A new invariant that's a lower bound of LS-category

Let $X$ be a simply connected CW-complex of finite type and $\mathbb{K}$ any field. A first known lower bound of LS-category $cat(X)$ is the Toomer invariant $e_{\mathbb{K}}(X)$ (\cite{Too}). In $1980$’s F'elix et al. introduced the concept of {\it depth} in algebraic topology and proved the depth theorem: $depth(H_{\ast }(\mathrm{\Omega }X,\mathbb{K}))\le cat(X)$. In this paper, we use the Eilenberg-Moore spectral sequence of $X$ to introduce a new numerical invariant, denoted by $\text{textsc}r(X,\mathbb{K})$, and show that it has the same properties as those of $e_{\mathbb{K}}(X)$. When the evaluation map (\cite{FHT88}) is non-trivial and $char(\mathbb{K})≠2$, we prove that $\text{textsc}r(X,\mathbb{K})$ interpolates $depth(H_{\ast }(\mathrm{\Omega }X,\mathbb{K}))$ and $e_{\mathbb{K}}(X)$. Hence, we obtain an improvement of L. Bisiaux theorem (\cite{Bis99}) and then of the depth theorem. Motivated by these results, we associate to any commutative differential graded algebra $(A,d)$, a purely algebraic invariant $\text{textsc}r(A,d)$ and, via the theory of minimal models, we relate it with our previous topological results. In particular, if $(\mathrm{\Lambda }V,d)$ is a Sullivan minimal algebra such that $d={\sum }_{i\ge k}{d}_i$ and $d_i(V)\subseteq {\mathrm{\Lambda }}^{i}V$, a greater lower bound is obtained, namely $e_0(\mathrm{\Lambda }V,d)\ge \text{textsc}r(\mathrm{\Lambda }V,d)+(k-2)$.