Cross-sections of symplectic Stiefel manifolds

Sigrist, François; Suter, Ulrich W.

doi:10.1090/s0002-9947-1973-0326728-8

Cited by 19 publications

(9 citation statements)

References 12 publications

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“…In the papers [3,5,12,18,19,23] the problem of when the structure group G n in one of the fibrations (1.1)-(1.3) can be reduced to a group G = SU(k) or Sp(k) via a standard inclusion G → G n was solved and the results were expressed in terms of complex and quaternionic James numbers in a way similar to the result quoted above. The following theorem can be regarded as a generalization of these results.…”

Section: Resultsmentioning

confidence: 99%

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$G$-Structures on Spheres

Čadek

Crabb

2006

Proc. Lond. Math. Soc.

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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.

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Section: Resultsmentioning

confidence: 99%

“…Classical computations in [1,3,5,23] showed that t(r) = a(r), b(r) and c(r) in the real, complex and quaternionic cases, respectively. Classical computations in [1,3,5,23] showed that t(r) = a(r), b(r) and c(r) in the real, complex and quaternionic cases, respectively.…”

Section: Resultsmentioning

confidence: 99%

$G$-Structures on Spheres

Čadek

Crabb

2006

Proc. Lond. Math. Soc.

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“…Let c F k be the F-James number computed by Adams [1] for F = R, by Adams and Walker [2] for F = C, and by Sigrist and Suter [13] for F = H. The key result of this paper is the following desuspension lemma which was unnoticed by Namboodiri [9] and Önder [11].…”

Section: Lemma 4 If ρ Has a G-section Then S(m) G S(m ⊗ F ξ K−1 ) Tmentioning

confidence: 90%

“…Let F be one of the classical division algebras over R, namely, the real numbers R, the complex numbers C, or the quaternionic numbers H. Let G be a compact Lie group, M be a finite-dimensional right F-representation space of G with a G-equivariant F-inner [1] in 1962 for the real case, by Adams and Walker [2] in 1965 for the complex case, and by Sigrist and Suter [13] …”

Section: Introductionmentioning

confidence: 99%

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Real, complex and quaternionic equivariant vector fields on spheres

Obiedat

2006

Topology and its Applications

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The equivariant real, complex and quaternionic vector fields on spheres problem is reduced to a question about the equivariant J -groups of the projective spaces. As an application of this reduction, we give a generalization of the results of Namboodiri [U. Namboodiri, Equivariant vector fields on spheres, Trans. Amer. Math. Soc. 278 (2) (1983) 431-460], on equivariant real vector fields, and Ön-der [T. Önder, Equivariant cross sections of complex Stiefel manifolds, Topology Appl. 109 (2001) 107-125], on equivariant complex vector fields, which avoids the restriction that the representation containing the sphere has enough orbit types.

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On the groupsJ(ℂℙ m ) andJ(ℍℙ n )

Suter

1974

Bol. Soc. Bras. Mat

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Cross-sections of symplectic Stiefel manifolds

Abstract: The cross-section problem for the symplectic Stiefel manifolds is solved, using the now-proved Adams conjecture.

Cited by 19 publications

References 12 publications

$G$-Structures on Spheres

$G$-Structures on Spheres

Real, complex and quaternionic equivariant vector fields on spheres

On the groupsJ(ℂℙ m ) andJ(ℍℙ n )

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