An equivariant generalization of the Miller splitting theorem

Abstract

Let G be a compact Lie group. We build a tower of G -spectra over the suspension spectrum of the space of linear isometries from one G -representation to another. The stable cofibres of the maps running down the tower are certain interesting Thom spaces. We conjecture that this tower provides an equivariant extension of Miller's stable splitting of Stiefel manifolds. We provide a cohomological obstruction to the tower producing a splitting in most cases; however, this obstruction does not rule out a split tower in the case where the Miller splitting is possible. We claim that in this case we have a split tower which would then produce an equivariant version of the Miller splitting and prove this claim in certain special cases, though the general case remains a conjecture. To achieve these results we construct a variation of the functional calculus with useful homotopy-theoretic properties and explore the geometric links between certain equivariant Gysin maps and residue theory. 55P42, 55P91, 55P92

Global stable splittings of Stiefel manifolds

“…We write V K = V ⊗ R K for the scalar extension from R to K of a euclidean inner product space V , with K-inner product [−, −] obtained from the euclidean inner product −, − on V by (A. 24) [x ⊗ λ, y ⊗ µ] = λ • x, y • µ for x, y ∈ V and λ, µ ∈ K. The underlying euclidean inner product space uW of a K-inner product space W is the underlying R-vector space endowed with the euclidean inner product…”

“…He then obtains the stable splitting of O/O(m), U/U (m) and Sp/Sp(m) by passing to colimits. Crabb [5] and Ullman [24] have obtained certain equivariant refinements of some of Miller's splittings for certain Stiefel manifolds of finite-dimensional representations of specific compact Lie groups. Our results are in an entirely different direction.…”

Multiplicative structure in the stable splitting of ΩSLn(C)

“…What are the proper equivariant analogues of our result? See for example [Ull12], [Tyn17]. both for their mathematical expertise and their consistent encouragement; many of the ideas in this paper grew out of their suggestions.…”

Suspension Splittings and Self-maps of Flag Manifolds

“…These suspension splittings also fit into a larger framework that considers stable decompositions of homogeneous spaces. The classic example of this is Miller's stable splitting of Stiefel manifolds [16], which inspired a great many variants and refinements (e.g., [12,18,24,25]). In those cases, the stable feature is prominent in the sense that multiple suspensions are usually needed to realize the decomposition, whereas in our case the decomposition occurs after a single suspension.…”