Computing the Sullivan Milnor–Moore S.S. and the rational LS category of certain spaces

Let (ΛV,d) be the Sullivan model of an elliptic space S and (ΛV,dσ) be the associated pure model. We give an algorithm, based on Groebner basis computations, that computes the stage lσ=l0(ΛV,dσ) at which the (Sullivan version of the) Milnor–Moore spectral sequence of (ΛV,dσ) collapses. When (d−dσ)V⊂Λ>lσV we call S a Ginsburg space. We show that the rational LS category of any Ginsburg space S, cat0(ΛV,d), coincides with that of the associated pure space cat0(ΛV,dσ). A previous algorithm due to the author computes cat0(ΛV,dσ). So we obtain an algorithm that determines whether a space is Ginsburg and which in this case computes its rational LS category.