LS-Category and the Depth of Rationally Elliptic Spaces

TL;DR

This paper establishes a relationship between the rational category of elliptic spaces and the depth of their Sullivan minimal models, introducing new spectral sequences to generalize existing tools and recover known results.

Contribution

It introduces two new spectral sequences that generalize the Milnor-Moore spectral sequence, providing a novel algebraic approach to understanding the rational category of elliptic spaces.

Findings

The rational category equals the depth of the model if and only if the model is elliptic.

Recovered known results relating rational category to homotopy groups.

Provided an algebraic method for calculating the rational category over fields of characteristic p.

Abstract

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 $p> dim(X)/r$.