Computing Homotopy Classes for Diagrams

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Publikace nespadá pod Pedagogickou fakultu, ale pod Přírodovědeckou fakultu. Oficiální stránka publikace je na webu muni.cz.
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FILAKOVSKÝ Marek VOKŘÍNEK Lukáš

Rok publikování 2023
Druh Článek v odborném periodiku
Časopis / Zdroj Discrete and Computational Geometry
Fakulta / Pracoviště MU

Přírodovědecká fakulta

Citace
www https://doi.org/10.1007/s00454-023-00513-0
Doi http://dx.doi.org/10.1007/s00454-023-00513-0
Klíčová slova Equivariant homotopy; Algorithm; Tverberg-type problem
Popis We present an algorithm that, given finite diagrams of simplicial sets X, A, Y, i.e., functors $${\mathcal {I}}^\textrm{op}\rightarrow {\textsf {s}} {\textsf {Set}}$$ I op › s Set , such that (X, A) is a cellular pair, $$\dim X\le 2\cdot {\text {conn}}Y$$ dim X ? 2 · conn Y , $${\text {conn}}Y\ge 1$$ conn Y ? 1 , computes the set $$[X,Y]^A$$ [ X , Y ] A of homotopy classes of maps of diagrams $$\ell :X\rightarrow Y$$ l : X › Y extending a given $$f:A\rightarrow Y$$ f : A › Y . For fixed $$n=\dim X$$ n = dim X , the running time of the algorithm is polynomial. When the stability condition is dropped, the problem is known to be undecidable. Using Elmendorf’s theorem, we deduce an algorithm that, given finite simplicial sets X, A, Y with an action of a finite group G, computes the set $$[X,Y]^A_G$$ [ X , Y ] G A of homotopy classes of equivariant maps $$\ell :X\rightarrow Y$$ l : X › Y extending a given equivariant map $$f:A\rightarrow Y$$ f : A › Y under the stability assumption $$\dim X^H\le 2\cdot {\text {conn}}Y^H$$ dim X H ? 2 · conn Y H and $${\text {conn}}Y^H\ge 1$$ conn Y H ? 1 , for all subgroups $$H\le G$$ H ? G . Again, for fixed $$n=\dim X$$ n = dim X , the algorithm runs in polynomial time. We further apply our results to Tverberg-type problem in computational topology: Given a k-dimensional simplicial complex K, is there a map $$K\rightarrow {\mathbb {R}}^d$$ K › R d without r-tuple intersection points? In the metastable range of dimensions, $$rd\ge (r+1) k+3$$ r d ? ( r + 1 ) k + 3 , the problem is shown algorithmically decidable in polynomial time when k, d, and r are fixed.
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