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# Computing Homotopy Classes for Diagrams

Autoři FILAKOVSKÝ Marek 2023 Článek v odborném periodiku Discrete and Computational Geometry https://doi.org/10.1007/s00454-023-00513-0 http://dx.doi.org/10.1007/s00454-023-00513-0 Equivariant homotopy; Algorithm; Tverberg-type problem 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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