STABLE SPLITTINGS AND THE DIAGONAL
NICHOLAS J. KUHN
October 14, 1999. Revised January 6, 2000.
Abstract.Many approximations to function spaces admit natural stable spl*
*it-
tings, with a typical example being the stable splitting of a space CdX *
*ap-
proximating ddX. With an eye towards understanding cup products in the
cohomology of such function spaces, we describe how the diagonal interac*
*ts
with the stable splitting. The description involves group theoretic tran*
*sfers.
In an appendix independent of the rest of the paper, we use ideas from
Goodwillie calculus to show that such natural stable splittings are uniq*
*ue, and
discuss three different constructions showing their existence.
1.Introduction
Many mapping space functors of a topological space X admit homotopy equiv-
alent combinatorial models CX constructed in the following way. Let be the
category with objects the finite sets 0 = ; and n = {1; 2; : :;:n}, n 1, and m*
*or-
phisms the injective functions. Following the terminology of [CMT ], a coeffic*
*ient
system is a contravariant functor
C : op! topological spaces:
Meanwhile, a based space X defines a covariant functor
Xx : ! topological spaces
by the formula Xx (n) = Xn. Then CX is defined to be the coend of C and Xx :
a1
CX = ( C(n) x Xn)=(~)
n=0
where (ff*(c); x) ~ (c; ff*(x)) for all c 2 C(n); x 2 Xm ; and ff : m ! n.
An important family of examples is the case when C(n) = Cd(n), the space of
`n disjoint little d-cubes in a big d-cube', i.e. the nth space in the Boardma*
*n-
Vogt little d-cubes operad Cd. Here the associated space CX maps naturally to
the iterated loopspace ddX, and this is a homotopy equivalence if X is path
connected.
The coefficient systems C arising in this family are -free: for all n, the ac*
*tion
of the symmetric group n on the space C(n) is free. This is typical of coeffici*
*ent
____________
1991 Mathematics Subject Classification. Primary 55P35; Secondary 55P42.
Partially supported by the N.S.F.
1
2 KUHN
systems arising in both of the most common ways: from -free (unital) operads,
and from configuration spaces. For the rest of the paper we will assume that all
our coefficient systems are -free.
The space CX has a natural increasing filtration, with
na
FnCX = ( C(m) x Xm )=(~):
m=0
W1
Letting DnX = FnCX=Fn-1CX = C(n)+ ^n X^n; and DX = n=0 DnX, there
is a natural stable equivalence1 [CMT ] that splits the filtration
s : (CX)+ -!~DX:
In the author's experience, many applications of the combinatorial models CX
arise from an analysis of how the stable splitting interacts with unstable stru*
*cture.
By and large, such interactions were analyzed by the early 1980's (see e.g. [K1*
* , K2,
K3, K4, LMMS ]), but the most fundamental unstable structure, the diagonal on
CX, seems to have been not previously studied from this point of view. It is the
point of this note to remedy this.
Our immediate interest in this question comes from the fact that recent proje*
*cts
by D. Tamaki [T ], N. Strickland [S1], and the author (joint with S. Ahearn), h*
*ave
all involved studying the product structure on h*(CX) for various C as above and
for various multiplicative generalized cohomology theories h*. Understanding su*
*ch
cup product stucture amounts to understanding the homomorphisms induced by
the maps l in h*-theory, where we let l: CX ! (CX)l be the l-fold diagonal
map. For this, it suffices to study lstably.
Definitions 1.1.Let l: DX ! (DX)^l be the stable composite
-1 l+ s^l
DX s-!~(CX)+ -! (CX)l+-!~(DX)^l:
Given n and I = (i1; : :;:il), let DIX = Di1X ^ . .^.DilX, and let nI: DnX !
DIX be the (n; I)thcomponent of l.
We will often identify DIX with the homeomorphic space C(I)+ ^I X^|I|, where
C(I) = C(i1) x . .x.C(il), I = i1x . .x.il, and |I| = i1 + . .+.il.
Our main theorem, Theorem 2.4, is a description of each nIas a sum of more
familiar maps arising from transfers and diagonals, with the sum indexed by equ*
*iv-
alence classes of covers of n by i1; i2; : :;:il. An immediate corollary is ea*
*sy to
state:
Theorem 1.2. Suppose I = (i1; : :;:il).
(1) nI' * unless |I| n and it n, for all t.
(2) If |I| = n, then nI: DnX ! DIX is the composite
C(n)+ ^n X^n ! C(n)+ ^I X^n ! C(I)+ ^I X^n
____________
1We will, in general, not distinguish a space Z from its suspension spectrum*
* 1 Z, but will
try to alert readers as to when maps are defined only stably.
STABLE SPLITTINGS AND THE DIAGONAL 3
where the first map is the (stable) transfer induced by the inclusion I n and
the second map is induced by the I-equivariant map C(n) ! C(I) arising from the
coefficient system structure.
(3) If X is a coH-space, then furthermore nI' * if |I| > n.
Remarks 1.3.
(1) If l = 2 and X is a coH-space, the theorem says that the composite
-1 l+ s^s
DX s-!~(CX)+ -! (CX x CX)+ -!~DX ^ DX
is the wedge over all i + j = n of the maps DnX ! DiX ^ DjX which are the
composites
C(n)+ ^n X^n -Tr!C(n)+ ^ixj X^n ! (C(i) x C(j))+ ^ixj X^n:
Here Tr is the transfer associated to ix j n.
When X is a suspension and C = Cd, so that CX ' ddX, a closely related
statement was given by D. Tamaki [T , Prop. 3.5]. More recently, N. Strickland *
*has
also noted and used a version of the C = C1 case [S1, S2] (with X = S2). In th*
*is
last situation, the second map in the composite can be viewed as the identity in
homotopy. The methods used by both of these authors heavily use that their X is*
* a
coHspace, and we hope that our paper offers some perspective on their arguments.
(2) In [A ], G. Arone gave an explicit model for the Goodwillie tower associate*
*d to
the functor
X 1 Map*(K; X);
where K is a fixed finite complex. Working with this, the author, together with
S. Ahearn, have been studing multiplicative properties of the associated spectr*
*al
sequences computing h*(Map*(K; X)), where h* is a multiplicative cohomology
theory. This joint project overlaps with the present paper in that we can recov*
*er
the first two parts of Theorem 1.2 in the many classical situations (see [B ]) *
*for
which there is a coefficient system C and natural number m, both depending on K,
and a natural map
CX ! Map*(K; m X)
which is an equivalence for all connected X. It was partially a desire to `chec*
*k our
work' that led to the present note.
(3) Related to (2), we observe that the first statement of Theorem 1.2 is predi*
*cted
by Goodwillie calculus.
The organization of the paper is as follows. The main theorem is stated in se*
*ction
2, after the maps arising in its statement have been carefully defined. This is*
* then
proved in section 3, following the strategy followed in [K4 ], where we analyze*
*d the
James-Hopf maps in the same way that we here analyze diagonals. In the short
Appendix A, we discuss statement (3) of Theorem 1.2 and similar simplifications
of the results in [K4 ] which arise when X is a coH-space.
In proving Theorem 2.4, we use a simple characterization from [K4 ] of the st*
*able
equivalence of [CMT ]. A reader might wonder if other people's constructions *
*of
stable splittings yield the same maps. Appendix B, which can be read independen*
*tly
of the rest of the paper, addresses this question. Using ideas from Goodwillie
4 KUHN
calculus, we show that there is a unique equivalence that `splits the filtratio*
*n' and is
appropriately `natural', and discuss three different constructions of such spli*
*ttings.
The author would like to thank the referee for a careful critique of an earlier*
* version
of this section.
2. The main theorem
We begin with various definitions of a combinatorial nature.
Definitions 2.1.Fix n and I = (i1; : :;:il). A partial I cover of n is an l-tup*
*le
S = (S1; : :;:Sl) of subsets of n so that St has cardinality it for all t. Such*
* an S is
a cover if n is the union of S1; : :;:Sl.
We let S(n; I) be the set of all partial I covers of n. The tautological acti*
*on of
n on n induces an action on S(n; I) in the obvious way, and two partial I covers
of n are said to be equivalent if they are in the same n-orbit.
Definitions 2.2.Fix S = (S1; : :;:Sl) 2 S(n; I).
(1) For 1 t l, let ffS;t: it! n be the unique monic order preserving map with
image St.
(2) Let ffS : |I| ! n be the map defined by the maps ffS;t, viewing |I| as the *
*disjoint
union i1 + . .+.il.
(3) Let S n be the stablizer subgroup.
(4) Let aeS : S ! I be the group homomorphism with tthcomponent aet: S !
itdefined by aet(oe)(x) = ff-1S;t(oeffS;t(x)).
Given a partial cover S 2 S(n; I), we now proceed to construct stable maps
S : (C(n) xn Xn)+ ! (C(I) xI X|I|)+ :
natural in both X and -free coefficient systems C.
Firstly, the maps ffS;t: it! n induce
pS : C(n) -! C(I) = C(i1) x . .x.C(il):
Then we define
S : Xn -! X|I|;
by letting, the tth component of S(x) be xffS(t), if x = (x1; : :;:xn). Both th*
*ese
maps are equivariant with respect to aeS : S ! I. Thus they assemble into an
unstable map
fS : (C(n) xS Xn)+ ! (C(I) xI X|I|)+ :
The stable map S is then the stablization of this map precomposed with the stab*
*le
transfer
TrnS : (C(n) xn Xn)+ ! (C(n) xS Xn)+ :
If the partial cover S is a cover, then ffS : |I|! n is onto. Then S prolongs*
* to
a natural map
S : X^n -! X^|I|;
and thus S prolongs to a natural stable map
S : DnX -! DIX:
STABLE SPLITTINGS AND THE DIAGONAL 5
The proof of [K4 , Lemma 2.6], based on the naturality of the transfer and the
behavior of extended power constructions under conjugation, generalizes to the
setting here to prove the next lemma.
Lemma 2.3. If S is equivalent to S0 then S is homotopic to S0.
Our main theorem now goes as follows.
X
Theorem 2.4. nI' S : DnX ! DIX, with the sum ranging over equiva-
S
lence classes of I covers of n.
Note that the first two statements of Theorem 1.2 follow. There are no I cove*
*rs
of n unless |I| n and it n, for all t. If |I| = n, there is a single equivalen*
*ce class
of I covers of n represented by the disjoint union decomposition n = i1 + . .+.*
*il,
and for this S, S is the composite of Theorem 1.2(2).
Let S be an I cover of n. In the appendix we check that if n < |I| and X is a
coH-space, then the diagonal S : X^n ! X^|I|is S-equivariantly null. Thus
S will be null, and so Theorem 1.2(3) also follows from Theorem 2.4.
3.Proof of the main theorem
We begin this section by describing a characterization of s : (CX)+ - ! DX
which we noted in [K4 ].
To explain this, we first note that, since Df : D(Y ) ! D(Z) is defined for a*
*ll
stable maps f : Y ! Z, the fact that X+ ! X is stably split epic implies the ne*
*xt
lemma.
Lemma 3.1. For all spaces X, the projection D(X+ ) ! D(X) is stably split epic,
up to homotopy.
Thus, to understand s : (CX)+ -! DX, one need just study s : (C(X+ ))+ -!
D(X+ ), and to do this, we make the following observation. There is a natural
(unstable) homeomorphism of spaces
a1 1_
s0: C(X+ )+ = ( C(n) xn Xn)+ = (C(n)+ ^n (X+ )^n) = D(X+ ):
n=1 n=1
As in [K4 ], we have the next definitions.
Definitions 3.2.Let : D(X+ ) ! D(X+ ) be the composite of stable equiva-
lences
0)-1 s
D(X+ ) _____-(s~C(X+ )+ _____-~D(X+ ):
Given n and m, let n;m : Dn(X+ ) ! Dm (X+ ) be the (n; m)thcomponent of .
Thanks to the last lemma, the next proposition, describing the maps n;m in
terms of more fundamental maps, can be regarded as a characterization of s.
6 KUHN
Proposition 3.3.n;m ' * if n < m, and for n m, n;m is the composite
Dn(X+ ) ! (C(n) xm xn-m Xn)+ ! Dm (X+ ) ^ Dn-m (X+ ) ! Dm (X+ )
where the first map is the stable transfer associated to m xn-m n, the second
map is induced by the map C(n) ! C(m) x C(n - m) associated to the coefficient
system, and the last map is the projection.
In the case when C = C1 this was shown in [K4 ]. In fact, this was shown
twice. For s constructed as the adjoint to the James-Hopf maps of [CMT ], this*
* is
[K4 , Proposition 4.5]. For s constructed using R. Cohen's `stable proofs of st*
*able
splittings' [C ], this is [K4 , Theorem A.2]. In both cases, the proofs given *
*there
generalize easily to general coefficient systems C. (This was already hinted a*
*t in
[K4 ]: in that paper, see the last paragraph of section 5.)
We now turn to proving Theorem 2.4, following the strategy used in [K4 , x4].
Recall that we are trying to identify nI, the (n; I)thcomponent of the compos*
*ite
-1 l+ s^l
l: DX s-!~(CX)+ -! (CX)l+-!~(DX)^l:
The observations above show that it suffices to identify this component with X
replaced by X+ . In this case, nIis the (n; I)thcomponent of the composite
-1 l+ ^l
l: D(X+ ) -!~D(X+ ) -! D(X+ )^l-!~ D(X+ )^l;
where is the stable map described by Proposition 3.3.
Let nI be the (n; I)thcomponent of the composite
l+ ^l^l ^l
D(X+ ) -! D(X+ ) -!~ D(X+ ) :
X
Proposition 3.4.nI ' S : (C(n) xn Xn)+ ! (C(I) xI X|I|)+ ; with the
S
sum ranging over equivalence classes of partial I covers of n.
Assuming this key result for the moment, we finish the proof of Theorem 2.4.
If S is a partial I cover of n, we let S + k denote S viewed as a partial cov*
*er
of n + k, under the inclusion n ! n + k. Note that every partial I cover of n is
equivalent to one of the form S + n - m with S an I cover of m, and S is unique
up to equivalence. (The number n - m will be the number of `uncovered' points of
the original partial cover.)
Lemma 3.5. If n m and S is an I cover of m, then
S+n-m ' S O n;m : (C(n) xn Xn)+ ! (C(I) xI X|I|)+ :
This follows from standard properties of the transfer as in [K4 , Proof of 4.*
*13].
TheoremX2.4 now follows. Let "l : D(X+ ) ! D(X+ )^l have (n; I)thcomponent
given by S : (C(n) xn Xn)+ ! (C(I) xI X|I|)+ ; with the sum ranging over
S
equivalence classes of I covers of n. Proposition 3.4 and Lemma 3.5 combine to *
*say
^lO l+' "lO :
STABLE SPLITTINGS AND THE DIAGONAL 7
Precomposing with -1 yields the main theorem: l' "l.
It remains to prove Proposition 3.4. If I = (i1; : :;:il), let I0= (n-i1; : :*
*;:n-il).
One can view Ix I0as a subgroup of lnin the obvious way, and |I| + |I0| = nl.
Then nI will be the composite of the stable map
(C(n) xn Xn)+ -! (C(n)lxln Xnl)+ -Tr!(C(n)lxIxI0 Xnl)+
and the stablization of the unstable map
(C(n)lxIxI0 Xnl)+ ! ((C(I) x C(I0)) xIxI0 Xnl)+ ! (C(I) xI X|I|)+ :
The first of these maps, Tr O , can be analyzed using the naturality of the
transfer with respect to pullbacks of covering spaces. To describe our pullbac*
*k,
we need yet more notation. If S = (S1; : :;:Sl) is a partial I cover of n, let *
*S0 =
(n - S1; : :;:n - Sl) denote the complementary partial I0 cover of n. Note that
S0= S and that (S0)0= S.
Lemma 3.6. There is a pullback diagram
` n l nl
SC(n) xS X - ! C(n) xIxI0 X
# #
C(n) xn Xn - ! C(n)lxln Xnl
where S runs through the partial I covers of n, and the component maps
C(n) xS Xn ! C(n)lxIxI0 Xnl
are induced by aeS x aeS0: S ! I x I0and S x S0: Xn ! Xnl:
Note that Proposition 3.4 follows from this.
By naturality, to verify the lemma, it suffices to check that there is a pull*
*back
diagram `
` aeSxaeS0
SBS ________-SB(I x I0)
# #
Bn ________- Bln
where S runs through the partial I covers of n.
This we check using the double coset formula, i.e. we check that there is an
isomorphism of n-sets
a
n=S ' ln=(I x I0);
S
where n acts diagonally on ln. But this is easy to see. By definition, there is*
* an
isomorphism of n-sets a
n=S ' S(n; I):
S
Now note that S(n; I) = S(n; i1) x . .x.S(n; il) is acted on transitively by ln*
*with
isotropy group precisely I x I0.
8 KUHN
Appendix A. Simplifications when X is a coHspace.
In this appendix, we verify the intuitively reasonable fact stated at the end*
* of
x2: if S is an I cover of n and |I| > n, so that the diagonal is involved in an
interesting way in the construction of S : DnX ! DIX, then S is null if X is a
coH-space. We verify this by proving a more general lemma, which applies also to
the constructions of [K4 ].
If A is a finite set, and X is a pointed space, we let WA (X) XA denote the
fat wedge, and X^A = XA =WA (X). Then X^A is a covariant functor of X, and a
contravariant functor of A if we restrict maps in the category of finite sets t*
*o just
the epimorphisms. Given an epimorphism of finite sets ff : B ! A, we write the
induced map as ff: X^A ! X^B to emphasize that it is a generalized diagonal
map.
It is easy to check that there are natural homeomorphisms X^(AqB) ' X^A ^
X^B and X^(AxB) ' (X^A )^B .
If a group G acts on a finite set A, then X^A will naturally be a G-space. Gi*
*ven
an epimorphism of G-sets ff : B ! A, and a 2 A, let Ga G be the stabilizer
subgroup, and let Ba = ff-1(a) B. Then Ba is a Ga-set.
Lemma A.1. Let ff : B ! A be an epimorphism of G-sets. Suppose that there
exists an element a 2 A such that Ga acts trivially on Ba, and Ba contains more
than one element. Then ff: X^A ! X^B will be G-equivariantly null if X is a
coH-space.
To see this, we first note that there is a decomposition of ff : B ! A as a G*
*-map:
a a
G xGa Ba ! G=Ga:
Ga2G\A Ga2G\A
Thus ffis the smash product, over the G-orbits in A, of G-maps
ffa: X^G=Ga ! X^(GxGaBa);
where ffa : G xGa Ba ! G=Ga is the evident epimorphism.
Choosing a 2 A as in the hypotheses, we will have G xGa Ba = G=Ga x Ba, so
that ffacan be identified with
^G=Ga : X^G=Ga ! (X^Ba )^G=Ga;
where : X ! X^Ba is a diagonal map into a smash product of more than one
copy of X. If X is a coH-space, then will be (nonequivariantly) null, so that
^G=Ga will be G-equivariantly null, establishing the lemma.
For the purposes of verifying the statement at the end of x2, let A = n, B = *
*|I|,
ff = ffS : |I| ! n, and G = S, where S is an I cover of n. Then, for all a 2 A,*
* Ga
acts trivially on Ba, so the lemma applies.
As a second application, in [K4 ] we defined an (r; q)-set S with |S| = n to *
*be a
cover of n by r distinct subsets each of cardinality q. As discussed there, if *
*S n
is the stabilizer of this configuration, one gets a homomorphism S ! ro q, and
an epimorphism ffS : rq ! n which is S-equivariant. Letting A = n, B = rq,
G = S, and ff = ffS, we again have that for all a 2 A, Ga acts trivially on Ba,*
* so
the lemma applies.
STABLE SPLITTINGS AND THE DIAGONAL 9
In this case, the topological consequence, a simplification of the main resul*
*t of
[K4 ] in the coH-space case, goes as follows. Let C1 and D1 denote the C and
D constructions associated to the E1 operad C1 . Given an arbitrary coefficient
system C, let jq : CX ! C1 DqX be the qth James-Hopf invariant, i.e. the
unstable map adjoint to the stable splitting projection sq : CX ! DqX. Then let
Jnr;q: DnX ! D1;rDqX be the (n; r)thcomponent of the stable map
-1 jq+ s
DX s-!~(CX)+ -! (C1 DqX)+ -!~D1 DqX:
In [K4 ], we described Jnr;qas a sum of maps associated to the equivalence clas*
*ses
of (r; q)-sets S with |S| = n. If X is a coH-space, and rq > n, our lemma appli*
*es,
and we have the following corollary of [K4 , Theorem 2.3].
Theorem A.2. If X is a coH-space, then Jnr;q' * unless rq = n, and Jrqr;qis the
composite
C(rq)+ ^rq X^rq ! C(rq)+ ^roq X^rq ! (C1 x C(q)r)+ ^roq X^rq:
Here the first map is the (stable) transfer induced by the inclusion r o q rq,
and the second map is induced by maps C(rq) ! C(q)r arising from the coefficient
system structure, together with any r-equivariant map C(rq) ! C1 (r). (As C1 (r)
is contractible, this will exist and be unique up to equivariant homotopy.)
Appendix B. Characterizations of stable splittings
Given a stable map s : (CX)+ ! DX, we let sn : (CX)+ ! DnX denote the
composite (CX)+ -s!DX ! DnX, where the second map is projection onto the
nthwedge summand.
Implicit in the statement `the stable map s : (CX)+ ! DX splits the filtration
of CX' is that s satisfies the following property:
(B.1) For all n 0, the composite(FnCX)+ in,!(CX)+ -sn!DnX
is homotopic to the stablization of the projection
ssn : (FnCX)+ ! FnCX=Fn-1CX = DnX:
We will also insist that s preserve the increasing filtrations on (CX)+ and D*
*X:
(B.2) For all n 0, the composite(FnCX)+ ,! (CX)+ -s! DX
Wn
factors, up to homotopy, throughm=0Dm X ,! DX:
Equivalently, this last property says that the composite
_1
(FnCX)+ ,! (CX)+ -s! DX ! Dm X
m=n+1
is null homotopic.
Lemma B.1. If s satisfies ( B.1) and ( B.2), then s is an equivalence. Furthe*
*rmore,
such an s is uniquely determined by the maps sn.
10 KUHN
Wn
Proof.Let "sn: (FnCX)+ ! m=0 Dm X have `mthcomponent' sm O in.
By assumption, there are homotopy commutative diagrams
(Fn-1CX)+ -! (FnCX)+ -! DnX
# "sn-1 # "sn # Id
Wn-1 W n
m=0 Dm X -! m=0Dm X -! Dm X;
and
(FnCX)+ -! (CX)+
# "sn # s
W n
m=0Dm X -! DX:
By induction on n, the first diagram implies that each "snis a homotopy equival*
*ence.
One consequence of this is that {(CX)+ ; DX} = limn{(FnCX)+ ; DX}; i.e. there
are no nonzero lim1Wterms, as the Mittag-Leffler condition clearly holds in the
inverse system {( nm=0Dm X); DX}. From the second diagram, we thus conclude
that s can be viewed as hocolimn"sn. The map s is thus determined by the maps s*
*n,
and will be a homotopy equivalence. __|_|
For a fixed space X, there can be many maps s satisfying properties (B.1) and
(B.2). Thus it appears ambiguous to refer to the stable splitting of CX. Howeve*
*r, as
we will see below, this ambiguity disappears if one assumes that s is appropria*
*tely
natural in X. Since our proof of this evokes T.Goodwillie's only partially publ*
*ished
theory of the `calculus' of homotopy functors, categories of spaces and spectra*
* that
we work with need to have enough structure to support his proofs.2
Given two functors F and G from pointed spaces to spectra, it is useful to
generalize the notion of a natural transformation from F to G in the following *
*way.
Definition B.2.A weak natural transformation h : F ! G will be a triple
(H; f; g), with H a functor from spaces to spectra, g : H ! G a natural transfo*
*r-
mation, and f : H ! F a natural transformation such that f(X) : H(X) ! F (X)
is a homotopy equivalence for all X.
Note that, if F and G are homotopy functors, then a weak natural transformati*
*on
h : F ! G induces a well defined natural transformation in the homotopy categor*
*y:
h(X) = g(X) O f(X)-1 2 {F (X); G(X)}. Furthermore, using homotopy pullbacks,
one can define the composition of weak natural transformations.3
In particular, viewing CX and DX as functors from the category of pointed
spaces to spectra, it makes sense to say that a weak natural transformation s :
(CX)+ ! DX satisfies properties (B.1) and (B.2).
Our uniqueness result goes as follows.
____________
2Since the details of [Goo] have not been forthcoming, it is a bit hard to k*
*now exactly what this
structure is. (But see the `short review of Goodwillie calculus' in the preprin*
*t [McC ].) However, it
seems clear that the category of coordinate free spectra of [LMMS ], or the cat*
*egory of symmetric
spectra of [HSS] each support more than enough structure.
3To blithely declare that the weak natural transformations lead to an actual*
* category of frac-
tions raises set theoretic questions that we would rather not discuss.
STABLE SPLITTINGS AND THE DIAGONAL 11
Proposition B.3. Let s and t be two weak natural transformations from (CX)+
to DX satisfying properties ( B.1) and ( B.2). Then s and t agree up to homotop*
*y:
for all X,
s(X) ' t(X) : CX ! DX:
Sketch proof.We will show that, up to homotopy, there is a formula for the com-
ponents sn : (CX)+ ! DnX that is independent of s.
Let pn : (CX)+ ! Pn((CX)+ ) be projection onto the nth degree polynomial
functor approximation to (CX)+ in the Goodwillie sense. His general theory says
that Dn(X) is homogeneous of polynomial degree n and that there is a weak natur*
*al
transformation
Pn((CX)+ ) -sn!DnX;
unique up to homotopy, such that in the homotopy category
sn : (CX)+ ! DnX
factors as the composite
(CX)+ -pn!Pn((CX)+ ) -sn!DnX:
Furthermore, using that Pn preserves fibrations up to homotopy, one deduces
that (FnCX)+ has polynomial degree at most n and that the composite
(FnCX)+ -in!(CX)+ -pn!Pn((CX)+ )
is a homotopy equivalence. Thus
sn ' snO pn
' snO (pn O in) O (pn O in)-1 O pn
' (sn O pn) O in O (pn O in)-1 O pn
' sn O in O (pn O in)-1 O pn
' ssn O (pn O in)-1 O pn;
a formula that is evidently independent of s. __|_|
We now discuss the existence of stable splittings.
One construction comes from the proof of the last proposition: with notation *
*as
there, one defines a weak natural transformation sn : (CX)+ ! DnX as the com-
posite of pn : (CX)+ ! Pn((CX)+ ) with Pn((CX)+ ) pnOin-~(FnCX)+ -ssn!DnX.
A second construction is to use the natural transformations arising as adjoin*
*ts
of the James-Hopf maps of [CMT ]. These are combinatorially defined, unstable
maps jn : CX ! 1 1 DnX which generalize James' original construction of
global Hopf invariants. These are natural in X, and are quite explicit, except *
*for
embeddings of certain configuration spaces in R1 .
Finally, a third construction uses the idea of R. Cohen [C ]. (See also [LMMS*
* ,
VII.5].) We briefly review this, tweaked slightly so that the splittings are na*
*tural.
12 KUHN
One starts by observing that, working in a category of spectra (e.g. [EKMM *
*])
with an associative, commutative smash product with unit S = 1 S0, a map of
spectra j : S ! E defines a covariant functor
E^ : ! Spectra
by the formula E^(n) = E^n. Thus, given a coefficient system C, we can define a
filtered spectrum CE to be the coend of 1 C and E^: CE is the coequalizer of
the two apparent maps
_ -! _
C(n)+ ^ E^m C(n)+ ^ E^n:
ff:m!n -! n
This is related to the construction of the introduction by
1 (CX)+ = C(1 (X+ )):
Furthermore, there is an isomorphism of spectra
C(Z _ S) = D(Z);
natural in spectra Z.
The construction CE is functorial with respect to maps under S.
The construction of a natural splitting s now goes as follows. Let X0 denote
the homotopy fiber of the projection X+ ! S. Note that the composite X0 !
X+ ! X is a natural homotopy equivalence. A weak natural transformation s :
(CX)+ ! DX satisfying properties (B.1) and (B.2) is then obtained by applying
the construction C to the natural maps under (and over) S
X+ - X0_ S -! X _ S:
Remark B.4. These days, some readers might find our first, Goodwillie calculus
based, construction of the stable splittings the preferred one. However, the au*
*thor
knows a proof of Proposition 3.3, the characterization of the stable splittings*
* needed
in the body of the note, only using one of the other two constructions. Of the *
*three
constructions given here, it is our last one that seems best suited to use when
studying further structure of the stable splittings.
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[B]C.-F. B"odigheimer Stable splitting of mapping spaces, in Algebraic Topology*
* (Seattle, Wash.,
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Department of Mathematics, University of Virginia, Charlottesville, VA 22903