LS-Category and the Depth of Rationally Elliptic Spaces

Let $X$ be a finite type simply connected rationally elliptic CW-complex with Sullivan minimal model $(\Lambda V, d)$ and let $k\geq 2$ the biggest integer such that $d=\sum_{i\geq k}d_i$ with $d_i(V)\subseteq \Lambda ^iV$. We show that: $cat(X_{\mathbb{Q}}) = depht(\Lambda V, d_k)$ if and only if $(\Lambda V,d_{k})$ is elliptic. This result is obtained by introducing tow new spectral sequences that generalize the Milnor-Moore spectral sequence and its $\mathcal{E}xt$-version \cite{Mur94}. As a corollary, we recover a known result proved - with different methods - by L. Lechuga and A. Murillo in \cite{LM02} and G. Lupton in \cite{Lup02}: If $(\Lambda V,d_{k})$ is elliptic, then $cat(X_{\mathbb{Q}}) = dim(\pi_{odd}(X)\otimes\mathbb{Q}) + (k-2)dim(\pi_{even}(X)\otimes\mathbb{Q})$. In the case of a field ${IK}$ of $char({IK})=p$ (an odd prim) we obtain an algebraic approach for $e_{IK}(X)$ where $X$ is an $r$-connected ($r\geq 1$) finite CW-complex such that $p> dim(X)/r$.