SAMELSON PRODUCTS IN p-REGULAR SO(2n) AND ITS HOMOTOPY NORMALITY

Abstract

Abstract.A Lie group is called p-regular if it has the p-local homotopy type of a product of spheres. (Non)triviality of the Samelson products of the inclusions of the factor spheres into p-regular SO(2n) (p) is determined, which completes the list of (non)triviality of such Samelson products in p-regular simple Lie groups. As an application, we determine the homotopy normality of the inclusion SO(2n − 1) → SO(2n) in the sense of James at any prime p.2010 Mathematics Subject Classification. 55Q15.

Note on Samelson Products in Exceptional Lie Groups

“…Since P 1 H 11 (B 1 ) = 0 as in [9], we also have (P 1 ) 2 H 16 ( B (11) 1 × B (19) 1 ) = 0. Then by Corollary 3.7, we can also apply Lemma 4.1, so that (7) be the inclusions. Then for each i, j, i , j = 0.…”

“…Let μ : The homotopy groups of the factor spaces of the mod 7 decomposition of E 8 are calculated in [9]. In particular, one has A j| A / / (BE 8 ) (7) where the map j is the adjoint of the inclusion B → (E 8 ) (7) .…”

“…Since A ∧ A has cells in dimension 12i + 6 for 0 ≤ i ≤ 6, it follows from Lemma 4.4 that 1 | A , 1 | A = 0. Then the corresponding Whitehead product [ j| A , j| A ] is trivial so that the map j| A ∨ j| A : A ∨ A → (BE 8 ) (7) extends over A × A, where j : B 1 → (BE 8 ) (7) denotes the adjoint of 1 as in Lemma 4.5. Thus, there is a homotopy commutative diagram:…”

Homotopy commutativity in Hermitian symmetric spaces

“…In this section, we prove that the loop spaces of the irreducible Hermitian symmetric spaces of type AIII, BDI, EIII by applying the following lemma. The lemma was proved by Kono and Ōshima [17] when A and B are spheres and p is odd, and its variants are used in [6,7,8,9,13,14,15,16,23]. For an augmented graded algebra A, let QA n denote the module of indecomposables of dimension n. Lemma 3.1.…”

“…In particular, H Note that F 4 /Spin(9) is the Cayley plane OP 2 . Then since OP 2 = S 8 ∪ e16 , a generator u ∈ H 8 (F 4 /Spin(9); Z/2) ∼ = Z/2 is mod 2 spherical, and so a generator v ∈ H 8 (E 6 /Spin(10)) ∼ = Z/2 is mod 2 spherical too. By the Gysin sequence associated to the fibration S 1 → E 6 /Spin(10)…”