Computing the Sullivan Milnor-Moore S. S. and the rational LS category of certain spaces
The computation of Lyusternik-Shnirel'man category of a space, \(\text{ cat}(X)\), is often very difficult. Estimates of this numerical invariant have been obtained using the Eilenberg-Moore spectral sequence by \textit{M. Ginsburg} [Ann. Math. (2) 77, 538-551 (1963; Zbl 0148.17003)] and \textit{G. H. Toomer} [Math. Z. 138, 123-143 (1974; Zbl 0284.55012)]. A refined version of these estimates has been found through the use of rational homotopy methods in the work of \textit{Y. Felix} and \textit{S. Halperin} [Trans. Am. Math. Soc. 273, 1-37 (1982; Zbl 0508.55004)]. The author considers elliptic spaces, that is, \(X\) for which \(\pi_*(X)\otimes \mathbb{Q}\) has finite dimension over \(\mathbb{Q}\). The algebraic analysis of the minimal model for an elliptic space is aided when there is a pure model, that is, a minimal model of the form \((\Lambda V, d)\) where \(d V^{\text{odd}} = 0\) and \(d V^{\text{even}} \subset \Lambda V^{\text{odd}}\). In this case, for example, there is a cocycle \(w_1\) that represents the top class of \(H(\Lambda V,d)\), a Poincaré duality algebra, for which \(w_1\) has length at least the rational category of \((\Lambda V, d)\). The algebraic clarity of a pure model admits the use of computational commutative ring theory, in particular, the use of Groebner bases for analysis. The author shows that the rational Ginsburg invariant of \((\Lambda V,d)\), a pure model for an elliptic space, is computable from reduced Groebner bases for the modules of boundaries and \(d\)-cocycles. An algorithm is given and several examples presented.
scientific article; zbMATH DE number 1879149