## Algebraic & Geometric Topology

### An equivariant generalization of the Miller splitting theorem

Harry E Ullman

#### 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.

#### Article information

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

Dates
Revised: 28 November 2011
Accepted: 20 December 2011
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715367

Digital Object Identifier
doi:10.2140/agt.2012.12.643

Mathematical Reviews number (MathSciNet)
MR2914615

Zentralblatt MATH identifier
06035490

#### Citation

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. https://projecteuclid.org/euclid.agt/1513715367

#### References

• G Arone, The Mitchell–Richter filtration of loops on Stiefel manifolds stably splits, Proc. Amer. Math. Soc. 129 (2001) 1207–1211
• G Arone, The Weiss derivatives of $B{\rm O}(-)$ and $B{\rm U}(-)$, Topology 41 (2002) 451–481
• M F Atiyah, F Hirzebruch, Vector bundles and homogeneous spaces, from: “Proc. Sympos. Pure Math., Vol. III”, Amer. Math. Soc. (1961) 7–38
• M C Crabb, On the stable splitting of ${\rm U}(n)$ and $\Omega {\rm U}(n)$, from: “Algebraic topology, Barcelona, 1986”, (J Aguadé, R Kane, editors), Lecture Notes in Math. 1298, Springer, Berlin (1987) 35–53
• T G Goodwillie, Calculus II: Analytic functors, $K$–Theory 5 (1991/92) 295–332
• M Hovey, J H Palmieri, N P Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128, no. 610, Amer. Math. Soc. (1997)
• N Kitchloo, Cohomology splittings of Stiefel manifolds, J. London Math. Soc. 64 (2001) 457–471
• L G Lewis, Jr, J P May, M Steinberger, J E McClure, Equivariant stable homotopy theory, Lecture Notes in Math. 1213, Springer, Berlin (1986)
• M A Mandell, J P May, Equivariant orthogonal spectra and $S$–modules, Mem. Amer. Math. Soc. 159, no 755, Amer. Math. Soc. (2002)
• H Miller, Stable splittings of Stiefel manifolds, Topology 24 (1985) 411–419
• D Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75 (1969) 1293–1298
• N P Strickland, Common subbundles and intersections of divisors, Algebr. Geom. Topol. 2 (2002) 1061–1118
• N P Strickland, Multicurves and equivariant cohomology, Mem. Amer. Math. Soc. 213, no. 1001, Amer. Math. Soc. (2011)
• H E Ullman, The equivariant stable homotopy theory around isometric linear map, PhD thesis, University of Sheffield (2010)
• M Weiss, Orthogonal calculus, Trans. Amer. Math. Soc. 347 (1995) 3743–3796