Equivariant cross sections of complex Stiefel manifolds

Real, complex and quaternionic equivariant vector fields on spheres

“…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].…”

“…as it is shown in Namboodiri [9] and Önder [11]. In this section, G will be a finite group, M will be a right F-representation space of G satisfying the k-dimension condition on fixed point sets, and p will be a prime number.…”

“…We will use the results of McClure [8] on the equivariant J -groups to give necessary and sufficient conditions on M ⊗ F ξ k−1 − M to vanish in JO G (FP k−1 ). Then, using Theorem 1, we give a method for computing ρ F G (M) and generalizations of the results of Namboodiri [9] and Önder [11] in the real and complex cases. Finally, we give a summary of the remaining open cases for computing ρ F G (M).…”

“…If G is a finite group and M satisfies the k-dimension condition on fixed point sets, then ρ R G (M) was computed by Namboodiri [9] in 1983 by assuming that M has enough orbit types, and ρ C G (M) was computed by Önder [11] in 2001 by assuming that M has enough orbit types and G has no type-H real irreducibles.…”

“…Now, we are ready to give our generalizations of the results of Namboodiri [9] for equivariant real vector fields and Önder [11] for equivariant complex vector fields which avoids the assumption that M has enough orbit types. 4, 8, or 16 in the real case, and dim C M G = 2 in the complex case.…”