Pairing Rank in Rational Homotopy Group

Abstract

Let $X$ be a simply connected CW complex of finite rational LS-category with $\dim H_n(X;{\mathbb Q})<\infty$ for all $n$. The dimension of rational Gottlieb group $G_*(X)\otimes {\mathbb Q}$ is upper-bounded by the rational LS-category $cat_0(X)$ the inequation $\dim G_*(X)\otimes {\mathbb Q}\leq cat_0(X)$ holds [2]. Then we introduce a new rational homotopical invariant between them, denoted as the pairing rank $v_0(X)$ in the rational homotopy group $\pi_*(X)\otimes {\mathbb Q}$ such that $\dim G_*(X)_{\mathbb Q}\leq v_0(X)\leq cat_0(X)$. If $\pi_*(f)\otimes {\mathbb Q}$ is injective for a map $f:X\to Y$, then we have $v_0(X)\leq v_0(Y)$. Also it has a good estimate for a fibration $X{\to} E{\to} Y$ as $v_0(E)\leq v_0(X) +v_0(Y)$.