This paper presents two algorithms. In their simplest form, the first algorithm decides the existence of a pointed homotopy between given simplicial maps f , g : X → Y and the second computes the group [ΣX, Y ] * of pointed homotopy classes of maps from a suspension; in both cases, the target Y is assumed simply connected and the algorithms run in polynomial time when the dimension of X is fixed. More generally, these algorithms work relative to A ⊆ X, fibrewise over a simply connected B and also equivariantly when all spaces are equipped with a free action of a fixed finite group G. * The research of M. F. was supported by Masaryk University project MUNI/A/0838/2012. The research of L. V. was supported by the Center of Excellence -EduardČech Institute (project P201/12/G028 of GAČR).

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In the paper weak sufficient conditions for the reduction of the chain complex of a twisted cartesian product F × τ B to a chain complex of free finitely generated abelian groups are found.2010 Mathematics Subject Classification. Primary 18G35; Secondary 55U15.