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}} {textsf {Set}}, such that (X, A) is a cellular pair, dim Xle 2cdot {text {conn}}Y, {text {conn}}Yge 1, computes the set [X,Y]^A of homotopy classes of maps of diagrams ell :Xrightarrow Y extending a given f:Arightarrow Y. For fixed 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 of homotopy classes of equivariant maps ell :Xrightarrow Y extending a given equivariant map f:Arightarrow Y under the stability assumption dim X^Hle 2cdot {text {conn}}Y^H and {text {conn}}Y^Hge 1, for all subgroups Hle G. Again, for fixed 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 Krightarrow {mathbb {R}}^d without r-tuple intersection points? In the metastable range of dimensions, rdge (r+1) k+3, the problem is shown algorithmically decidable in polynomial time when k, d, and r are fixed.