On some Lusternick-Schnirelmann type invariants

Abstract

In this paper, we show that the invariant $\mathrm{R}_0(X)$, introduced in [15], coincides with $cat_0(X)$ for any rationally elliptic space $X$. Additionally, we define, for any space $X$ over an arbitrary field $\mathbb{K}$, an ${\it Ext-version}$ homotpy invariant $\mathrm{L}_{\mathbb{K}}(X)$ of the Ginsburg invariant $l_{\mathbb{K}}(X)$. Then, we establish the equality between $\mathrm{L}_{0}(X):=\mathrm{L}_{\mathbb{Q}}(X)$ and $l_0(X)$ in the case where $X$ is rationally elliptic.