1997American Mathematical Society THE MITCHELL-RICHTER FILTRATION OF LOOPS ON STIEFEL MANIFOLDS STABLY SPLITS GREG ARONE Abstract.We prove that the Mitchell-Richter filtration of the space of loo* *ps on complex Stiefel manifolds stably splits. The result is obtained as a sp* *ecial case of a more general splitting theorem. Another special case is H. Mille* *r's splitting of Stiefel manifolds. The proof uses the theory of orthogonal ca* *lculus developed by M. Weiss. The argument is inspired by an old argument of Goodwillie for a different, but 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 ve* *ctor 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) be O(n) or U(n) if F is R or C respectively. For V an object of J , let SnV be the one-poi* *nt 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 ) *, and for all n 1 the functor V 7! Fn(V )=Fn-1(V ) := homotopy cofiber of theFmapn-1(V ) ! Fn(V ) is (up to a natural weak equivalence) of the form V 7! (Xn^ SnV)hAut(n):= (Xn^ SnV^ 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 equivalence 1_ F ' Fn=Fn-1 n=1 One immediate consequence of theorem 1.1 is H. Miller's stable splitting of Sti* *efel manifolds [C86, Mi85]. To recall what Miller's splitting is, let V be a fixed o* *bject of J , and consider the functor W 7! Mor(V; V W) where Morstands for the space of morphisms in J , i.e., linear isometric inclus* *ions. Following [C86], elements of Mor(V; V W) will be written as pairs (g; h) with g* * 2 hom(V; V ) and h 2 hom(V; W). The Stiefel manifold is filtered by the subspaces Rn(V ; W) = {(g; h) 2 Mor(V; V W) | dim(ker(g - 1))? n} where n = 0; : :;:dim(V ). It is well known (see [C86, page 42], for instance) * *that the maps Rn-1(V ; W) ! Rn(V ; W) are cofibrations for all n = 1; : :;:dim(V ) a* *nd __________ 1 Received by the editors June 3, 1999. 1991 Mathematics Subject Classification. 55P35. Key words and phrases. stable splitting, Stiefel manifolds, Weiss' calculus. The author is partially supported by the NSF. 2 GREG ARONE that the n-th subquotient of this filtration Rn(V ; W)=Rn-1(V ; W) is homotopy equivalent (in fact, homeomorphic) to the Thom space n;W) Gn(V )Adnhom(F where Gn(V ) is the grassmanian of n-dimensional subspaces of V , Adnis the ad- joint representation of Aut(n) (considered as a vector bundle over Gn(V )). Cle* *arly, another way to write this Thom space is (SAdn^ Mor(Fn; V )+ ^ SnW)hAut(n) and it is easy to verify that all the identifications in sight are functorial i* *n W . It follows that the filtration of Mor(V; V W) by the subspaces Rn(V ; W) satisfies the hypothesis of theorem 1.1 and therefore stably splits. Thus one obtains Mil* *ler's splitting ([C86, theorem 1.16]). Another consequence of theorem 1.1 is a stable splitting of the Mitchell-Richter filtration of the space of loops on complex Stiefel manifolds. Until the end of the section, F = C and J is the category of complex inner-product spaces. The Mitchell-Richter filtration of the space Mor(V; V W) is a filtration by subspa* *ces Sn(V ; W) described in [C86, page 50] (strictly speaking, the spaces Sn(V ; W) * *do not filter Mor(V; V W), but a certain space of algebraic loops that is weakly equivalent to it if dim(W) > 0. If W = 0 then the space Mor(V; V ) is the group completion of the space of algebraic loops). The maps Sn-1(V ; W) ! Sn(V ; W) are closed cofibrations, and the quotient Sn(V ; W)=Sn-1(V ; W) is naturallynho* *me- omorphic to a certain Thom space ([C86, proposition 2.20]) Sn(V )hom(C ;W). Here Sn(V ) ,! Gn(V ) is a certain subspace of the Grassmanian, and the bundle over Sn(V ) is the pullback of the obvious hom(Cn; W)-bundle over Gn(V ). Let "Sn(V ) be the pullback of the diagram Sn(V ) ,! Gn(V ) Mor(Cn; V ) The space "Sn(V ) as a free action of U(n), and it is clear that there is a nat* *ural homotopy equivalence n;W) nW Sn(V )hom(C ' (S"n(V )+ ^ S )hU(n) Again, it is quite obvious that all the constructions and identifications that * *we used are functorial in W . It follows that the Mitchell-Richter filtration sati* *sfies the hypothesis of theorem 1.1 and therefore stably splits. We obtained the following theorem (originally conjectured by M. Mahowald): Theorem 1.2.The Mitchell-Richter filtration of Mor(V; V W) stably splits. Thus there is a stable equivalence 1_ Mor(V; V W) ' Sn(V ; W)=Sn-1(V ; W) n=1 To the best of our knowledge, theorem 1.2 is new when dim(W) > 1, and this may be the main justification for writing this note. When dim(W) = 1 then there is a homeomorphism Mor(V; V W) ~=SU(V W) and thus Mor(V; V W) can be identified with the space of loops on the special unitary group. So, in the * *case dim(W) = 1 theorem 1.2 recovers the Mitchell-Richter splitting of SU(n). LOOPS ON STIEFEL MANIFOLDS 3 2.Proof of theorem 1.1 It is well known that there exists a large family of functorial filtrations, of* * which the May-Milgram filtration of kkX is the prime example, that stably split. The most general published result in this direction is probably due to Cohen, May a* *nd Taylor [CMT78]. In fact, it turns out that all these splitting results are imme* *diate consequences of the existence of Goodwillie's calculus of homotopy functors. Th* *is was known to Goodwillie for many years. The argument, unfortunately, remains unpublished, but it can be found in some versions of [G96]. It also turns out that theorem 1.1 follows from the existence of Weiss' orthogo* *nal calculus [W95] in almost exactly the same way as the Cohen-May-Taylor theorem follows from Goodwillie's homotopy calculus. Our argument is, therefore, nothing but a straightforward adaptation of an old (and unpublished) argument of Good- willie. Thus, there are no new ideas in this paper (but there is a new result, * *which we hope justifies writing it). Let us recall some definitions and results from [W95]. Definition 2.1.Let F be a continuous functor from J to based spaces. F is polynomial of degree n if the natural map F(V ) ! holim{F(U V ) | 0 6= U Fn+1} is a homotopy equivalence for all V . Note that the indexing category for the homotopy limit, i.e., the category of n* *on- zero vector subspaces of Fn+1, is a topological category, and the homotopy limit depends on this topology in an obvious way. The following proposition is obvious from the definition Proposition 2.2.Let F1 ! F2 be a natural transformation of two polynomial functors of degree n. Then the homotopy fiber of this transformation is a poly- nomial functor of degree n. Next we introduce an important class of polynomial functors. Theorem 2.3.Let be a spectrum with an action of Aut(n). Define F(V ) by i j F(V ) := 1 ^ SnV hAut(n) The functor F is polynomial of degree n. Proof.This is the content of example 5.7 in [W95]. Only the real case is consid* *ered there, but it is clear that the same analysis works in the complex case. In fac* *t,_the functor F is homogeneous of degree n, as will be explained below. |_| In particular,iif Xn is a based space with an action of Aut(n) then the functor j V 7! 1 1 Xn^ SnV hAut(n)is polynomial of degree n. Now suppose F satisfies the hypothesis of theorem 1.1. Consider the functor 1 1 (Fn). We claim that it is polynomial of degree n. In fact, for any k 0, the functor 1 1+k(Fn) is polynomial of degree n. We prove it by induction on 4 GREG ARONE n. For n = 0 there is nothing to prove. Suppose it is true for n - 1. Consider * *the fibration sequence of functors i j 1 1+k(Fn) ! 1 1+k Xn^ SnVhAut(n)! 1 1+k+1(Fn-1) The last two functors are polynomial of degree n by the remark following the- orem 2.3 and by the induction hypothesis. It follows, by proposition 2.2, that 1 1+k(Fn) is polynomial of degree n. The next step is to recall that given a functor F , there exists, in some sense* *, a best possible approximation of F by a polynomial functor of degree n (the n-th "Taylor polynomial" of F ). Given a functor F , define the functor TnF by TnF(V ) = holim{F(V U) | 0 6= U Fn+1} the functor TnF comes equipped with a canonical natural transformation F ! TnF . Define the functor PnF to be the homotopy colimit TnF ! TnTnF ! . .!.TknF ! . . . For a general functor F , the functor PnF is polynomial of degree n, and it is, in some sense, the best possible approximation of F by a polynomial functor of degree n. But we will not need this. We do need the observation that if F is polynomial of degree n then, by the very definitions, the map F(V ) ! TnF(V ) (and therefore also the map F(V ) ! PnF(V )) is an equivalence for all V . We w* *ill also need the fact if F is as in theorem 2.3 then F is homogeneous of degree n, i.e., Pn-1F ' *. This is proved in [W95] in the real case, and the proof for the complex case is similar. We will also need the following proposition, whose pro* *of is obvious (similar to the proof of proposition 2.2) Proposition 2.4.Let F1 ! F2 ! F3 be a fibration sequence of functors. It induces fibration sequences TnF1! TnF2! TnF3 and PnF1! PnF2! PnF3. Now suppose again that F satisfies the hypothesis of theorem 1.1. Consider the fibration sequence of functors 1 1 Fn-1! 1 1 Fn ! 1 1 Fn=Fn-1 We saw that the first functor in this sequence is polynomial of degree n - 1, t* *he second one is polynomial of degree n, and the third one is homogeneous of degree n. Consider the following diagram 1 1 Fn-1 ! 1 1 Fn ! 1 1 Fn=Fn-1 #' # # Pn-11 1 Fn-1 !' Pn-11 1 Fn ! Pn-11 1 Fn=Fn-1' * In this diagram, the rows are fibration sequences, the bottom right space is co* *n- tractible, and the arrows marked with ' are weak equivalences. It follows that 1 1 Fn-1is a homotopy retract of 1 1 Fn. It is easy to see from the defini- tions that the retraction is an infinite loop map, and thus 1 Fn ' 1 Fn-1_ Fn=Fn-1 The proof of theorem 1.1 is completed by induction on n. Presumably, it is possible to write explicit splitting maps, by writing explici* *t models for the functors PiFn, but it seems more trouble than it is worth, at this stag* *e. LOOPS ON STIEFEL MANIFOLDS 5 Acknowledgement:_I would like to express warmest thanks to Bill Richter for many helpful emails and for his enthusiasm for the questions discussed here. Bill de* *clined to be a co-author, but without him this paper (for better or for worse) would n* *ot have been written. References [CMT78]F.R. Cohen, J.P. May and L.R. Taylor, Splitting of certain spaces CX, Ma* *th. Proc. Cambridge Philos. Soc. 84 (1978) 3, 465-496. [C86] M.C. Crabb, On stable splitting of U(n) and U(n), Lecture Notes in Math. * *1298 (1986), 35-53. [G96] T.G. Goodwillie, Calculus III: the Taylor series of a homotopy functor, p* *reprint, 1996. [Mi85]H. Miller, Stable splittings of Stiefel manifolds, Topology 24 (1985), 41* *1-419. [W95] M. Weiss, Orthogonal calculus, Trans. Amer. Math. Soc. 347, Number 10 (19* *95), 3743- 3796. [Erratum: Trans. Amer. Math. Soc. 350, Number 2 (1998), 851-855.] (G. Arone) The University of Chicago, Department of Mathematics, Chicago, IL 60* *637 E-mail address: arone@math.uchicago.edu