LS-Category and the Depth of Rationally Elliptic Spaces

Abstract

Let $X$ be a finite type simply connected rationally elliptic CW-complex with Sullivan minimal model $(\Lambda V,d)$ and let $k\ge 2$ the biggest integer such that $d={\sum }_{i\ge k}{d}_i$ with $d_i(V)\subseteq {\Lambda }^{i}V$. 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 ($r\ge 1$) finite CW-complex such that $p>dim(X)/r$.

arXiv commentThis paper has been withdrawn by the author. it was replaced by the preprint with the number arXiv:1211.5068