A new lower bound for LS-category

Abstract

Let X be a simply connected CW-complex of finite type and {mathbb {K}} an arbitrary field. In this paper, we use the Eilenberg–Moore spectral sequence of C_*(Omega (X), mathbb K) to introduce a new homotopical invariant textsc {r}(X, {mathbb {K}}). If X is a Gorenstein space with nonzero evaluation map, then textsc {r}(X, {mathbb {K}}) turns out to interpolate mathrm {depth}(H_*(Omega (X), {mathbb {K}})) and mathrm {e}_{{mathbb {K}}}(X). We also define for any minimal Sullivan algebra (Lambda V,d) a new spectral sequence and make use of it to associate to any 1-connected commutative differential graded algebra (A, d) a similar invariant textsc {r}(A,d). When (Lambda V,d) is a minimal Sullivan model of X, this invariant fulfills the relation textsc {r}(X, {mathbb {K}}) = textsc {r}(Lambda V,d).