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