Determination of the 2-primary components of the 32-stem homotopy groups of $$S^n$$ S n

On the Homotopy Groups of the Suspended Quaternionic Projective Plane and the Applications

“…(3) By Proposition 6.1(4) and σ 9 ν 16 ∈ π 9 19 is of order 8, we know the Toda bracket {j 5 , ν 9 , 4σ 12 } is well defined, takeing 4σ 13 from it. By Lemma 6.2(2), we have Σ ∞ 4σ 13 ∈ ± j, ν, 4σ ∋ ±4â mod j • π S 11 (S 0 ) + π S 8 (HP 2 ) • 4σ ⊆ 16π S 15 (HP 2 ) = 16â .…”

“…The symbols of the generators of π n+k (S n ) we use are all from [1], [5], [6], [7], [8] and [9] , and mainly from [1], the only two differences are : we denote Toda's E by Σ, the suspension functor, and we denote Toda's ∆ by P , where ∆ are the boundary homomorphisms of the EH∆ sequence. And by abuse of notation, sometimes we use the same symbol to denote a map and its homotopy class, especially, for an element of a homotopy group, sometimes we use the same symbol to denote a map whose homotopy class is that element.…”

“…After localization at 2, ΣHP 2 ≃ S 5 ∪ ν5 e9 . After localization at 3, ΣHP 2 ≃ S 5 ∪ α1(5) e 9 .Proof.…”

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

“…We also have (ε 7 + σ ′ η 14 )ν 15 = 0 since ε 3 ν11 = η 3 ε4 (2.49) and σ ′ η 14 ν15 = η 7 ε8 (Lemma 2.3(3)). Therefore, by (2.54), the following matrix Toda bracket ( [20], [19], [28]) is well defined:…”