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2012 An equivariant generalization of the Miller splitting theorem
Harry E Ullman
Algebr. Geom. Topol. 12(2): 643-684 (2012). DOI: 10.2140/agt.2012.12.643

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.

Citation

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Harry E Ullman. "An equivariant generalization of the Miller splitting theorem." Algebr. Geom. Topol. 12 (2) 643 - 684, 2012. https://doi.org/10.2140/agt.2012.12.643

Information

Received: 31 March 2011; Revised: 28 November 2011; Accepted: 20 December 2011; Published: 2012
First available in Project Euclid: 19 December 2017

MathSciNet: MR2914615
zbMATH: 06035490
Digital Object Identifier: 10.2140/agt.2012.12.643

Subjects:
Primary: 55P42, 55P91, 55P92

Rights: Copyright © 2012 Mathematical Sciences Publishers

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