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An equivariant generalization of the Miller splitting
theorem
Harry E Ullman
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Algebraic & Geometric Topology 12
(2012) 643–684
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Abstract
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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.
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Keywords
isometry, Miller splitting, cofibre
sequence, functional calculus, Gysin map, residue
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Mathematical Subject Classification
Primary: 55P42, 55P91, 55P92
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Publication
Received: 31 March 2011
Revised: 28 November 2011
Accepted: 20 December 2011
Published: 8 April 2012
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