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

Abstract

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 Félix et al. introduced the concept of {\it depth} in algebraic topology and proved the depth theorem: $depth(H_{\ast }(\Omega X,\mathbb{K}))\le cat(X)$. In this paper, we use the Eilenberg-Moore spectral sequence of $X$ to introduce a new numerical invariant, denoted by $\textsc{r}(X, \mathbb{K})$, 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 $char(\mathbb{K})\ne 2$, we prove that $\textsc{r}(X, \mathbb{K})$ interpolates $depth(H_{\ast }(\Omega X,\mathbb{K}))$ 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 $\textsc{r}(A,d)$ and, via the theory of minimal models, we relate it with our previous topological results. In particular, if $(\Lambda V,d)$ is a Sullivan minimal algebra such that $d={\sum }_{i\ge k}{d}_i$ and $d_i(V)\subseteq {\Lambda }^{i}V$, a greater lower bound is obtained, namely $e_0(ΛV, d)\geq \textsc{r}(ΛV, d) + (k-2)$.

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