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An equivariant generalization of the Miller splitting theorem

Harry E Ullman

Algebraic & Geometric Topology 12 (2012) 643–684

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.

isometry, Miller splitting, cofibre sequence, functional calculus, Gysin map, residue
Mathematical Subject Classification 2010
Primary: 55P42, 55P91, 55P92
Received: 31 March 2011
Revised: 28 November 2011
Accepted: 20 December 2011
Published: 8 April 2012
Harry E Ullman
School of Mathematics and Statistics
University of Sheffield
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