Computing Homotopy Classes for Diagrams

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