The Mitchell-Richter filtration of loops on Stiefel manifolds stably splits

Author:
Greg Arone

Journal:
Proc. Amer. Math. Soc. **129** (2001), 1207-1211

MSC (1991):
Primary 55P35

Published electronically:
October 4, 2000

MathSciNet review:
1814154

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the Mitchell-Richter filtration of the space of loops on complex Stiefel manifolds stably splits. The result is obtained as a special case of a more general splitting theorem. Another special case is H. Miller's splitting of Stiefel manifolds. The proof uses the theory of orthogonal calculus developed by M. Weiss. The argument is inspired by an old argument of Goodwillie for a different, but closely related, general splitting result.

**[CMT78]**F. R. Cohen, J. P. May, and L. R. Taylor,*Splitting of certain spaces 𝐶𝑋*, Math. Proc. Cambridge Philos. Soc.**84**(1978), no. 3, 465–496. MR**503007**, 10.1017/S0305004100055298**[C86]**M. C. Crabb,*On the stable splitting of 𝑈(𝑛) and Ω𝑈(𝑛)*, Algebraic topology, Barcelona, 1986, Lecture Notes in Math., vol. 1298, Springer, Berlin, 1987, pp. 35–53. MR**928822**, 10.1007/BFb0082999**[CM88]**M. C. Crabb and S. A. Mitchell,*The loops on 𝑈(𝑛)/𝑂(𝑛) and 𝑈(2𝑛)/𝑆𝑝(𝑛)*, Math. Proc. Cambridge Philos. Soc.**104**(1988), no. 1, 95–103. MR**938454**, 10.1017/S0305004100065269**[G96]**T.G. Goodwillie,*Calculus III: the Taylor series of a homotopy functor,*preprint, 1996.**[Mi85]**Haynes Miller,*Stable splittings of Stiefel manifolds*, Topology**24**(1985), no. 4, 411–419. MR**816522**, 10.1016/0040-9383(85)90012-6**[W95]**Michael Weiss,*Orthogonal calculus*, Trans. Amer. Math. Soc.**347**(1995), no. 10, 3743–3796. MR**1321590**, 10.1090/S0002-9947-1995-1321590-3

Michael S. Weiss,*Erratum: “Orthogonal calculus” [Trans. Amer. Math. Soc. 347 (1995), no. 10, 3743–3796; MR1321590 (96m:55018)]*, Trans. Amer. Math. Soc.**350**(1998), no. 2, 851–855. MR**1458338**, 10.1090/S0002-9947-98-02130-8

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Additional Information

**Greg Arone**

Affiliation:
Department of Mathematics, The University of Chicago, Chicago, Illinois 60637

Address at time of publication:
Department of Mathematics, University of Aberdeen, Aberdeen, AB24 3UE, United Kingdom

Email:
arone@math.uchicago.edu, g.arone@maths.abdn.ac.uk

DOI:
http://dx.doi.org/10.1090/S0002-9939-00-05794-4

Keywords:
Stable splitting,
Stiefel manifolds,
Weiss' calculus

Received by editor(s):
June 9, 1999

Published electronically:
October 4, 2000

Additional Notes:
The author was partially supported by the NSF

Communicated by:
Ralph Cohen

Article copyright:
© Copyright 2000
American Mathematical Society