## Homotopy groups of the homogeneous spaces $$F_4/G_2$$ and $$F_4/\text{Spin}(9)$$.(English)Zbl 0998.55006

Let $$G_2$$ and $$F_4$$ be the classical exceptional Lie groups of rank 2 and 4 respectively. In this paper, the authors calculate 2-primary components of homotopy groups of homogeneous spaces $$F_4/G_2$$ and $$F_4/\text{Spin}(9)$$. More precisely, they determine $$\pi_i(F_4/G_2:2)$$ for $$i\leq 45$$ and $$\pi_i(F_4/\text{Spin}(9) : 2)$$ for $$i < 38$$. The main tools are the homotopy exact sequences associated with the 2-local fibration $$S^{15}\to F_4/G_2\to S^{23}$$ and with the fibration $$S^7\to \Omega(F_4/\text{Spin}(9))\to\Omega S^{23}$$ introduced by D. M. Davis and M. Mahowald [J. Math. Soc. Japan 43, No. 4, 661-672 (1991; Zbl 0736.57020)]. The determination of the group extensions arising from these sequences is done by using Toda brackets, as in Theorem 2.1 of [M. Mimura and H. Toda, J. Math. Kyoto Univ. 3, 217-250 (1964; Zbl 0129.15404)].

### MSC:

 55Q52 Homotopy groups of special spaces 57T20 Homotopy groups of topological groups and homogeneous spaces

### Keywords:

exceptional Lie groups

### Citations:

Zbl 0736.57020; Zbl 0129.15404