9855-6

Cited by 21 publications

(69 citation statements)

References 19 publications

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“…This is a classical and well-studied topic in geometric topology (see, e.g., [56] for a survey) and a natural higher-dimensional generalization of graph planarity (the case k = 1, d = 2). Recently, a significant effort has been devoted to the systematic study of embeddability in higher dimensions from an algorithmic viewpoint [36,39,59]. In particular, it is now known that embeddability of a kdimensional complex into R d is algorithmically decid-able for d ≤ 3 [39] and algorithmically undecidable for…”

Section: Computing Embeddingsmentioning

confidence: 99%

“…This paper forms part of a systematic effort [5,2,54,50,7,34,9,8,6,38,60,15,59,48,49] to understand the computational complexity of questions in homotopy theory such as computing π d (X) or [X, Y ] for given spaces X and Y ; for these algorithmic versions of the problems, the input spaces are assumed to be represented combinatorially as (finite) simplicial complexes.…”

Section: Introductionmentioning

confidence: 99%

“…Examples of such applications (some of which we will discuss in more detail below) include the embeddability of simplicial complexes (a higherdimensional version of graph planarity) [36,39,59], questions in topological combinatorics [37,35], and solving equations and optimization with uncertainty [18,17,16].…”

mentioning

confidence: 99%

“…Building on the framework of objects with effective homology of Sergeraert et al, a variety of new results in computational homotopy theory have been obtained in recent years [7,34,9,8,60,15,59,48,49]. Some of these papers obtain the first polynomial-time algorithms for the problems in question, by using a refined framework of objects with polynomial-time homology [34,9] that allows for a computational complexity analysis.…”

mentioning

confidence: 99%

“…In a subsequent step, we then hope to generalize this further to the equivariant setting [X, Y ] G of [59], in which a finite group G of symmetries acts on the spaces X, Y and all maps and homotopies are required to be equivariant, i.e., to preserve the symmetries. Thus, the goal is to obtain an algorithm that, given an element of [X, Y ] G , computes an explicit equivariant map X → Y (represented as a simplicial map on a suitable subdivision of X, say).…”

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confidence: 99%

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Computing Simplicial Representatives of Homotopy Group Elements

Filakovský¹,

Franek²,

Wagner³

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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A central problem of algebraic topology is to understand the homotopy groups π d (X) 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 π 1 (X) 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 π 1 (X) trivial), compute the higher homotopy groupHowever, these algorithms come with a caveat: They compute the isomorphism type of π d (X), d ≥ 2 as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of π d (X). Converting elements of this abstract group into explicit geometric maps from the d-dimensional sphere S d to X has been an important open question in the emerging field of computational homotopy theory.Here we present an algorithm that, given a simply connected simplicial complex X, computes π d (X) and represents its elements as simplicial maps from a suitable triangulation of the d-sphere S d to X. For fixed d, the algorithm runs in time exponential in size(X), the number of simplices of X. Moreover, we prove that this is optimal: For every fixed d ≥ 2, we construct a family of simply connected simplicial complexes X such that for any simplicial map representing a generator of π d (X), the size of the triangulation of S d on which the map is defined is exponential in size(X).

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Section: Computing Embeddingsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Computing Simplicial Representatives of Homotopy Group Elements

Filakovský¹,

Franek²,

Wagner³

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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Topologically 4‐chromatic graphs and signatures of odd cycles

Simons

Tardif

Wehlau

2020

Journal of Graph Theory

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We investigate group-theoretic "signatures" of odd cycles of a graph, and their connections to topological obstructions to 3-colourability. In the case of signatures derived from free groups, we prove that the existence of an odd cycle with trivial signature is equivalent to having the coindex of the hom-complex at least 2 (which implies that the chromatic number is at least 4). In the case of signatures derived from elementary abelian 2-groups we prove that the existence of an odd cycle with trivial signature is a sufficient condition for having the index of the hom-complex at least 2 (which again implies that the chromatic number is at least 4).

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