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THE MITCHELL-RICHTER FILTRATION OF LOOPS ON STIEFEL MANIFOLDS STABLY SPLITS

By Greg Arone and Communicated Ralph Cohen

Abstract

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. 1. Statement of main results Let F be R or C. Let U be an infinite-dimensional vector space over F with a positive-definite inner product. Let J be the category of finite-dimensional vector subspaces of U with morphisms being linear maps respecting the inner product. We will use the letters U, V, W to denote objects of J. Let Aut(n) beO(n) or U(n) ifF is R or C respectively. For V an object of J,letSnV be the one-point compactification of Fn ⊗ V. SnV is a sphere with a natural action of Aut(n). Here is our main theorem: Theorem 1.1. Let F: J → Spaces ∗ be a continuous functor from J to based spaces. Suppose that there exists a filtration of F by sub-functors Fn such that F0(V) ≡∗,andforalln ≥ 1 the functor V ↦ → Fn(V)/Fn−1(V): = homotopy cofiber of the map Fn−1(V) → Fn(V) is (up to a natural weak equivalence) of the form V ↦ → (Xn ∧ S nV) h Aut(n): = (Xn ∧ S nV ∧ EAut(n)+) Aut(n) where Xn is a based space equipped with a based action of Aut(n). Then the filtration stably splits, i.e., there is a natural stable equivalenc

Year: 2000
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