A quick note about the homotopy groups of the Cayley plane: . Mimura computed some of them. Specifically for i=8,9,10,...,23 he computed that $\pi_i(\mathbf{CaP}^2)$ \pi_i\mathbf{CaP}^2$equals Z, Z/2, Z/2, Z/24, 0, 0, Z/2, Z/120, (Z/2)$^{\oplus3}$, (Z/2)$^{\oplus4}$, Z/24$\oplus$Z/2, Z/504$\oplus$Z/2, 0, Z/6, Z/4, Z$\oplus$Z/120$\oplus$(Z/2)$^{\oplus2}$, respectively. See Theorem 7.2 of his 1967 paper The homotopy groups of Lie groups of low rank: http://www.ams.org/mathscinet-getitem?mr=206958 3 added 4 characters in body; added 3 characters in body A quick note about the homotopy groups of the Cayley plane: Mimura computed some of them. Specifically for i=8,9... 8,9,10,...,23 he computed that$\pi_i(\mathbf{CaP}^2)$equals Z, Z/2, Z/2, Z/24, 0, 0, Z/2, Z/120, (Z/2)$^{\oplus3}$, (Z/2)$^{\oplus4}$, Z/24$\oplus$Z/2, Z/504$\oplus$Z/2, 0, Z/6, Z/4, Z$\oplus$Z/120$\oplus$(Z/2)$^{\oplus2}$, respectively. See Theorem 7.2 of his 1967 paper The homotopy groups of Lie groups of low rank: http://www.ams.org/mathscinet-getitem?mr=206958 http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.kjm/1250524375 2 deleted 105 characters in body; deleted 2 characters in body A quick note about the homotopy groups of the Cayley plane: Mimura computed some of them. Specifically for$i=8,9\dots$i=8,9... he computed that$\pi_i(\mathbf{CaP}^2)$equals$\mathbf{Z}$,$\mathbf{Z}/2$,$\mathbf{Z}/2$,$\mathbf{Z}/24$,$0$,$0$,$\mathbf{Z}/2$,$\mathbf{Z}/120$,$(\mathbf{Z}/2)^{\oplus3}$,$(\mathbf{Z}/2)^{\oplus4}$,$\mathbf{Z}/24\oplus \mathbf{Z}/2$,$\mathbf{Z}/504 \oplus \mathbf{Z}/2$,$0$,$\mathbf{Z}/6$,$\mathbf{Z}/4$, Z, Z/2, Z/2, Z/24, 0, 0, Z/2, Z/120, (Z/2)$^{\oplus3}$, (Z/2)$^{\oplus4}$, Z/24$\oplus$Z/2, Z/504$\oplus$Z/2, 0, Z/6, Z/4, Z$\mathbf{Z} \oplus \mathbf{Z}/120 \oplus \oplus$Z/120$\oplus$(\mathbf{Z}/2)^{\oplus2}$, Z/2)$^{\oplus2}$, respectively. See Theorem 7.2 of his 1967 paper The homotopy groups of Lie groups of low rank: