Some homotopy groups of the rotation group $R\sb n$

The 23-rd and 24-th homotopy groups of the n-th rotation group

“…Steenrod [34], Sugawara [35], Kervaire [17] and Mimura [21] had studied the k-th homotopy groups π k (R n ) of the n-th rotation group R n . The group structures of π k (R n ) for k ≤ 22 had been determined in this period [34], [35], [17], [41], [18], [14], [15], [16] and [11].…”

“…By using this choice, we have ∆(ν 2 18 ) = [ε 15 ] 18 (Proposition 4.13). For any element [ε 15 ] ∈ R 16 23 , we have ∆(ν 2 18 ) ≡ [ε 15 ] 18 mod [σ 2 9 ] 18 . At best we will be able to obtain the relation [ε 15 ] 22 = 0…”

Higher homotopy associativity in the Harris decomposition of Lie groups

“…In [11,16], ν 6 + 6 and [ν 6 + 6 ] are chosen such that π * ( ν 6 + 6 ) = π * ([ν 6 + 6 ]) =ν 6 + 6 , where π 14 (S 6 ) = Z/8{ν 6 } ⊕ Z/2{ 6 }. Then by lemma 4.4,…”

“…By the definition of [η 5 6 ] 7 in [11], π * ([η 5 6 ] 7 ) = 0, and so s 1 (ν 6 + 6 ) + s 3 π * ([ι 7 ]σ ) = 0. Consider a homotopy exact sequence of the lower fibration of (4.2).…”

“…As in [11], there is x ∈ π 8 (Spin(6)) such that π * (x) = ν 5 , and [ν 5 ] 7 ∈ π 8 (Spin(7)) is chosen as [ν 5 ] 7 = j * (x), where π n+3 (S n ) = Z/8{ν n } for n 5. On the other hand, in [16], [ν 2 5 ] ∈ π 11 (SU (3)) is chosen to be any y ∈ π 11 (SU (3)) satisfyingπ * (y) = ν 2 5 .…”