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

TL;DR

This paper introduces a new algebraic invariant, r(X, \u211d), derived from the Eilenberg-Moore spectral sequence, which improves existing lower bounds for the LS-category of simply connected CW-complexes by interpolating between known invariants.

Contribution

The paper defines r(X, ) as a new invariant that refines lower bounds of LS-category, connecting algebraic and topological properties through spectral sequences and minimal models.

Findings

r(X, ) matches properties of the Toomer invariant.

r(X, ) interpolates depth and Toomer invariant when certain conditions hold.

An improved lower bound for LS-category is established using r().

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 F\'elix et al. introduced the concept of {\it depth} in algebraic topology and proved the depth theorem: $depth (H_*(\Omega X, \mathbb{K})) \leq cat(X)$. 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 $char(\mathbb{K})\not = 2$, we prove that $\textsc{r}(X, \mathbb{K})$ interpolates $depth(H_*(\Omega X, \mathbb{K}))$ 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 $d=\sum_{i\geq k}d_i$ and $d_i(V)\subseteq \Lambda ^iV$, a greater lower bound is obtained, namely $e_0(\Lambda V, d)\geq \textsc{r}(\Lambda V, d) + (k-2)$.