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

Arone, Greg

doi:10.1090/S0002-9939-00-05794-4

The Mitchell-Richter filtration of loops on Stiefel manifolds stably splits
HTML articles powered by AMS MathViewer

by Greg Arone PDF

Proc. Amer. Math. Soc. 129 (2001), 1207-1211 Request permission

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.

References

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, DOI 10.1017/S0305004100055298
M. C. Crabb, On the stable splitting of $\textrm {U}(n)$ and $\Omega \textrm {U}(n)$, Algebraic topology, Barcelona, 1986, Lecture Notes in Math., vol. 1298, Springer, Berlin, 1987, pp. 35–53. MR 928822, DOI 10.1007/BFb0082999
M. C. Crabb and S. A. Mitchell, The loops on $\textrm {U}(n)/\textrm {O}(n)$ and $\textrm {U}(2n)/\textrm {Sp}(n)$, Math. Proc. Cambridge Philos. Soc. 104 (1988), no. 1, 95–103. MR 938454, DOI 10.1017/S0305004100065269
T.G. Goodwillie, Calculus III: the Taylor series of a homotopy functor, preprint, 1996.
Haynes Miller, Stable splittings of Stiefel manifolds, Topology 24 (1985), no. 4, 411–419. MR 816522, DOI 10.1016/0040-9383(85)90012-6
Hermann Kober, Transformationen von algebraischem Typ, Ann. of Math. (2) 40 (1939), 549–559 (German). MR 96, DOI 10.2307/1968939