Algebraic & Geometric Topology

An equivariant generalization of the Miller splitting theorem

Harry E Ullman

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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.

Article information

Algebr. Geom. Topol., Volume 12, Number 2 (2012), 643-684.

Received: 31 March 2011
Revised: 28 November 2011
Accepted: 20 December 2011
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P42: Stable homotopy theory, spectra 55P91: Equivariant homotopy theory [See also 19L47] 55P92: Relations between equivariant and nonequivariant homotopy theory

isometry Miller splitting cofibre sequence functional calculus Gysin map residue


Ullman, Harry E. An equivariant generalization of the Miller splitting theorem. Algebr. Geom. Topol. 12 (2012), no. 2, 643--684. doi:10.2140/agt.2012.12.643.

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