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

Abstract

Let ( ΛV, d) be the Sullivan model of an elliptic space S and ( ΛV, d σ ) be the associated pure model. We give an algorithm, based on Groebner basis computations, that computes the stage l σ = l 0( ΛV, d σ ) at which the (Sullivan version of the) Milnor–Moore spectral sequence of ( ΛV, d σ ) collapses. When ( d− d σ ) V⊂ Λ > l σ V we call S a Ginsburg space. We show that the rational LS category of any Ginsburg space S, cat 0( ΛV, d), coincides with that of the associated pure space cat 0( ΛV, d σ ). A previous algorithm due to the author computes cat 0( ΛV, d σ ). So we obtain an algorithm that determines whether a space is Ginsburg and which in this case computes its rational LS category.