## African Diaspora Journal of Mathematics

### Pairing Rank in Rational Homotopy Group

T. Yamaguchi

#### 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 $v_0(E)\leq v_0(X) +v_0(Y)$.

#### Article information

Source
Afr. Diaspora J. Math. (N.S.), Volume 17, Number 1 (2014), 85-92.

Dates
First available in Project Euclid: 20 October 2014

Mathematical Reviews number (MathSciNet)
MR3270015

Zentralblatt MATH identifier
1322.55005

#### Citation

Yamaguchi, T. Pairing Rank in Rational Homotopy Group. Afr. Diaspora J. Math. (N.S.) 17 (2014), no. 1, 85--92. https://projecteuclid.org/euclid.adjm/1413810104