LSCategory and the Depth of Rationally Elliptic Spaces
Abstract
Let $X$ be a finite type simply connected rationally elliptic CWcomplex 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 MilnorMoore 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 $cat(X_{\mathbb{Q}}) = dim(\pi_{odd}(X)\otimes\mathbb{Q}) + (k2)dim(\pi_{even}(X)\otimes\mathbb{Q})$. 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 CWcomplex such that $p> dim(X)/r$.
 Publication:

arXiv eprints
 Pub Date:
 October 2009
 DOI:
 10.48550/arXiv.0910.4660
 arXiv:
 arXiv:0910.4660
 Bibcode:
 2009arXiv0910.4660R
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Commutative Algebra;
 55P62;
 55M30
 EPrint:
 This paper has been withdrawn by the author. it was replaced by the preprint with the number arXiv:1211.5068