## African Diaspora Journal of Mathematics

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

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

#### 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

**Permanent link to this document**

https://projecteuclid.org/euclid.adjm/1413810104

**Mathematical Reviews number (MathSciNet)**

MR3270015

**Zentralblatt MATH identifier**

1322.55005

**Subjects**

Primary: 55P62: Rational homotopy theory 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirelʹman) category of a space 55R05: Fiber spaces 55Q70: Homotopy groups of special types [See also 55N05, 55N07]

**Keywords**

Gottlieb group LS-category n-pairing pairing rank Sullivan minimal model rational homotopy

#### 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