$G$-Structures on Spheres

Abstract

This paper deals with the problem of determining those groups G and the homomorphisms ρ to which the structure group G n in the fibrations above can be reduced. The problem has been solved in many interesting special cases. Considering standard inclusions ρ : G = G k → G n we get the famous problem on sections of Stiefel manifolds over spheres resolved in [1, 3, 5, 23]. The other standard inclusions SU(k) → SO(n), Sp(k) → SO(n) and Sp(k) → SU(n) are dealt with in [12], [18] and [19], respectively. In these cases the question was to find a minimal standard subgroup to which G n can be reduced.In [15] Leonard asked an opposite question: find all maximal proper subgroups to which G n can be reduced. He solved it in the cases when G is a reducible maximal subgroup of G n . Moreover, he proved that G n cannot be reduced to any proper subgroup (G, ρ) if (1) n is even and G n = SO(n) or SU(n), unless G n = SO(6) and G = SU(3);(2) n ≡ 11 mod 12 and G n = Sp(n);(3) G is a non-simple irreducible maximal proper subgroup of G n . 793 can be reduced to G via a homomorphism ρ : G → G n if and only if one of the following cases occurs: (A) G n = SO(n), G = SO(k), n = m − 1, m ≡ 0 mod a(m − k) and, up to conjugation, ρ is the standard inclusion SO(k) → SO(n); (B) G n = SO(n), G = SU(k), n = 2m − 1, m ≡ 0 mod 2 ν2(b(m−k)) and, up to conjugation, ρ is the composition of the standard inclusions SU(k) → SO(2k) → SO(n) or the composition SU(4) → SO(8) × SO(6) → SO(15) where the first homomorphism is given on the first factor by the standard inclusion and on the second factor by the double covering SU(4) ∼ = Spin(6) → SO(6); (C) G n = SO(n), G = Sp(k), n = 4m − 1, m ≡ 0 mod 2 ν2(c(m−k)) and, up to conjugation, ρ is the composition of the standard inclusions Sp(k) → SO(4k) → SO(n) or the exterior square Sp(3) → SO(15); (D) G n = SU(n), G = SU(k), n = m − 1, m ≡ 0 mod b(m − k) and, up to conjugation, ρ is the standard inclusion SU(k) → SU(n); (E) G n = SU(n), G = Sp(k), n = 2m − 1, m ≡ 0 mod c(m − k) and, up to conjugation, ρ is the composition of the standard inclusions Sp(k) → SU(2k) → SU(n); (F) G n = Sp(n), G = Sp(k), n = m − 1, m ≡ 0 mod c(m − k) and, up to conjugation, ρ is the standard inclusion Sp(k) → Sp(n).As a consequence of Theorem 2.

On the complex Banach conjecture

“…Proof. The proof follows closely that of Proposition 3.1 in [7]. First note that the homotopy long exact sequence such that ig is homotopic to the inclusion j : G ֒→ SU (n).…”

“…Statement (3) is Theorem 3 of Leonard [11], when G n = SU (n). The proof of (4) follows from Corollary 2.2 of Cadec-Crabb [7].…”

“…Proof. We follow the ideas of Corollary 3.2 of Cadek-Crabb in [7]. Consider the standard fibration SU (n − 2) → SU (n − 1)…”

Convex bodies all whose sections (projections) are equal

“…If this is the case, dim(G) is not two small with respect to n and hence G is not an exceptional Lie group, with the possible exception of the Lie group E 7 ⊂ O 133 . This is so, because it can be proved that in this case, dim(G) ≥ 2n − 3 (see Proposition 3.1 of [8]). Hence to rule out the exceptional groups one can simply check (e.g., in Wikipedia) the following table in which we list the smallest irreducible representation for them, and the smallest irreducible representation congruent to 1 mod 4 is in boldface, verifying that in all the cases, with the exception of .…”

M-Theory with Framed Corners and Tertiary Index Invariants

“…There is, up to isomorphism, exactly one stably trivial, but not trivial, 11-dimensional vector bundle τ over Y 11 . It may be described as the pullback of the tangent bundle of S 11 by a map f : Y 11 → S 11 of degree one (collapsing the complement of an open disk) [20]. It follows from [82] that the structure group of τ can be reduced to SO(k) by the standard inclusion SO(k) → SO (11) if and only if 12 ≡ 0 mod a(12 − k), where a(r) is the Hurwitz-Radon number of r. The special case in which τ is the tangent bundle is considered in [17].…”