Real, complex and quaternionic equivariant vector fields on spheres

Abstract

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.

Mass partitions via equivariant sections of Stiefel bundles

“…Indeed, ρ(C, n) = 1 if n is odd and ρ(C, 2k) = 2 for 1 ≤ k ≤ 11 (see [2,4] for an explicit formula for arbitrary n). In fact, the only (see, e.g., [15,20,21]) explicitly known example of a section s : S 4n−1 → V 2 (C 2n ) is s(x) = (x, jx), where j is the usual unit quaternion and S 4n−1 ⊆ H n . This map is S 1 -equivariant, with s(λx) = (λx, λ −1 jx) for each λ ∈ S 1 and x ∈ S 4n−1 , and hence is Z q -equivariant for each q.…”

A note on the construction of complex and quaternionic vector fields on spheres

“…In [3], Becker solved several important cases of the equivariant real vector fields problem, both the maximal number and the construction, on spheres with free group action by using the known Clifford algebras constructions of the non-equivariant real vector fields on spheres. As noted in [6] and [8], both sides of the equivariant complex and quaternionic vector fields problem on spheres with free group action is still completely open. By using a method similar to that used by Becker, the construction of the non-equivariant complex and quaternionic vector fields on spheres might lead to the solution of the equivariant complex and quaternionic vector fields problem on spheres with free group action.…”

“…The situation is completely different with the construction of complex and quaternionic vector fields; there is no explicit construction that gives two or more linearly independent complex vector fields on S(C n ), and there is no known construction that gives even a single quaternionic vector field on S(H n ). In addition to their self importance, the actual construction of complex and quaternionic vector fields on spheres might lead to the solution of several open problems in the equivairant complex and quaternionic vector fields on spheres (see [6,8]).…”

Замечание О Построении Комплексных И Кватернионных Векторных Полей На Сферах

“…Помимо того, что задача о построении важна сама по себе, явная конструкция комплексных и кватернионных векторных полей на сфере может привести к решению нескольких открытых проблем теории эквивариантных комплексных и кватернионных векторных полей на сферах (см. [8], [9]).…”

“…В [11] Беккер решил проблему эквивариантных вещественных векторных полей (как задачу о максимальном числе, так и задачу о построении) в некоторых важных случаях, а именно, на сферах со свободным действием группы, при помощи известных конструкций неэквивариантных вещественных векторных полей на сферах, использующих алгебры Клиффорда. Как отмечено в статьях [8] и [9], оба вопроса (о максимальном числе и построении) полностью открыты для эквивариантных комплексных и кватернионных векторных полей на сферах со свободным действием группы. Возможно, с помощью метода, подобного методу Беккера, из конструкции неэквивариантных комплексных и кватернионных векторных полей на сферах удастся получить решение проблемы об эквивариантных комплексных и кватернионных векторных полях на сферах со свободным действием группы.…”