FILTERED SPECTRA ARISING FROM PERMUTATIVE
CATEGORIES
GREGORY ARONE AND KATHRYN LESH
Abstract.Given a special category C satisfying some mild hypotheses, we
construct a sequence of spectra interpolating between the spectrum assoc*
*iated
to C and the EilenbergMac Lane spectrum HZ. Examples of categories to
which our construction applies are: the category of finite sets, the cat*
*egory
of finitedimensional vector spaces, and the category of finitelygenera*
*ted free
modules over a reasonable ring. In the case of finite sets, our construc*
*tion
recovers the filtration of HZ by symmetric powers of the sphere spectrum*
*. In
the case of finitedimensional complex vector spaces, we obtain an appar*
*ently
new sequence of spectra, {Am }, that interpolate between bu and HZ. We
think of Am as a "buanalogue" of Spm (S) and describe farreaching form*
*al
similarities between the two sequences of spectra. For instance, in both*
* cases
the mth subquotient is contractible unless m is a power of a prime, and *
*in
vkperiodic homotopy the filtration has only k + 2 nontrivial terms. The*
*re is
an intriguing relationship between the buanalogues of symmetric powers *
*and
Weiss's orthogonal calculus, parallel to the not yet completely understo*
*od rela
tionship between the symmetric powers of spheres and the Goodwillie calc*
*ulus
of homotopy functors. We conjecture that the sequence {Am }, when rewrit*
*ten
in a suitable chain complex form, gives rise to a minimal projective res*
*olution
of the connected cover of bu. This conjecture is the buanalogue of a th*
*eorem
of Kuhn and Priddy about the symmetric power filtration. The calculus of
functors provides substantial supporting evidence for the conjecture.
1.Introduction
In this paper we define and investigate a filtration on spectra that arise fr*
*om
the application of Segal's infinite loop space machine to certain permutative c*
*ate
gories. The input to our construction is an "augmented" permutative category C
(Definition 2.1), and the output is a sequence of spectra interpolating between*
* the
spectrum associated to C and the EilenbergMac Lane spectrum H Z. Examples of
categories to which the construction applies are: the category of finite sets,*
* the
category of finitedimensional vector spaces over a field, the category of fini*
*tely
generated free modules over a ring satisfying the dimension invariance property,
and the category of finitelygenerated free groups. The motivating example is t*
*he
category of finite sets, to which Segal's infinite loop space machine associate*
*s the
sphere spectrum, S. For this example, the construction turns out to give the sy*
*m
metric power filtration,
S = Sp1(S) ! Sp2(S) ! . .!.Sp1 (S) ' H Z,
____________
Date: December 24, 2005.
1991 Mathematics Subject Classification. Primary 55P47; Secondary 55P42, 55P*
*48.
The authors were partially supported by the National Science Foundation, Gra*
*nt DMS
0307069.
1
2 GREGORY ARONE AND KATHRYN LESH
which interpolates between the sphere spectrum S and the infinite symmetric pow*
*er
Sp1 (S) ' H Z. The example of finite sets was studied extensively by the second
author in [11, 12], and this paper is, among other things, a generalization of *
*that
work. In effect, we show that Lesh's model for the symmetric power filtration c*
*an
be generalized by lifting the grouptheoretic constructions of [12] to a catego*
*rical
level.
Throughout the paper, we work with a (possibly topological) permutative`cate
gory C that happens to be a disjoint union of small categories, C = nCn. We c*
*all
Cn the "nth component" of C. Let A be the spectrum associated to C.
The first part of the paper gives a general categorical construction of a seq*
*uence
of spectra that interpolate between A and H Z. In Construction 3.8, we define
inductively a sequence of permutative categories
(1.1) C = K0C ! K1C ! . .!.K1 C ' Z 0,
each of which, like C itself, is a disjoint union of components indexed by nonn*
*egative
integers. The category Km C is obtained from Km1 C by using a suitable homotopy
pushout in the category of permutative categories to "kill" the mth component of
Km1 C, which is denoted Km1 Cm and is the bottom nontrivial component. The
category K1 C is defined as the colimit, and the nerve of each component of K1 C
is contractible, so K1 C ' Z 0.
After applying Segal's infinite loop space machine to the sequence (1.1), we
obtain a sequence of spectra
(1.2) A = A0 ! A1 ! . .!.A1 ' H Z,
where Am is the spectrum associated to Km C. We observe in Theorem 3.9 that the
inductive definition of Km C leads to the expression of Am as a homotopy pushou*
*t:
0 1
1 (BKm1 Cm )+ ! Am1
B ? C
(1.3) Am ' hocolimB@ ?y CA.
S
Thus it follows that we have a formula for the subquotients,
Am =Am1 ' 1 (BKm1 Cm ).
In particular, it is a basic property of our construction that the subquotients
Am =Am1 are always suspension spectra.
In the middle part of the paper, which is the most technical portion, we cons*
*ider
the spaces BKm1 Cm that appear in (1.3). Because of the inductive definition
of the categories Km C, the classifying space BKm1 Cm is not easy to analyze *
*in
general. However, by making two sets of simplifying assumptions, we can identify
these spaces in more concrete terms.
The first simplification allows us to construct models for the categories Km *
*C that
are not inductively defined, but rather are obtained by filtering a single larg*
*e com
binatorial construction. We consider categories C that have an underlying "mono
genic" category B, meaning that B is generated under coproducts by a single ob
ject g, the generator (Definition 4.1). We assume that C is a suitable category*
* of
"admissible" isomorphisms of B (Definition 4.2), and we also call C "monogenic."
The categories of finite sets, finitedimensional vector spaces, finitelygener*
*ated
free modules and finitelygenerated free groups are all monogenic. We also as
sume that the underlying category B satisfies an axiom that we call the Injecti*
*vity
FILTERED SPECTRA 3
Axiom (Axiom 5.1). Roughly speaking, the axiom says that the canonical map
from the coproduct to the product is injective. The categories of finite sets, *
*finite
dimensional vector spaces, and finitelygenerated free modules satisfy the Inje*
*ctivity
Axiom, but the category of free groups does not. The assumption of an underly
ing monogenic category that satisfies the Injectivity Axiom allows us to introd*
*uce a
combinatoriallydefined permutative category F1 C with a filtration by permutat*
*ive
categories Fm C (Definition 4.5). These categories model the permutative catego*
*ries
Km C in the sense that Fm C and Km C are related by a chain of monoidal functors
each of which induces an equivalence of classifying spaces (Theorem 7.1).
The second simplification uses further assumptions on C to show that the cat
egory Fm C, which models Km C, can be described as a union of orbit categories,
generalizing the description in [12]. More concretely, if g denotes the generat*
*or of
the monogenic category C and gn denotes its nfold coproduct with itself, we co*
*n
sider the groups Gn = AutC(gn). We define a collection of "standard" subgroups
H(Gn) (Definition 8.1), which correspond in an appropriate sense to partial par*
*ti
tions of gn. We filter this collection by collections Hm (Gn), which correspond*
* to
partitions whose "blocks" have size no larger than m. In the symmetric group ca*
*se,
the filtration corresponds to orbit length and was called the "complexity filtr*
*ation"
in [12].
The following theorem is the main technical result of the middle part of the
paper; it expresses the space BKm C in terms of the classifying spaces of the
collections Hm (Gn). (See Section 8.)
Theorem 8.2. Let C be a monogenic category`satisfying Axioms 5.1, 8.5, and 8.7.
Then BKm C is monoidally equivalent to nBHm (Gn).
The conclusion of the theorem holds for the category of finite sets (with minor
fiddling_see Corollary 8.4 and its proof), the category of finitedimensional v*
*ector
spaces, and the category of finitelygenerated free modules over an integral do*
*main
in which 2 6= 0. On the other hand, the category of free groups does not satisfy
any of our additional axioms. Thus, Construction 3.8 produces a filtration of t*
*he
"Ktheory of free groups" spectrum as in (1.2), but Theorem 8.2 does not apply.
Combining Theorem 8.2 with (1.3), we obtain the following corollary. In the
case of finite sets, it essentially recovers [12], Theorem 1.2.
Corollary 8.3. Given a category C as in Theorem 8.2, let Rm be the collection
Hm1 (Gm ), and let ffl : 1 (BRm )+ ! S be induced by the map BRm ! *. Then
0 1
1 (BRm )+ ! Am1
B ? C
Am ' hocolimB@ ffl?y CA.
S
Corollary 8.3 allows us to show that in the example of finite sets, we obtain*
* the
same filtration as that studied by the second author in [12], and we recover the
filtration from S0 to H Z by symmetric powers of the sphere spectrum.
Corollary 8.4. Let A0 ! A1 ! . .b.e the sequence of spectra resulting from the
application of Construction 3.8 to the category of finite sets and isomorphisms.
Then Am ' Spm(S).
The middle part of the paper is indispensable for the applications, but other*
*wise
is probably rather tedious. One reason for this is that we worked quite hard to
4 GREGORY ARONE AND KATHRYN LESH
achieve sufficient generality to include the case of algebraic Ktheory in Theo*
*rem 8.2
and Corollary 8.3, though we do not establish any further results about it at p*
*resent.
For now we confine ourselves to the remark that our filtration is reminiscent o*
*f, but
not identical to, the rank filtration studied by J. Rognes in [17].
The remainder of the paper is devoted to a detailed study of the example of f*
*inite
dimensional complex vector spaces and unitary isomorphisms, which gives rise to
a filtration of the connective complex Ktheory spectrum, bu. Our construction
gives a sequence of spectra_new, as far as we can tell_interpolating between bu
and H Z:
(1.4) bu = A0 ! A1 ! . .!.A1 ' H Z.
From here to the end of the introduction, we use Am to refer specifically to t*
*he
mth spectrum in (1.4), the unitary case. In view of Corollary 8.4, we think of *
*Am
as a "buanalogue" of Spm (S), terminology that will be further justified below.
Perhaps the most interesting point made in this paper is that the sequence {A*
*m }
has a striking similarity to the sequence {Spm (S)}, a similarity that goes con*
*sid
erably beyond the formal consequences of the general construction. The following
facts about symmetric powers of spheres are the most relevant in describing the
similarity. The notation X refers to the unreduced suspension of a space X. The
notation X"hGdenotes the based homotopy orbits of a based Gaction on X, i.e.,
X"hG= (EG xG X)=(EG xG *).
o There is an equivalence of spectra
(1.5) Spm (S)= Spm1(S) ' 1 (Pm  ^ Sm )"h m,
where Pm is the category of proper, nontrivial partitions of the set m_.*
* ([2],
Theorem 1.14)
o Let p be a prime. The successive quotients Spm (S)= Spm1(S) are trivial
at p except when m = pk for some k, and rationally trivial for m 2.
(This goes back at least to Nakaoka in the fifties, but also follows fro*
*m [2],
Theorems 8 and 1.14.)
o Let p be a prime, k > 0 an integer, and let L(k) be the pcompletion of
k pk1
k Spp (S)= Sp (S). Then L(k) is a wedge summand of a suspension
spectrum. ([14])
o If K(n) denotes the nth Morava Ktheory, then
k pk1
K(n)*Sp p(S)= Sp (S) = 0
for n < k. Moreover, the pcompletion of the sequence
2 1
S = Sp1(S) ! Spp(S) ! Spp (S) ! . .!.Sp (S) ' H Z
k
terminates at Spp (S) in vkperiodic homotopy. (An explicit reference for
this statement is hard to find, but it is folklore knowledge among exper*
*ts.)
The following four theorems summarize some of our main results about the
filtration (1.4)of bu, each analogous to one of the points above about the symm*
*etric
powers of spheres.
Theorem 9.5. There is an equivalence of spectra
Am =Am1 ' 1 (Lm  ^ S2m )"hU(m),
where Lm is the category of proper directsum decompositions of Cm .
FILTERED SPECTRA 5
Theorem 9.7. Let p be a prime. The successive quotients Am =Am1 are trivial at
p except when m = pk for some k 1, and are rationally trivial for m 2.
Because of Theorem 9.7 and itskanaloguekforsymmetric1powers, we will often
work at a prime p and study Spp (S)= Spp (S) and Apk=Apk1. We let T (k) denote
the pcompletion of (k+1)Apk=Apk1, which turns out to be the buanalogue of
L(k), and we write k for an irreducible subgroup of the unitary group U(pk) th*
*at
is described in Section 10.
Theorem 11.1. The spectrum T (k) is a wedge summand of the suspension spec
trum 1 (S2pk)"h k.
Theorem 11.3. Let p be a prime and let K(n) denote the nth Morava Ktheory.
Then K(n)*(Apk=Apk1) = 0 for n < k, and the sequence bu = A0 ! A1 ! Ap !
Ap2! . .!.A1 ' H Z terminates at Apk in vkperiodic homotopy.
To further pursue the analogies between the symmetric powers of spheres and
the spectra Am , we need to bring the GoodwillieWeiss calculus of functors int*
*o the
picture. It turns out that just as the symmetric power filtration is related to*
* the
Goodwillie tower of the identity, so the buanalogue is related to the Weiss to*
*wer
of the functor V 7! BU (V ).
We recall that it was shown in [3] that the Goodwillie tower of the identity
functor, evaluated at S1 and completed at a prime p, has the layers
i kj
(1.6) DpkId(S1) = 1 map S1 ^ Ppk , 1 Sp"
h pk
and the tower converges to S1 up to pcompletion. Similarly, the Weiss tower for
V 7! BU (V ), evaluated at V = C and completed at p, has the layers
i kj
(1.7) DpkBU (C) = 1 map Lpk , 1 Sadpk^ S2p"
hU(pk)
and the tower converges to BU(1) up to pcompletion. To shorten notation, we set
DIk = DpkId(S1)
DU k = DpkBU (C).
It turns out that the infinite loop spaces of the subquotients of our filtrat*
*ion
actually appear (up to a shift) as the layers in the Goodwillie and Weiss tower*
*s, as
described in Theorem 11.2.
Theorem 11.2. For all k 0, there are homotopy equivalences
Bk1 DIk' 1 L(k)
Bk1 DU k' 1 T (k).
The first part of the theorem is not new. It is proved in [2] using equations*
* (1.5),
(1.6), and a selfduality result for Ppk . The proof of the second part of t*
*he
theorem follows a similar plan, depending on Theorem 9.5, equation (1.7), and a
selfduality result for Lpk (Corollary 10.2). The selfduality results for *
*Ppk and
Lpk follow in turn from the relationships of these complexes to Tits buildin*
*gs for
the general linear group and the symplectic group, respectively.
We close the paper with a series of conjectures that are based on the interac*
*tion
of our construction and the calculus of functors. In the example of finite sets*
*, the
6 GREGORY ARONE AND KATHRYN LESH
conjectures relate the Goodwillie tower of the identity functor, the symmetric *
*pow
ers of the sphere spectrum, and the Whitehead conjecture (now a theorem of Kuhn,
and Kuhn and Priddy). In the example of finitedimensional complex vector spaces
and unitary isomorphisms, the conjectures relate the Weiss tower for V 7! BU (V*
* ),
the spectra Apk, and a "buWhitehead conjecture." An interpretation of our con
jectures is that our construction provides "minimal projective resolutions" of *
*H Z
and bu<0> (the fiber of the map bu ! H Z), respectively. (For H Z, this consequ*
*ence
is the main theorem of [7] and [9].) We further conjecture that the Taylor and
Weiss towers themselves provide the contracting homotopies to establish acyclic*
*ity.
The remainder of the paper is organized as follows.
The first part of the paper sets up our general categorical construction. In
Section 2, we recall Thomason's construction of homotopy colimits of permutative
categories and discuss two models for homotopy pushouts. Section 3 introduces
our main construction, which associates with a suitable permutative category C a
sequence of permutative categories Km C.
In the middle part of the paper, we do the main technical work for identifying
the spectra resulting from our construction. In Section 4, we construct a seque*
*nce
of combinatorially defined categories Fm C, and in Sections 5, 6, and 7 we show
them to be equivalent, under mild hypotheses, to the sequence Km C. In Section *
*8,
we show that under further hypotheses, the categories Fm C can be interpreted
in grouptheoretic terms, in the style of [12]. We also observe that if C is t*
*he
category of finite sets then we recover the constructions of [12]. In particula*
*r, the
sequence Fm C gives rise to the filtration of the EilenbergMac Lane spectrum H*
* Z
by symmetric powers of spheres.
In the last part of the paper, we focus our attention on the specific example*
* of
finitedimensional complex vector spaces and its relationship with the example *
*of
finite sets. In Section 9, we describe the subquotients of our filtration in th*
*e unitary
case in terms of the complex of directsum decompositions of a complex vector s*
*pace
introduced in [1]. Section 10 relates the complex of directsum decompositions *
*to
the symplectic Tits building, and uses this to establish a shifted selfduality*
* result
for the complex of directsum decompositions. In Section 11 we use this selfdu*
*ality
to show that the subquotients in our sequence of spectra are equivalent, up to a
suitable shift, to the layers in certain Taylor towers. Finally, in Section 12*
* we
give our conjectures on the relationship of the Taylor towers to our constructi*
*on.
We conjecture that the Taylor towers provide contracting homotopies to a "chain
complex of spectra" that one obtains from our filtration, and we suggest a bu
analogue of the Whitehead Conjecture.
We would like to thank the referee for a careful reading of this paper.
Notation:
If n is a nonnegative integer, we write n_for the set {1, 2, . .,.n}.
If P is a poset, then we write P for its geometric realization, i.e., the n*
*erve of
P regarded as a category. If C is a category, we write either BC or C for its*
* nerve,
depending on context. We also use the letter B to indicate delooping.
If C and D are categories, we write C ' D and say that the categories are wea*
*kly
equivalent if there exists a chain of functors beginning at C and ending at D, *
*possibly
going in both directions, such that the functors induce weak equivalences of sp*
*aces
on the nerves.
FILTERED SPECTRA 7
If F : C ! D is a functor and D is an object of D, we write C # D for the
category over D: objects are pairs (C, f) where f is a morphism F (C) ! D in
D, and morphisms are commuting triangles. We use repeatedly the result that
if C # D is contractible for every object D in D, then F  induces a homotopy
equivalence of nerves ([16], Theorem B).
Throughout the paper, when we refer to a subgroup of a Lie group, we always
mean a closed subgroup.
2. Homotopy colimits of permutative categories
In this section, we recall Thomason's construction of homotopy colimits of di
agrams of permutative categories [20]. Our goal is to present a slightly simpli*
*fied
version of his model of the homotopy pushout for use in our construction in Sec
tion 3. We also compare this construction with the bar construction, which will*
* be
needed in Section 7.
Throughout the paper, we restrict ourselves to "augmented" categories, as spe*
*c
ified in the following definition. Let Z 0 indicate the symmetric monoidal cate*
*gory
whose objects are nonnegative integers with only identity morphisms, with addi
tion as the monoidal operation. We use the term "monoidal functor" to mean "lax
monoidal functor" in the sense of [20].
Definition 2.1. Let C be a symmetric monoidal category. An augmentation of C
is a monoidal functor ffl : C ! Z 0. We write Cn for the full subcategory of C *
*whose
objects are in ffl1(n), and we call Cn the nth component of C. We will say tha*
*t C
is reduced if C0 consists only of the zero object and the identity morphism.
`
We note that if C is augmented, then it is a disjoint union C = n 0Cn. Exam
ples of reduced augmented symmetric monoidal categories are the category of fin*
*ite
sets, with the augmentation given by cardinality, the category of finitedimens*
*ional
vector spaces over a field, with the augmentation given by dimension, the cat
egory of free modules over a ring possessing the dimension invariance property,
with augmentation given by the dimension, and the category of free groups, with
augmentation given by the number of generators.
Recall that a symmetric monoidal category is called "permutative" if the unit
and associativity isomorphisms are actually identity morphisms. Any symmetric
monoidal category is equivalent to a permutative category; for example, take the
equivalent category that has just one object in each isomorphism class. For re*
*a
sons of convenience, we will always work with permutative, rather than symmetric
monoidal, categories. From now on, we assume unless otherwise stated that our
categories are permutative, augmented, and reduced.
The nerve of a permutative category groupcompletes to an infinite loop space,
and hence there is an associated spectrum. We will be interested in looking at
homotopy pushouts of spectra arising in this manner, and we would like to have
a categorical model for this pushout. In other words, we would like to have a
homotopy pushout of permutative categories that corresponds to the homotopy
pushout of spectra. Such a homotopy colimit was constructed by Thomason in
[20].1 For the reader's convenience, we recall the general definition.
____________
1This construction should not be confused with another, more elementary homo*
*topy colimit
of categories considered by Thomason in an earlier paper (the socalled "lax co*
*limit"). The more
8 GREGORY ARONE AND KATHRYN LESH
Definition 2.2 ([20], Construction 3.22). Let F be a functor from a small categ*
*ory
L to the category of permutative categories. Then hocolimF is defined to be the
following category.
o The objects of hocolimF have the form [(L1, X1), . .,.(Ln, Xn)] where n *
*is
a positive integer, Li is an object of L for i = 1, . .,.n, and Xi is an*
* object
of F (Li) for i = 1, . .,.n.
o A morphism ff : [(L1, X1), . .,.(Ln, Xn)] ! [(L01, X01), . .,.(L0m, X0m)*
*] is
given by the following data: (i) a surjection ff : n_! m_, (ii) for each
1 i n, aLmorphism fii: Li! L0ff(i)in L, and (iii) for each 1 j m,
a morphism i2ff1(j)fii(Xi) ! X0jin F (L0j).
In Thomason's words, it is "painful but straightforward" to check that this de
fines a category. There is a monoidal structure on hocolimF , given on objects *
*by
concatenation, and it makes hocolimF into a permutative category.
We will use a slightly simplified version of this construction for a homotopy
pushout square. Let A, B, and C be augmented permutative categories, and assume
that A ! B and A ! C are monoidal functors that preserve the augmentation.
Definition 2.3. Given a diagram
A ! C
?
(2.1) ?y
B
of augmented, permutative categories, we define the augmented permutative cate
gory B A C as follows.
(1) Objects consist of an object in B, an object in C, and a list of objects*
* in A,
i.e., a nonnegative integer n and a map f : n_[ {b, c} ! Obj(A) [ Obj(B)*
* [
Obj(C) such that f(b) 2 Obj(B), f(c) 2 Obj(C), and f(i) 2 Obj(A) for all
i 2 n_.
(2) If [B; A1, . .,.An; C] and [B0; A01, . .,.A0k; C0] are defined by maps f*
* and
f0, respectively, then morphisms [B; A1, . .,.An; C] ! [B0; A01, . .,.A0*
*k; C0]
consist of:
o a surjective map g : n_[ {b, c} ! k_[ {b, c} such that b 7! b and c *
*7! c;
o for all j 2 k_[ {b, c}, a morphism
x2g1(j)f(x) ! f0(j).
(3) The symmetric monoidal product [B; A1, . .,.An; C] [B0; A01, . .,.A0k;*
* C0]
is given by [B B0; A1, . .,.An, A01, . .,.A0k; C C0].
(4) The augmentation is defined by
ffl[B; A1, . .,.An; C] = ffl(B) + ffl(A1) + . .+.ffl(An) + ffl(C).
Proposition 2.4. The diagram
A ! C
?? ?
y ?y
B ! B A C
____________
elementary homotopy colimit corresponds to a homotopy colimit of spaces, while *
*the homotopy
colimit that we are interested in corresponds to a homotopy colimit of spectra.
FILTERED SPECTRA 9
passes (upon application of Segal's construction) to a homotopy pushout diagram*
* of
spectra.
Proof.Let D denote the category resulting from applying Thomason's hocolim
construction to (2.1); upon application of Segal's construction, the square
A ! C
?? ?
y ?y
B ! D
passes to a homotopy pushout diagram of spectra by [20], Theorem 4.1. Up to iso
morphism, a typical object of D has the form [B1, . .,.Bk; A1, . .,.Am ; C1, . *
*.,.Cn],
and there is a monoidal retraction D ! B A C by
[B1, . .,.Bk; A1, . .,.Am ; C1, . .,.Cn] 7! [B1 . . .Bk; A1, . .,.Am ; C1 .*
* . .Cn]
that is compatible with the inclusions of B and C. Since the overcategory D #
[B; A1, . .,.Am ; C] has [B; A1, . .,.Am ; C] with the identity map as a termin*
*al ob
ject, and so has contractible nerve, we know that D and B A C have equivalent
nerves. Therefore D and B A C give rise to the same spectrum upon the applicati*
*on
of Segal's construction, which completes the proof.
The attraction of Thomason's pushout is that it keeps one in the category of
permutative categories, without introducing a simplicial dimension with each ap
plication. It is ideally suited for our general construction in Section 3. The *
*analysis
of specific examples, however, will involve the combinatorial model that we int*
*ro
duce in Section 4, and to establish the relationship of the model to our constr*
*uction,
we will need to use a different model for the pushout, namely the bar construct*
*ion.
Given a pushout diagram as in (2.1), the bar construction is defined to be the *
*sim
plicial permutative category [q] 7! B xAqxC, with the evident face and degenera*
*cy
maps. Taking nerves degreewise, one obtains a simplicial space, which we den*
*ote
Bar(B, A, C); taking geometric realization, one again obtains a space.
We may also consider the version of the bar construction in which there are o*
*nly
face maps, but no degeneracy maps. Thus we represent the bar construction via
the contravariant functor [q] 7! B x Aq x C where [q] ranges over the category *
*of
standard ordered sets and injective maps that are monotonic increasing.
Proposition 2.5. There is an augmentationpreserving monoidal functor
f : hocolimqB x Aq x C ! B A C,
where hocolimis the Thomason homotopy colimit of the face maps in the bar con
struction. This functor induces a homotopy equivalence of spaces
f :  Bar(B, A, C) ! B A C.
Proof.There is an obvious inclusion functor fq : B x Aq x C ! B A C. This
functor is monoidal, meaning that it is lax monoidal in the sense of [20], page
1593, and therefore it passes to a map of spaces upon realization. For an
injective0monotonic map ff : [q] ! [q0], which corresponds to a face map @ff:
B x Aq x C ! B x Aq x C, there is a natural transformation fq0 ! fq O @ff.
These natural transformations satisfy the conditions of [20], Proposition 3.21,*
* and
therefore the functors fq assemble to a monoidal functor, which we denote f:
f : hocolimqB x Aq x C ! B A C.
10 GREGORY ARONE AND KATHRYN LESH
The geometric realization of this homotopy colimit is equivalent to  Bar(B, *
*A, C),
and so we have a Gamma map f :  Bar(B, A, C) ! B A C.
To prove that f is a homotopy equivalence, one may mimic Thomason's proofs
of his main theorems. Namely, one first proves that f is an equivalence when the
diagram B A ! C is a diagram of free permutative categories and free functors,
in which case it is a straightforward calculation. For the general case, one us*
*es the
fact that permutative categories have canonical free resolutions.
3. The general construction of the filtration
In this section, we use the homotopy pushouts of Section 2 to construct a fil
tration on a general augmented permutative category C with associated spectrum
A. The result is a sequence of augmented permutative categories and monoidal
functors,
C = K0C ! K1C ! . .!.K1 C ' Z 0,
and when we apply Segal's infinite loop space machine, we obtain a sequence of
spectra
A = A0 ! A1 ! . .!.H Z.
The filtration is obtained by "killing" the components of C, one at a time. To
set this up, we begin with a definition and then recall some technical results *
*that
we will need.
Definition 3.1. Let C be a small category. The free unital permutative category*
* on
C, denoted Free(C), is the category whose objects are (n, f) where n is a posit*
*ive
integer and f : n_! Obj(C). There are morphisms from (n, f) to (k, g) only if
n = k; a morphism (n, f) ! (n, g) is given by (i) an isomorphism oe : n_! n_, a*
*nd
(ii) for each i 2 n_, a morphism f(i) ! g(oei) in C. In addition, there is a un*
*it object
0_.
In brief a
Free(C) := C o n.
n 0
The following lemma and proposition are routine.
Lemma 3.2.
(1) Free(C) is a permutative category.
(2) Given a permutative category D and a functor F : C ! D, there is a unique
extension of F to a monoidal functor Free(C) ! D.
Proposition 3.3 ([20], Lemma 2.5). The spectrum associated to the symmetric
monoidal category Free(C) is 1 BC+ .
Given an augmented permutative category C whose mth component is Cm , we
want to construct an augmented permutative category that will play the role of
the quotient of C by Cm . To this end, endow Free(Cm ) with the augmentation
ffl(n, f) = mn, so that the functor Free(Cm ) ! C extending the inclusion is au*
*g
mentation preserving. Let S denote the category of finite sets and isomorphisms.
There is a natural projection Free(Cm ) ! S, and in order for the projection to
preserve augmentation, we endow S with the augmentation ffl(S) = mS and call
the resulting augmented category Sm . Recall that the tensor product of categor*
*ies
is defined in Definition 2.3.
FILTERED SPECTRA 11
Definition 3.4. The homotopy cofiber of the map Free(Cm ) ! C, is the augmented
permutative category C==Cm := Sm Free(CmC).
We think of C==Cm as "the category obtained from C by killing the mth compo
nent." This is justified by the following lemma.
Lemma 3.5. The inclusion functor C ! C==Cm is an isomorphism on components
of augmentation less than m. The mth component of C==Cm has contractible nerve.
Proof.Let D = C==Cm . We already know that D is an augmented permutative
category. Objects in D have one of the following forms:
o D = [n_; (n1, f1), (n2, f2), . .,.(nk, fk); C], where each niis a positi*
*ve integer,
fiis a map from n_ito the objects of Cm , and C is an arbitrary object i*
*n C;
o D = [n_; 0_; C], where 0_is the unit object in Free(Cm ) and C is an arb*
*itrary
object in C.
Suppose that i < m; we must prove that Ci~=Di. For an object D to have aug
mentation i, it must have the second form and also have n = 0, because otherwise
a nonzero multiple of m would be contributed to the augmentation of D. Thus
such an object must have the form [0_; 0_; C], where ffl(C) = i. The full subca*
*tegory
of such objects is isomorphic to Ci.
We must also show that Dm  ' *. There are three types of objects of D that
have augmentation equal to m.
(1) [1_; 0_; 0_], since 1_2 Sm has augmentation m;
(2) [0_; (1_, 1 7! C); 0_], where C 2 Cm and so has augmentation m. We will,*
* until
the end of this proof, denote such an object more simply as [0_; C; 0_];
(3) [0_; 0_; C], where C 2 Cm .
There are five types of morphisms, which we categorize by the type of the obj*
*ect
in the target.
(1) (a) [1_; 0_; 0_] has only the identity selfmap; it has no morphisms to *
*objects
of other types.
(b) For each object C 2 Cm , there is a unique morphism
[0_; C; 0_] ! [1_; 0_; 0_].
(2) For each morphism C ! C0 in Cm , there is a morphism
[0_; C; 0_] ! [0_; C0; 0_].
(3) For each morphism C ! C0 in Cm , there are morphisms
[0_; C;!0_][0_; 0_; C0]
[0_; 0_;!C][0_; 0_; C0].
We define the following subcategories of Dm . Let D1mbe the full subcategory *
*of
objects of the form [1_; 0_; 0_] or [0_; C; 0_], let D2mbe the full subcategory*
* of objects of
the form [0_; C; 0_] or [0_; 0_; C0], and let D12m= D1m\ D2mbe the full subcate*
*gory of
objects of the form [0_; C; 0_]. The nerve of Dm is the union of the nerve of D*
*1mand
the nerve of D2m. Since the inclusion D12m ,! D1m is a cofibration, it foll*
*ows that
there is a homotopy pushout square
D12m! D1m
?? ?
y ?y
D2m ! Dm .
12 GREGORY ARONE AND KATHRYN LESH
Contractibility of Dm  now follows from two claims: D1m is contractible, an*
*d the
map D12m ! D2m is a homotopy equivalence. For the first claim, note that D1*
*mhas
[1_; 0_; 0_] as a final object. For the second claim, consider the functor : *
*D2m! D12m
defined on objects by ([0; C; 0]) = [0; C; 0] and ([0; 0; C]) = [0; C; 0]. To*
* define
on morphisms, send morphisms of type (2)or (3)to the corresponding morphisms
of type (2). Then provides a homotopy inverse to the inclusion D12m ! D2m.
Definition 3.6. We say that an augmented permutative category C is "mreduced"
if C is reduced and Ci ' * for i m.
We have the following corollary.
Corollary 3.7. If C is (m  1)reduced, then C==Cm is mreduced.
Construction 3.8. Let C be a reduced augmented permutative category with
associated spectrum A. We define a sequence of permutative categories Km C in
ductively, beginning with K0C = C. Once Km1 C has been defined, let Km1 Cm
be the mth component of Km1 C, and define Km C = Km1 C==Km1 Cm . It follows
immediately from Corollary 3.7 and induction that Km C is mreduced. There are
inclusion functors Km1 C ! Km C. Letting K1 C be the colimit of the constructi*
*on,
we obtain a sequence
C = K0C ! K1C ! . .!.K1 C ' Z 0,
and when we apply Segal's infinite loop space machine, we obtain a sequence of
spectra
A = A0 ! A1 ! . .!.H Z.
This is our filtration of the spectrum A associated to C. Strictly speaking,*
* it
is a relative filtration interpolating between A and H Z. The following theorem
follows immediately from the definition of the categories Km C, Proposition 2.4*
*, and
Proposition 3.3.
Theorem 3.9. Let 1 (BKm1 Cm )+ ! Am1 be induced by extending the in
clusion Km1 Cm ! Km1 C to a functor Free(Km1 Cm ) ! Km1 C. Let the map
ffl : 1 (BKm1 Cm )+ ! S be induced by the map BKm1 Cm ! *. Then
0 1
1 (BKm1 Cm )+ ! Am1
B ? C
Am ' hocolimB@ ffl?y CA
S
Thus by construction the subquotient Am =Am1 is equivalent to 1 BKm1 Cm ,
and so the subquotients of our filtration are suspension spectra.
4.Blowing up C to F1 C
In this section, we begin the development of a simplified combinatorial model
for the sequence of categories described in Construction 3.8. In the case that *
*C is
a "monogenic" category (Definition 4.1), we embed C in a much larger category
F1 C that is monoidally equivalent to Z 0. Then in Sections 5, 6, and 7, we pro*
*ve
that if C satisfies a mild condition (Axiom 5.1), then a certain filtration of *
*F1 C
models Construction 3.8, with our main goal being Theorem 7.1. In Section 8 we
show that further assumptions allow us to recast our combinatorial description *
*in
grouptheoretic terms, as in [12].
FILTERED SPECTRA 13
Our goal in this section is to define the category F1 C and to study the morp*
*hism
spaces, both through examples and by proving Lemma 4.12, which says that F1 C
can be thought of as a category of epimorphisms. We will use a category C that
consists of isomorphisms of an underlying category B that is itself generated u*
*nder
coproducts by a single object, as detailed in the following definition.
Definition 4.1. Let B be a (possibly topological) category with strictly associ*
*ative
coproduct.
(1) We call B pointed if it has a zero object, 0_, that is both initial and *
*final.
(2) We call a pointed category B monogenic if it has an object g (the genera*
*tor)
such that the objects of B consist of 0_and the coproducts of g with its*
*elf
n times for n = 1, 2, 3.... In this case, we write gn for the objects of*
* B,
where g0 means the zero object, and we give B the permutative structure
arising from the coproduct. We require that a monogenic category B have
the property of "dimension invariance," namely that gn ~=gr implies n = *
*r.
We will be particularly interested in the category of finitedimensional comp*
*lex
vector spaces (generated by C). Other examples of interest are the category of *
*finite
pointed sets (generated by 1_), the category of finitelygenerated free modules*
* over
a ring R that has the property of dimension invariance (generated by R), and the
category of finitelygenerated free groups (generated by Z). In each case, we f*
*orce
the coproducts to be strictly associative by taking the equivalent category tha*
*t has
only one object in each isomorphism class.
We need to consider certain subcategories of isomorphisms in B.
Definition 4.2. Let B be a monogenic category. A category of admissible isomor
phisms in B is a subcategory C satisfying the following conditions.
(1) Obj (C) = Obj(B).
(2) Morphisms in C are isomorphisms in B, and C is closed under inverses.
(3) All the automorphisms of gn induced by permutations are in C.
(4) Let f : A ! B and g : C ! D be morphisms in B. Then f and g are both
in C if and only if the morphism f g : A C ! B D is in C.
We call the morphisms of C the Cadmissible isomorphisms in B. We give the cate
gory C the permutative structure inherited from B and the augmentation ffl(gn) *
*= n.
By abuse of terminology, we also call C a monogenic category.
For the remainder of this section, let B be a monogenic category, as above, a*
*nd
let C be a subcategory of admissible isomorphisms. In most specific cases that *
*we
consider in this paper, C is the full subcategory of isomorphisms of B. The only
exception is the category of real or complex vector spaces. In this case, we en*
*dow
the vector spaces with a positivedefinite inner product and let C be the categ*
*ory
of isometric isomorphisms.
The category C is a category of isomorphisms, in other words a groupoid, and
since each isomorphism class`of C has just one object, we can write C as a disj*
*oint
union of groups, namely nAut C(gn). We use the notation Gn for the (possi
bly topological) group AutC(gn). For example, in the case that B is the category
of finitedimensional complex vector`spaces and C consists of the unitary isomo*
*r
phisms, we have Gn = U(n), and C = nU(n), the disjoint union of unitary
groups.
14 GREGORY ARONE AND KATHRYN LESH
Although the category F1 C will be constructed out of isomorphisms in C, we w*
*ill
also need some of the structure of the underlying category B. We follow Waldhau*
*sen
for the definition of cofibration.
Definition 4.3. A morphism i : V ! W in B is a Ccofibration~if there exists
an object V 0and a Cadmissible isomorphism V V 0=!W such that i is the
composite V = V 0_! V V 0~=W . A subobject of W 2 Obj(B) is an equivalence
class of cofibrations V ! W under the equivalence relation of precomposition by*
* a
Cadmissible isomorphism.
Typically, the category C will be clear from context, and we simply refer to
"cofibrations in B." We often use the notation V ,! W for a cofibration.
Remark 4.4. The group AutC(W ) acts transitively on the set (or space) of cofi
brations V ,! W in the following sense: if i1, i2 are two cofibrations from V t*
*o W ,
then there exists an element o 2 AutC(W ) such that i2 = o O i1.
In the case of finitedimensional real or complex vector spaces, the cofibrat*
*ions
are linear isometric inclusions, and the subobjects correspond, as one expects,*
* to
vector subspaces. In the case of finite sets, cofibrations are simply inclusion*
*s, and
a subobject is the same as a subset. In the case of finitelygenerated free mo*
*d
ules, cofibrations are Quillen's "admissible monomorphisms," and two admissible
monomorphisms represent the same subobject if and only if they have the same
image, in the usual sense of the word.
The following definition gives the combinatorial construction that we will fi*
*lter
to obtain combinatorial models for the categories Km C defined in Construction *
*3.8.
We will work extensively with this construction in Sections 58.
Definition 4.5. Let B and C be as above. The reduced, augmented, permutative
category F1 C is defined as follows.
(1) Objects of F1 C have the form [n; n1, . .,.nt], where n and t are nonneg*
*ative
integers, and n1, . .,.ntarePpositive integers. The augmentation is give*
*n by
ffl([n; n1, . .,.nt]) = n + ni.
(2) There are morphisms [n; n1, . .,.nt] ! [r; r1, . .,.ru] only between obj*
*ects
with the same augmentation. A morphism [f] is given by an equivalence
class of the following data.
(a) A function fff : t_! u_.
(b) A Cadmissible isomorphism
f : gn gn1 . . .gnt ! gr gr1 . . .gru
such that
(i)for 1 j u, the composite
i2ff1(j)gni ,! gn gn1 . . .gnt ! gr gr1 . . .gru
factors as a composite
i2ff1(j)gni ,! grj,! gr gr1 . . .gru
where the first map is a Ccofibration and the second map is the
standard inclusion; P
(ii)if we define ej = rj i2ff1(j)ni, then the composite
gn ,! gn gn1 . . .gnt ! gr gr1 . . .gru
FILTERED SPECTRA 15
factors through the composite of a Cadmissible isomorphism
`f : gn ! gr ge1 . . .geu and a coproduct of the identity on
gr with u cofibrations in B,
gr ge1 . . .geu ,! gr gr1 . . .gru.
We call ej the jth excess of f, and when necessary for specific*
*ity,
we also use the notation e(rj), or even e[f](rj).
We give the name compatible with fff to isomorphisms
gn gn1 . . .gnt ! gr gr1 . . .gru
satisfying 2(b)(i) and 2(b)(ii). A morphism in F1 C from [n; n1, . .*
*,.nt]
to [r; r1, . .,.ru] consists of a function fff together with an equi*
*valence
class of Cadmissible isomorphisms that are compatibleQwith fff. The
equivalence relation is given by the action of Grj on the set of C
admissible isomorphisms gn gn1 . . .gnt ! gr gr1 . . .gru,
where, as usual, Grj= AutC(grj).
(3) The monoidal structure on F1 (C) is defined by
[n; n1, . .,.nt] [r; r1, . .,.ru] = [n + r; n1. .,.nt, r1, . .,.ru],
with [0; ;] as the unit.
In general, we think of an object [n; n1, . .,.nt] of F1 C as representing th*
*e ob
ject gn+n1+...+nttogether with a partial partition into "disjoint" or "orthogon*
*al"
subobjects of dimensions n1, . .,.nt. Once a piece is split off into the partit*
*ion, we
remember only its dimension. We think of gn as the "free" part of the object. A
morphism between two objects can combine splitoff subobjects (Condition 2(b)(i*
*)),
and possibly enlarge them to larger splitoff subobjects by taking a further su*
*bob
ject from the free part to combine into a larger subobject (Condition 2(b)(ii)).
We think of F1 C as a model for H Z, which is justified by the following easy
proposition.
Proposition 4.6. The augmentation ffl : F1 C ! Z 0 induces a homotopy equiva
lence of classifying spaces.
Proof.The full subcategory of F1 C consisting of objects of augmentation n has
[0; n] as a terminal object.
Filtering F1 according to the following definition gives us combinatorially *
*de
fined categories that turn out to be models for the inductively defined categor*
*ies
Km C and are easier to analyze.
Definition 4.7. Let Fm C be the full subcategory of F1 C whose objects are of t*
*he
form [n; n1, . .,.ns] such that ni m for all i. (There is no restriction on n.)
It is clear that each Fm C is a monoidal augmented subcategory of F1 C, and
that there are monoidal inclusion functors Fm1 C ! Fm C. Our goal in subsequent
sections is to prove that under suitable assumptions, the classifying space of *
*the
category Fm C is equivalent to that of the category Km C (Theorem 7.1). We wi*
*ll
prove it by showing that the categories Fm1 C and Fm C are related in the same
way as Km1 C and Km C are. It will follow that the sequence of spectra arising
from the sequence of categories F0C = C ! F1C ! F2C ! . .i.s equivalent to the
filtration A = A0 ! A1 ! A2. . .given by Construction 3.8.
16 GREGORY ARONE AND KATHRYN LESH
We conclude this section by giving examples of morphism sets in F1 , and we
describe the relationship of our construction to an idea of Neil Strickland.
Remark 4.8. Note that because of the compatibility condition, a morphism [f] is
completely determined by the map fff and the equivalence class of the isomorphi*
*sm
`f : gn ! gr ge1 . . .geu, where two such isomorphisms give rise to the
same morphism in F1 C exactly when they are in the same orbit of the group
Ge1x . .x.Geu acting by postcomposition.
Examples 4.9. Suppose we take B to be the category of finitedimensional`complex
vector spaces Cn and isometric inclusions, so that C = nU(n).
(1) The object [n; ;] has automorphisms given by Gn = AutC(Cn), which in
this case is U(n), and there are no morphisms [m; ;] ! [n; ;] if n 6= m.*
* In
fact, in general we have an embedding of the category C in F1 C as the f*
*ull
subcategory with objects [m; ;], that is, as F0C.
(2) Consider the space of morphisms [f] : [3; 1, 2] ! [1; 2, 3], which has t*
*hree
connected components, corresponding to three possible functions fff. (See
Remark 4.11 about the topology we put on Fm C when C is a topological
category.)
If fff(1) = 1 and fff(2) = 2, then we need an equivalence class of iso
metric isomorphisms
f : C3 C1 C2 ! C1 C2 C3
under the action of U(2) x U(3). Since ff(1) = 1, the map f is a cofibra*
*tion
from C1 on the left to C2 on the right, and the choice of cofibration is
immaterial because of the equivalence relation. Similar consequences fol*
*low
from ff(2) = 2, so the equivalence class of f is actually determined by *
*what
happens on C3, i.e., by the equivalence classes of isomorphisms
`f : C3 ! C1 C1 C1
under the action of U(1) x U(1) on the last two factors. Thus we obtain
U(3)=(U(1) x U(1)) for this connected component of the morphism space.
There are two more connected components, corresponding to ff(1) =
2 and ff(2) = 1, and ff(1) = ff(2) = 2. Each of these components is
homeomorphic to U(3)=U(2).
Altogether, MorphF1 C([3; 1, 2], [1; 2, 3]) is homeomorphic to the spa*
*ce of
U(6)equivariant maps from U(6)=(U(1) x U(2)) to U(6)=(U(2) x U(3)).
We will see in Section 8 that this is a part of a general theorem.
(3) Consider the endomorphisms [f] of [4; 1, 1, 1, 5, 5], an object of augme*
*nta
tion 17. The function fff must consist of an element of 3x 2, permuting
the 1's and the 5's, and the values e1, . .,.e5 are all zero. Similar co*
*nsid
erations to those above tell us that endomorphisms of [4; 1, 1, 1, 5, 5]*
* are
all automorphisms, and correspond to elements of U(4) x 3 x 2. This
is related to the fact that the Weyl group in U(17) of U(1)3 x U(5)2 is
U(4) x 3 x 2.
Example 4.10. Let B be the category of finite pointed sets and C the subcategory
of isomorphisms of B. Let M be the full subcategory of F1 C with objects of
the form [0; n1, . .,.nk]. The nerves of M and F1 C are equivalent as spaces.
Indeed, the functor F1 C ! M that sends [n; n1, . .,.nk] to [0; 1,_._.,.1z__",*
* n1, . .,.nk]
n
FILTERED SPECTRA 17
induces a homotopy inverse to the map of classifying spaces induced by the incl*
*usion
M ! F1 C.
The category M can be interpreted as a category of "sets with multiplicities,"
a definition proposed by Neil Strickland. A set with multiplicities is a finit*
*e set
where every element is assigned a positive integer as its multiplicity. A morph*
*ism
f : S ! T is a map of sets such that for every t 2 T , the multiplicities of the
elements of f1 (t) add up to the multiplicity of t. The category M is isomorph*
*ic
to the category of sets with multiplicities, with [0; n1, . .,.nk] correspondin*
*g to the
set with k elements, of multiplicities n1, . .,.nk. Similarly, the category Fm*
* C of
Definition 4.7 is equivalent to the category of sets with muliplicities at most*
* m.
Remark 4.11. Notice that the morphism set
MorphF1 C([n; n1, . .,.nt], [r; r1, . .,.ru])
is a subset of
Y
MorphC(gn+n1+...+nt, gr+r1+...+ru)Q Grj~=Gr+r1+...+ru=( Grj).
If B is a topological category, i.e., if Gr is a topological group, then we mak*
*e F1 C a
topological category as well, by endowing Morph F1 C([n; n1, . .,.nt], [r; r1, *
*. .,.ru])
with the subspace topology. Notice that if we set G = Gn+n1+...+nt= Gr+r1+...+r*
*u,
Gno = Gn1 x . .x.Gnt, and Gro = Gr1x . .x.Gru, then there is an inclusion of
subspaces
Morph F1 C([n; n1, . .,.nt], [r; r1, . .,.ru]) mapG (G=Gno, G=Gro) = (G=Gro)G*
*no.
In Section 8, we will see that under some further assumptions on C, this inclus*
*ion
is in fact an isomorphism, in other words F1 C can be identified with a union of
orbit categories.
We close this section with Lemma 4.12, which can be interpreted to say that
F1 C is a "category of epimorphisms."
Lemma 4.12. Suppose we are given a diagram
[n; n1, . .,.nt][f]![r; r1, . .,.ru]
?
[h]?y
[k; k1, . .,.ki]
in F1 C. Then there exists at most one morphism [p] : [r; r1, . .,.ru] ! [k; k1*
*, . .,.ki]
in F1 C such that [p] O [f] = [h].
Proof.Suppose there exists a morphism [p] in F1 C such that [p]O[f] = [h]. We c*
*an
choose representative isomorphisms p, f, and h in C for [p], [f], and [h], resp*
*ectively,
such that p O f = h. The choice of p is determined by the choices of f and h, s*
*ince
all three maps are isomorphisms in C. However, it is conceivable that picking
new representatives f1 and h1 of [f] and [h], respectively, will lead to a new *
*map
p1 = h1O f11with [p] 6= [p1]. Thus we need to show that if f1 and h1 are any o*
*ther
representatives of [f] and [h], and if p1 has the property that p1 O f1 = h1, t*
*hen
[p1] = [p].
We begin with h1. By definition, [h1] = [h] if and only if there exists o 2
Gk1x. .x.Gkisuch that h1 = o Oh, in which case we also have h1Of1 = o OhOf1 .
Thus [h1 O f1 ] = [h O f1 ], that is, [p1] = [p].
18 GREGORY ARONE AND KATHRYN LESH
Finally, suppose [f1] = [f], and let p1 = hOf11. There exists oe 2 Gr1x. .x.*
*Gru
such that f1 = oeOf. Because of Remark 4.4 and Condition 2(b)(i) of Definition *
*4.5,
we know that there exists o 2 Gk1x . .x.Gki such that p O oe1 = o O p. Thus
p1 = h O f11= h O f1 O oe1 = p O oe1 = o O p, and therefore [p] = [p1].
5.Technical lemmas on morphisms in F1 C
In this section, we establish technical lemmas about decompositions of mor
phisms in F1 C that are used in later sections of the paper. The category C
is assumed to be a category of admissible isomorphisms of an underlying mono
genic category B, as in Section 4. The first decomposition result of this sect*
*ion
is Lemma 5.4, which says that a morphism in F1 C that looks like the coproduct
of two morphisms is in fact a coproduct of two morphisms in F1 C. The main
results are Lemma 5.6, which gives a particular type of coproduct decomposition
of a morphism in F1 C, and Lemma 5.9, which is a weak naturality result for the
construction of Lemma 5.6.
We need an assumption on the coproduct in B, and a resulting lemma and
corollary that will be used in the proofs of the decomposition lemmas that comp*
*rise
the rest of the section. Since B is assumed to be pointed, we have a projection
ssA : A B ! A 0 = A with the property that the composite A ,! A B ! A is
the identity, and that the composite B ,! A B ! A is the null map. Similarly,
there is a projection ssB : A B ! B.
We assume the following axiom on the coproduct in B for the remainder of the
paper.
Axiom 5.1 (Injectivity Axiom). The pointed category B has the property that if
f, f0 : A ! C D satisfy ssC O f = ssC O f0 and ssD O f = ssD O f0, then f = f*
*0.
The axiom means, essentially, that the canonical map B C ! B x C from
the coproduct in B to the product is an injection (provided these notions make
sense in B). The axiom is satisfied by the categories of finite pointed sets, f*
*initely
generated vector spaces, and finitelygenerated free modules. However, the cate*
*gory
of finitelygenerated free groups does not satisfy the axiom, and therefore not*
*hing
we do from here on has any relevance to this example. In fact, the reason we
mentioned the example of free groups is to show that not every reasonablelooki*
*ng
monogenic category satisfies our additional axioms.
The most important immediate consequence of the Injectivity Axiom is the fol
lowing lemma.
Lemma 5.2. Let f : A B ! C D have the property that A ,! A B !
C D ! D and B ,! A B ! C D ! C are null. Then f is the coproduct of
two maps f0 : A ! C and f00: B ! D.
Proof.Let f0 be the composite A ,! A B ! C D ! C and let f00be the
composite B ,! A B ! C D ! D. To compare f with f0 f00, we first
compose with projection to C,
0 f00
A B f!C D
? ?
f?y ?y
C D ! C,
FILTERED SPECTRA 19
and note that both ways around the square are f0 on A and null on B. Likewise,
when we replace the lower right corner with D, we find that both ways around the
square are null on A and f00on B. This concludes the proof, by the Injectivity
Axiom.
Corollary 5.3. A morphism f : A ! C D has the property that the composite
A ! C D ! D is null if and only if f factors through the inclusion C ,! C D.
The first decomposition lemma of this section, Lemma 5.4, is actually in serv*
*ice
of Lemma 5.9, and has to do with recognizing a morphism of F1 C as a coproduct *
*of
two other morphisms in F1 C. Here is the setting. Let A = [a; n1, . .,.nt, k1, *
*. .,.ki]
and B = [b; r1, . .,.ru, l1, . .,.lj] be objects of F1 C, and let [f] : A ! B b*
*e a
morphism in F1 C. Let no denote the sequence n1, . .,.nt, and define ko, ro, and
lo similarly. Define gno = gn1 . . .gnt, and define gko, gro, and glosimilarl*
*y.
Suppose given integers n, k, r, and l such that a = n+k and b = r +l, and suppo*
*se
that a representative f of [f] factors as a coproduct in the following sense: t*
*here are
Cadmissible isomorphisms f1, f2, h, and h0in B such that the following diagram
commutes,
ga gno gko f! gb gro glo
? ?
(5.1) h?y~= h0?y~=
f1 f2
(gn gno) gk gko! (gr gro) gl glo,
where the vertical arrows use Cadmissible isomorphisms h : ga ! gn gk and
h0: gb ! gr gl, together with with the identity on the gno, gko, gro, and glo*
*and
the commutativity isomorphisms of B.
Lemma 5.4. The maps f1 and f2 satisfy the conditions of Definition 4.5 to rep
resent morphisms in F1 C, namely [f1] : [n; no] ! [r; ro] and [f2] : [k; ko] ! *
*[l; lo],
respectively.
Proof.We need to check Conditions 2(b)(i) and 2(b)(ii) in Definition 4.5 in ord*
*er
to know that f1 and f2 actually do represent morphisms in F1 C. We concentrate
our attention on f1, the proof for f2 being exactly analogous.
Condition 2(b)(i) is essentially inherited from f itself. We know that f1 *
*f2
restricted to gno is null into glo, and thus f also has this property. Likewis*
*e, f
restricted to gko is null into gro. It follows from Condition 2(b)(i) for f tha*
*t the
set map fff : t_+_i! u_+_j_is in fact the coproduct of two maps fff1 : t_! u_and
fff2: i_! j_. Likewise, the necessary factoring property in Condition 2(b)(i) f*
*or f1
follows from the corresponding factoring property for f.
Now for the real issue. We need to produce a map `f1 that satisfies Condi
tion 2(b)(ii) for f1. Suppose that we are given a representative isomorphism
`f : ga ! gb ge(r1) . . .ge(ru) ge(l1) . . .ge(lj).
20 GREGORY ARONE AND KATHRYN LESH
Consider the following sequence of morphisms, where all of the maps are either
cofibrations or Cadmissible isomorphisms.
1
gn gk______________h~=_______________//ga
 
 ~=`f
 fflffl
 b e(r1) e(ru) e(l1) e(lj)
 g g . . .g g . . .g

 ~=h0 id
 
 fflffl
 gr gl ge(r1) . . .ge(ru) ge(l1) . . .ge(lj)

 
 c
fflffl fflffl
(gn gno) (gk gko)___f1_f2_____//_(gr gro) (gl glo)
(The map c uses the identity on gr and gl, cofibrations ge(r1),! gr1, etc., and
the commutativity isomorphisms of the coproduct in the category B, while the le*
*ft
vertical inclusion is the standard inclusion.) Following gn counterclockwise f*
*rom
the top left to the bottom right shows that gn is null into gl glo, and since c*
* is the
coproduct of cofibrations and isomorphisms, this means that gn followed clockwi*
*se
from top left to bottom right is actually null into gl ge(l1) . . .ge(lj)in t*
*he
domain of c as well. Thus f1 restricted to gn factors as
gn ! gr ge(r1) . . .ge(ru),! gr gro,
and we define the first of these maps to be `f1. Following parallel reasoning f*
*or f2,
we find that f2 restricted to gk factors as
gk ! gl ge(l1) . . .ge(lu),! gl glo,
and we define the first of these maps to be `f2. Finally, when restricted to gn*
* gk,
the composite (h0 id)O`fOh1 is a Cadmissible isomorphism. Thus by Lemma 5.2
and Condition (4)of Definition 4.2, the maps `f1 and `f2 are both Cadmissible
isomorphisms, as required, which finishes the proof.
Our main goal for this section is a construction to decompose a morphism of
F1 C into a coproduct of two other morphisms once given a decomposition of the
target. In fact, we do not establish this in complete generality, for the sake *
*of ease
of notation, but only in the special case that we will need in later sections, *
*namely
the case where one summand (RB below) has no "free" component.
Let [f] : N = [n; n1, . .,.nt] ! R = [r;ir1, . .,.ru]jbe a morphism in F1 C w*
*ith
an associated isomorphism `f : gn ! gr j=uj=1gej. Let A [ B be a partition
P
of the set u_, and let e = j2B ej. If j1 < . .<.ju0 are the elements of A, l*
*et
rA denote the sequence rj1, . .,.rju0, and define grA = grj1 . . .grju0. Defi*
*ne
rB and grB similarly, and define RA = [r;nrA ] and RB = [0; rB ]. We make simil*
*ar
1(A) nff1(B)
definitions for nff1f(A), nff1f(B), g fff, and g f . Let NA = [n  e; nf*
*f1f(A)]
and NB = [e; nff1f(B)].
The isomorphism `f gives us a commuting diagram as follows.
FILTERED SPECTRA 21
Diagram 5.5.
gn gn1 . . .gnt f! gr gr1 . . .gru
? ?
~=?y`f id ~=?y
2 3 2 3
64gr ( ej nff1f(A)76 ej nff1f(B)7df r rA rB
______zj2Ag_") g 5 4(_j2Bg_)z___" g 5! (g g ) g
gne ge
Here the left vertical arrow uses `f and the commutativity isomorphisms for t*
*he
coproduct in the category B, the right vertical arrow uses only the commutativi*
*ty
isomorphisms in B, and df is defined to make the diagram commute.
Lemma 5.6. The isomorphism df in Diagram 5.5 splits as the coproduct of two
nff1(A) nff1(B)
isomorphisms fA : gne g f ! gr grA and fB : ge g f ! grB
that satisfy the conditions of Definition 4.5 to represent, respectively, morph*
*isms
[fA ] : NA ! RA and [fB ] : NB ! RB in F1 C. Further, [fA ] and [fB ] fit into a
commutative diagram
N  [f]! R
? ?
~=?y ~=?y
NA NB [fA][fB]!RA RB ,
where the vertical isomorphisms are represented by the vertical maps in Diagram*
* 5.5.
Proof.The restriction of df to gne gnff1(A)is null into the components grB, *
*and
nff1(B)
likewise df restricted to ge g f is null into the components grA. Thus by
Lemma 5.2, we find that df is the coproduct of morphisms fA : gne gnff1(A)!
grA and fB : ge gnff1(B)! grB. Since df is a Cadmissible isomorphism, Con
dition (4)of Definition 4.2 tells us that each of fA and fB are Cadmissible is*
*o
morphisms. By construction, they satisfy the conditions of Definition 4.5, whi*
*ch
finishes the proof of the lemma.
It is important to keep in mind in the discussion above that the choice of `f*
* is not
unique. In fact, the difficulty in using the construction of Lemma 5.6 stems fr*
*om the
fact that the isomorphism N ~=NA NB in Lemma 5.6 is not canonical, but depends
in an essential way on the particular choice used for `f. The existence of choi*
*ces
means that the decomposition will not be natural in a strict sense. Fortunately*
*, for
everything that we need to do, we only need to know that compatible choices can
be made so that appropriate diagrams commute. To be more precise, suppose that
we have a commutative diagram in F1 C,
0]
N0 = [n0; n01, . .,.n0t0][f!R0= [r0; r01, . .,.r0u0]
? ?
(5.2) [p]?y [q]?y
N = [n; n1, . .,.nt][f]!R = [r; r1, . .,.ru],
and suppose that A [ B is a partition of u_. Let A0= ff1q(A) and let B0= ff1q*
*(B).
We would like the construction of Lemma 5.6, applied to [f] and to [f0], to yie*
*ld a
commutative diagram as follows.
22 GREGORY ARONE AND KATHRYN LESH
Diagram 5.7.
~= [f0A0] [f0B0] ~=
N0  ! N0A0 N0B0! R0A0 R0B0! R0
? ? ? ?
[p]?y [pA] [pB]?y [qA] [qB]?y [q]?y
~= [fA fB] ~=
N  ! NA NB ! RA RB ! R.
(Here the composite across the top row should be [f0] and the composite across *
*the
bottom row should be [f].)
However, that turns out to be false, for essentially trivial reasons, unless *
*we make
the additional assumption that e[q](rj) = 0 for j 2 B. Hence we will make this
assumption as a hypothesis in the lemma below. The construction of Lemma 5.6
gives us a commuting diagram as follows, where the top square comes from applyi*
*ng
the lemma to [f0], and the bottom square comes from applying it to [f].
Diagram 5.8.
2 2 3 3
66r0 i e0j n0ff10(A0)7766ie0j n0ff10(B0)77f0A0if0B0r0r00jr00
4g______j2A0g_jz______" g 4f_j52B0g_z__j_" g f5! g g Ag B
gn0e0 x ge0 x
c`f0??~= ??~=
0 0 0 0
gn0 gn01 . . .gn0t0 f! gr gr1 . . .gru0
? ?
p?y q?y
gn gn1 . . .gnt f! gr gr1 . . .gru
? ?
c`f?y~= ~=?y
2 3 2 3
64gr ( ej nff1f(A)76 ej nff1f(B)7fA fB r rA rB
______zj2Ag_") g 5 4(_j2Bg_)z___" g 5! (g g ) g
gne ge
The naturality result we need for the use of Lemma 5.6 is that the corners of
this diagram form a square that is the coproduct of two squares in F1 C.
Lemma 5.9. Assuming that e[q](rj) = 0 for j 2 B, the map c`fO p O c1`f0is the
coproduct of two Cadmissible isomorphisms in B,
~ 0 ~ h i
0e0 nff1(A0) ne nff1(A)
pA : gn g f0 ! g g f
~ 0 ~ h i
0 nff1(B0) e nff1(B)
pB : ge g f0 ! g g f ,
and pA and pB represent morphisms in F1 C,
[pA ][:n0 e0; n0ff1 ]0! [n  e; nff1(A)]
f0(A ) f
[pB ][:e0; n0ff1 ]0! [e; nff1(B)].
f0(B ) f
The morphism [pA ] [pB ] makes Diagram 5.7 commute.
FILTERED SPECTRA 23
Proof.Combining the assumption that ff : u0_! u_is a coproduct of two maps
ffqB : B0! B and ffqA : A0! A with the assumption that q is a Cadmissible iso
morphism satisfying the conditions of Definition 4.5, together with the assumpt*
*ion
that e[q](rj) = 0 for j 2 B, we know that the composition from the top right co*
*rner
of Diagram 5.80to the bottom right corner is the coproduct of0a Cadmissible0is*
*omor
phism qB : grB0! grB with a Cadmissible isomorphism qA : gr grA0! gr grA.
We can abstract from Diagram 5.8 the following.
Diagram 5.10.
gn0 gn01 . . .gn0t0 p! gn gn1 . . .gnt
?? ?
y ?y
~ n0 1 ~ ~ 0 ~ c OpOc1h i h i
0) e0 nff10(B0)`f `f0 ne nff1(A) e nff1(B)
gn0e0 g fff0(A g g f ! g g f g g f
The bottom arrow is the same as going around Diagram 5.8 counterclockwise
from bottom left to top left, and therefore is the coproduct of two Cadmissibl*
*e iso
morphisms pA and pB as stated in the lemma, and pA and pB represent morphisms
in F1 C by Lemma 5.4.
6. Factoring morphisms in Fm C
In this section and the next, B is a monogenic category satisfying Axiom 5.1,
and C is a subcategory of admissible isomorphisms. Recall that Fm C is the full
subcategory of F1 C whose objects are of the form [n; n1, . .,.nt] where ni m *
*for
all i. Our goal in this middle part of the paper is Theorem 7.1, which establis*
*hes
that Fm C and the category Km C defined in Construction 3.8 are linked by a cha*
*in
of monoidal functors that induce homotopy equivalences on classifying spaces. By
definition, F0C ~=C = K0C, and we will show in Section 7 that Fm C is related to
Fm1 C in the same way that Km C is related to Km1 C, namely by a homotopy
pushout. The desired conclusion will follow by induction.
This section is entirely in service of Section 7 and establishes two technical
results. We begin with a definition involving two types of morphisms in Fm C
that, it turns out, need to be tweezed apart in order to analyze BFm C. We then
establish Lemma 6.2, a technical tool about factoring arbitrary morphisms in Fm*
* C
into a composite of these two kinds of morphisms. Finally, we apply Lemma 6.2 to
prove Proposition 6.3, which shows the equivalence of the nerves of two categor*
*ies
of chains in Fm C. These results will be used in Section 7.
Definition 6.1.
(1) We say that a morphism [n; n1, . .,.nt] ! [r; r1, . .,.ru] in Fm C creat*
*es no
components of augmentation m if the number of values equal to m in the
lists n1, . .,.nt and r1, . .,.ru is the same.
(2) We say that a morphism [n; n1, . .,.nt] ! [r; r1, . .,.ru] in Fm C creat*
*es only
components of augmentation m if the associated map of sets ff : t_! u_has
the property that every element of {j 2 u_ rj < m} has a single preimage
under ff, and nff1(j)= rj.
(3) We write F0mC for the subcategory of Fm C consisting of morphisms that
only create components of augmentation m.
24 GREGORY ARONE AND KATHRYN LESH
Note that F0mC is not a full subcategory of Fm C, but it is a monoidal subcat*
*egory.
Lemma 6.2. Let N = [n; n1, . .,.nt] and R = [r; r1, . .,.ru] be objects in Fm C.
(1) A morphism N ! R can be factored as a composite N ! R0! R in Fm C
such that N ! R0creates only components of augmentation m and R0! R
creates no components of augmentation m. The object R0 is unique up to
isomorphism.
(2) Given a commutative diagram
N = [n; n1, . .,.nt][f]!R = [r; r1, . .,.ru]
? ?
(6.1) [p]?y [q]?y
K = [k; k1, . .,.ki][h]!L = [l; l1, . .,.lj]
such that the vertical maps create no components of augmentation m, and
given factorings N ! R0! R and K ! L0! L as in (1)of [f] and [h],
respectively, there exists a unique morphism R0! L0 to expand (6.1)to a
commuting ladder
N ! R0 ! R
? ? ?
[p]?y ?y [q]?y
K ! L0 ! L,
and R0! L0creates no components of augmentation m.
Proof.The uniqueness of the object R0up to isomorphism is immediate from Def
inition 6.1. We will use Lemma 5.6 and Lemma 5.9 to construct the factoring and
to address the compatibility with commutative diagrams.
Let B = {s 2 u_ rs = m}, and let A = u_ B. Then using Lemma 5.6 (and
expanding its diagram slightly), we have a diagram
[f]
N _________________________//R
~= ~=
fflfflid [fB] [fA] id fflffl
NA NB _____//NA RB_____//RA RB
Then by construction, id [fB ] creates only components of augmentation m, whi*
*le
[fA ] id creates no components of augmentation m, and we define R0= NA RB
and factor [f] by going counterclockwise around the diagram from N to R.
Further, suppose given a commutative diagram (6.1)where the vertical mor
phisms create no components of augmentation m. Let D = {s 2 j_ ls = m}, and
let C = j_ D. Because [q] creates no components of augmentation m, we know
that ffq is a onetoone correspondence from B to D, and clearly for s 2 B we h*
*ave
rs = lffq(s), since both are m by hypothesis. Thus (6.1)satisfies the condition*
*s of
Lemma 5.9, and given fixed isomorphisms N ~= NA NB and K ~= KC KD ,
we get morphisms in F1 C between the pieces, namely [pA ] : NA ! KC and
[pB ] : NB ! KD . Then the following diagram establishes the existence of a map
FILTERED SPECTRA 25
R0! L0that satisfies the lemma.
[f]
N _____________________________//R
 
~= ~=
 R0 
fflfflid [fB]z___"____[fA] idfflffl
NA NB _____// NA RB _____//RA RB
 
 
[pA] [pB] [pA] [qD] [qC][qD]
 fflffl 
 
fflfflid [hD] [hA] id fflffl
KC OKDO ____//_KC___LD_z___"_//LCOO LD
 L0 
~= ~=
 
 [h] 
K _____________________________//L
Finally, the uniqueness of R0! L0follows from Lemma 4.12.
The second technical result of this section is in aid of the analysis of a ce*
*rtain
bisimplicial set in Section 7. Let q(Fm C) be the category whose objects are *
*q
chains N0 ! . .!.Nq in Fm C, and whose morphisms are commuting ladders
M0  ! M1 ! . ..! Mq
?? ? ?
y ?y ?y
N0  ! N1 ! . ..! Nq
where the vertical maps create no components of augmentation m. Let q(F0mC) be
the full subcategory of q(Fm C) where the horizontal maps create only componen*
*ts
of augmentation m. We consider bisimplicial sets in Section 7 whose levels are *
*the
nerves of these categories, and we will need the following proposition.
Proposition 6.3. The inclusion functor q(F0mC) ! q(Fm C) induces a weak
equivalence of classifying spaces.
Proof.Fix a chain N0 ! . .!.Nq, call it fl, in Fm C; it is sufficient to prove *
*that
the overcategory q(F0mC) # fl has contractible nerve. To prove this, we const*
*ruct
a chain fl0 that we assert is a terminal object in q(F0mC) # fl.
The chain fl0 is defined inductively. We begin by factoring N0 ! N1 as in
Lemma 6.2(1)to obtain N0 ! N01! N1, where N0 ! N01creates only components
of augmentation m, and N01! N1 creates no components of augmentation m.
Once N0iis defined, we consider the composite N0i! Ni ! Ni+1 and define N0i+1
by factoring N0i! Ni+1 as N0i! N0i+1! Ni+1 using Lemma 6.2. Here N0i!
N0i+1creates only components of augmentation m, and N0i+1! Ni+1 creates no
components of augmentation m. The construction gives us a qchain fl0in q(F0mC)
26 GREGORY ARONE AND KATHRYN LESH
and a commuting ladder mapping fl0 to fl:
N0 ! N01! N02 ! . ..! N0q
? ? ? ?
(6.2) ?y ?y ?y ?y
N0 ! N1 ! N2 ! . ..! Nq.
We assert that (6.2)is terminal in q(F0mC) # fl. For suppose that
M0 ! M1 ! M2 ! . ..! Mq
?? ? ? ?
y ?y ?y ?y
N0 ! N1 ! N2 ! . ..! Nq,
is another object in q(F0mC) # fl. Thus the morphisms Mi ! Mi+1 create only
components of augmentation m. As a result, if we apply the same construction to
M0 ! M1 ! M2 ! . .!.Mq that we applied to fl, we can choose M0i= Mi for
all i. Then the ladder
M0 ! M01 ! M02 ! . ..! M0q
?? ? ? ?
y ?y ?y ?y
N0 ! N01 ! N02 ! . ..! N0q.
that results from Lemma 6.2 (2)is actually a ladder
M0 ! M1 ! M2 ! . ..! Mq
?? ? ? ?
y ?y ?y ?y
N0 ! N01 ! N02 ! . ..! N0q.
Uniqueness of the vertical maps follows from Lemma 4.12, which finishes the pro*
*of.
7. Linking the combinatorial model to Construction 3.8
This section builds on the work of Sections 5 and 6 to establish the following
theorem, which relates the combinatorial construction of Section 4 to Construc
tion 3.8.
Theorem 7.1. Let B be a monogenic category satisfying Axiom 5.1, and let C be
a subcategory of admissible isomorphisms. Then Km C and Fm C are linked by a
chain of monoidal functors, each of which induces a homotopy equivalence of cla*
*s
sifying spaces. In particular, the associated spectra of Km C and Fm C are homo*
*topy
equivalent.
Corollary 7.2. Let C be either the category of finite sets, or the category of *
*finite
dimensional vector spaces, or the category of finitelygenerated free modules o*
*ver a
ring satisfying the dimension invariance property. Let A0 ! A1 ! . .!.H Z be
the sequence of spectra (1.2)associated with C. Then there is an equivalence
Am =Am1 ' 1 BFm1 Cm .
Proof of Corollary 7.2.All the categories named in the corollary satisfy the as
sumptions of Theorem 7.1. It follows from the proof of the theorem that if C is
one of the categories named in the corollary, then Fm C satisfies, up to homoto*
*py of
FILTERED SPECTRA 27
classifying spaces, the same inductive formula as Km C. In other words, there i*
*s a
chain of monoidal functors, each one inducing a homotopy equivalence of classif*
*ying
spaces, connecting Fm C and Fm1 C Free(Fm1CmS)m. On the level of spectra, th*
*is
translates to the assertion that there is a homotopy pushout square
1 (BFm1 Cm )+ ! Am1
?? ?
y ?y
S ! Am
The corollary follows.
The general strategy for proving Theorem 7.1 is to show that Fm C is related
to Fm1 C in the same way that Km C is related to Km1 C. Specifically, we will
show that BFm C is equivalent to the geometric realization of the bar construc
tion Bar(Fm1 C, Free(Fm1 Cm ), Sm ). It will then follow by Proposition 2.5 *
*and
induction that BFm C is equivalent to BKm C.
We analyze BFm C by introducing a bisimplicial set that we can study in the t*
*wo
different simplicial directions. Let E00(Fm C) be the bisimplicial set whose (*
*p, q)
simplices consist of diagrams in Fm C of the form
C0,0! . ..! C0,q
?? ?
y ?y
.. .
. ..
?? ?
y ?y
Cp,0! . ..! Cp,q,
where the vertical maps do not create components of augmentation m. Then
E00(Fm C) is the bisimplicial set of a bicategory, and its realization, diag(*
*E00(Fm C)),
is a space.
Lemma 7.3. There is a equivalence BFm C ' diag(E00(Fm C)).
Proof.The proof is exactly as in the proof of Lemma 3.1 of [11]. Namely, for ea*
*ch p,
there is an equivalence E00(Fm C)p,*' BFm C, and it is in fact a equivalence.
We still need to slim down E00(Fm C) a little bit. The goal is to tweeze apar*
*t the
morphisms that create only components of augmentation m from those that cre
ate no components of augmentation m. Although an arbitrary morphism in Fm C
is neither of these, we proved in Lemma 6.2 that an arbitrary morphism can be
factored into a composition of these two types of morphisms. Let E00(F0mC) be t*
*he
subbisimplicial set of E00(Fm C) consisting of (p, q)simplices where the hori*
*zontal
morphisms create only new components of augmentation m. Recall that q(F0mC)
is the category whose objects are qchains in F0mC (i.e., chains of morphisms t*
*hat
create only components of augmentation m), with morphisms consisting of com
muting ladders where the morphisms between the chains create no new components
of augmentation m.
Proposition 7.4. E00(F0mC)*,q,! E00(Fm C)*,qis a weak equivalence.
28 GREGORY ARONE AND KATHRYN LESH
Proof.The simplicial sets E00(F0mC)*,qand E00(Fm C)*,qare , respectively, the n*
*erves
of the categories q(F0mC) and q(Fm C). The result follows from Proposition 6.*
*3.
Our task is now to identify the bisimplicial set E00(F0mC), whose (p, q)simp*
*lices
consist of diagrams where the horizontal maps create only components of augmen
tation m and the vertical maps create no components of augmentation m. Let
Fm1 Cm denote the full subcategory of Fm1 C consisting of objects with augmen
tation exactly m. We assert in Proposition 7.8, below, that E00(F0mC) can actua*
*lly
be identified as the bar construction Bar(Fm1 C, Free(Fm1 Cm ), Smu)p to weak
equivalence. Since E00(F0mC)*,qis the nerve of q(F0mC), our strategy for provi*
*ng
this is to construct a functor
,q : Fm1 C x (Free(Fm1 Cm ))qx Sm ! q(F0mC)
that gives an equivalence on nerves. Notice that the domain of ,q is given by t*
*he
qsimplices of the bar construction. The functors ,o assemble to a map
Bar (Fm1 C, Free(Fm1 Cm ),!SmE)00(F0mC),
which, by Proposition 7.8, induces an equivalence of geometric realizations.
We need two auxiliary functors. For a nonnegative integer n, let n_m indicate
the object [0; m,_._.,.m_z___"], and note that this gives an embedding of Sm *
*as a full
n times
subcategory of Fm C. Also note that because Fm1 C is a permutative category,
there is a functor ~ : Free(Fm1 Cm ) ! Fm1 C taking (k_, f) to the monoidal
product of the objects f(1), . .,.f(k).
To define ,q, we must take an object in Fm1 C x (Free(Fm1 Cm ))qx Sm and
obtain a qchain from it. The data supplied are:
o an object C 2 Fm1 C,
o a list of objects (n1_, f1), . .,.(nq_, fq) in Free(Fm1 Cm ), and
o a set s_.
The functor ,q takes these data to the following qchain:
C ~(n1_, f1) . . .~(nq_, fq) s_m
! C ~(n1_, f1) . . .~(nq1_, fq1) nq_m s_m
! . . .
! C n1_m . . .nq_m s_m.
Remark 7.5. Notice that this qchain is the coproduct of q +2 others: the ident*
*ity
qchain on C, the identity qchain on s_m, and the qchains formed by taking the
morphism ~(ni_, fi) ! ni_m and lengthening it to a qchain by precomposing with
q  i identity maps and postcomposing with i  1 identity maps. Writing a chain
in q(F0mC) in the form just preceding this remark exhibits it as an object in *
*the
image of ,q.
Examples 7.6.
(1) If q = 0, then ,0 takes an object of Fm1 C x Sm and gives a 0chain in
F0mC. The pair consisting of [n; n1, . .,.nt] and the set s_goes to the *
*0chain
[n; n1, . .,.nt, m,_._.,.m_z___"].
s times
FILTERED SPECTRA 29
(2) If q = 1, then ,1 takes an object of Fm1 C x Free(Fm1 Cm ) x Sm and gi*
*ves
a 1chain in F0mC. The given data are [n; n1, . .,.nt] with each ni m *
* 1,
an object (C1, . .,.Ck) in Free(Fm1 Cm ), and a set s_. Note that since*
* each
object Ci has augmentation m, there is a unique morphism Ci! [0; m] in
Fm C. This input data gives the 1chain
[n; n1, . .,.nt] (C1 . . .Ck) [0; m,_._.,.m_z___"]
0 s times 1
! [n; n1, . .,.nt] @[0;_m]___._..[0;zm]_____"A [0; m,_._.*
*,.m_z___"].
k times s tim*
*es
Effectively, we are taking the monoidal product of the identity maps on *
*the
outer ends with the unique maps Ci! [0; m] in the middle.
Our goal is to establish that ,q induces an equivalence of nerves (Propositio*
*n 7.8).
Our first step is to establish a special case. Let ' be the restriction of ,1 *
*to
Free(Fm1 Cm ). Objects in the image of ' have the form [f] : N = [n; n1, . .,.*
*nt] !
[0; m, . .,.m] where N 2 Fm1 C, i.e., each ni m  1. Let D denote the full
subcategory of 1(F0mC) consisting of all objects having this form.
Proposition 7.7. The functor ' : Free(Fm1 Cm ) ! D induces a weak equivalence
of nerves.
Proof.Since ' is an embedding, it is sufficient to show that the image of ' con*
*tains
at least one object in every isomorphism class and is a full subcategory of D. *
*Let
[f] : N = [n; n1, . .,.nt] ! [0; m,_._.,.m_z___"] be a typical object of D, an*
*d consider the
u
canonical decomposition [0; m, . .,.m] = [0; m] . . .[0; m]. From Lemma 5.6, *
*we
obtain a decomposition
N [f]! [0; m, . .,.m]
? ?
~=?y ~=?y
N1 . . .Nu [fj]![0; m] . . .[0; m]
where N1, . .,.Nu are in Fm1 C. This diagram establishes that [f] is isomorphi*
*c to
an object in the image of ', namely the chain in the bottom row.
We must still show the the image of ' is a full subcategory of D. Suppose that
[ f0j]
N01 . . .N0u! [0; m] . . .[0; m]
?? ?
y ?y
N1 . . .Nu [fj]![0; m] . . .[0; m]
is an arbitrary morphism between objects (the rows) in the image of '. The right
vertical map is necessarily an isomorphism, given by a permutation of the u sum
mands, say oe 2 u. Thus by using the given decompositions on the right of the
diagram as input to Lemma 5.9, we find that the left vertical map must be the c*
*o
product of morphisms N0i! Noe(i)in Fm1 C, i.e., is a morphism in Free(Fm1 C),
as required.
30 GREGORY ARONE AND KATHRYN LESH
Proposition 7.8. The functor ,q : Fm1 C x (Free(Fm1 Cm ))qx Sm ! q(F0mC)
induces a weak equivalence of nerves.
Proof.As in the proof of Proposition 7.7, since ,q is an embedding, we show that
every object in q(F0mC) is isomorphic to one in the image of ,q, and that the *
*image
of ,q is a full subcategory of q(F0mC).
Let us consider a typical chain of q(F0mC),
(7.1) N0 f1!N1 f2!. .f.q!Nq.
Thus we are assuming that the maps f1, . .,.fq create only components of augmen
tation m. We write Ni = [ni; ni,1, . .,.ni,ti]. Our strategy is to decompose Nq*
* as
a coproduct by grouping nq,1, . .,.nq,tqinto blocks T1, T0, . .,.Tq, and then *
*to use
Lemma 5.6 to decompose (7.1)as a coproduct of chains. This will show that (7.1)
is isomorphic to a chain in the image of ,q by Remark 7.5.
To determine the blocks, we group all the elements of nq,1, . .,.nq,tqthat ar*
*e less
than m together into a block T1. The remaining blocks T0, . .,.Tq hold all the
elements nq,jsuch that nq,j= m. To determine which block an element nq,jgoes
into, we look back in the chain to see how many "preimages" of nq,jare also m.
(Since we are in Fm C, once ni,j= m we also know its "image" ni+1,fffi+1(j)= m,
because the image value cannot decrease but also cannot be more than m.) To be
precise, we partition tq_as follows:
T1 = {j 2 tq_ nqj< m}
T0 = {j 2 tq_ 9k 2 t0_s.t.n0,k= m and (fffq. .f.ff1)(k) = j}
T1 = {j 2 tq_ T0  9k 2 t1_s.t.n1,k= m and (fffq. .f.ff2)(k) = j}
..
.
Tq1 = {j 2 tq_ (T0 [ . .[.Tq2)  9k 2 tq1_s.t.nq1,k= m and fffq(k) = j}
Tq = {j 2 tq_ nq,j= m, but 6 9k 2 tq1_s.t.nq1,k= m and fffq(k) = j}
Thus Tq labels the components of augmentation m in Nq that are newly created
by the map Nq1 ! Nq, while Tq1 labels those that were created by the map
Nq2 ! Nq1, etc. Corresponding to this partition of tq_, we decompose Nq into
the coproduct of q + 2 objects T1(Nq), T0(Nq), . .,.Tq(Nq) namely:
(
Tj(Nq) = [nq; nT1]j = 1
[0; nTj] 0 j q.
(The notation nTj for a sequence is carried over from Section 5.) Then Nq ~=
T1(Nq) T0(Nq) . . .Tq(Nq), and we can apply Lemma 5.6 to decompose Nq1
into Nq1 ~=T1(Nq1) T0(Nq1) . . .Tq(Nq1), and the map fq as a coproduct
of q + 2 morphisms f(q,j): Tj(Nq1) ! Tj(Nq). Note that the summands Tj(Nq1)
are defined by the decomposition of Nq and the map fq, rather than by conditions
like those that define Tj(Nq), though in fact a similar list of conditions hold*
*s.
We make one observation about the maps f(q,j)before we continue. By hypoth
esis, fq : Nq1 ! Nq creates only components of augmentation m, and as a conse
quence, f(q,1)is an isomorphism. Further f(q,0), . .,.f(q,q1)are isomorphisms*
* by
construction, each one being an isomorphism of objects of the type [0; m, . .,.*
*m].
FILTERED SPECTRA 31
Thus at the possible cost of altering the left vertical isomorphism in the diag*
*ram
Nq1 ! Nq
? ?
~=?y ~=?y
f(q,j)q
qj=1Tj(Nq1)! j=1Tj(Nq),
we can even take f(q,1), f(q,0), . .,.f(q,q1)to be identity maps. This leaves*
* f(q,q)
as the only interesting morphism in the coproduct, having the form of a morphism
in the category D studied in Proposition 7.7.
Our next task is to indicate how this procedure will be iterated to decompose
the entire chain N0 ! N1 ! . .!.Nq into a coproduct of q + 2 chains, each of
which is easily analyzed. Let hq1 = fqOfq1, and decompose Nq2 as a coproduct
of summands Tj(Nq2) by using hq1 : Nq2 ! Nq ~= qj=1Tj(Nq) and applying
Lemma 5.6:
Nq2 hq1! Nq
? ?
~=?y ~=?y
h(q1,j)q
qj=1Tj(Nq2)! j=1Tj(Nq).
In this case, the same analysis as before allows us to take h(q1,1), h(q1,0)*
*, . .,.
h(q1,q2)to be the identity maps, while h(q1,q1)and h(q1,q), like f(q,q), h*
*ave
the form of morphisms in the category D.
There is a commutative diagram
Nq2 hq1!Nq
? ?
fq1?y =?y
Nq1 fq!Nq,
and we have decompositions of all of the terms,
h(q1,j)q
qj=1Tj(Nq2)! j=1Tj(Nq)
?
= ?y
f(q,j) q
qj=1Tj(Nq1) ! j=1Tj(Nq).
By Lemma 5.9, we can fill in to create a commuting square with a morphism
qj=1Tj(Nq2) ! qj=1Tj(Nq1) that is a coproduct of q + 2 morphisms, say
f(q1,j): Tj(Nq2) ! Tj(Nq1). Further, since we have chosen h(q1,1), h(q1,0*
*),
. .,.h(q1,q2)and f(q,1), f(q,0), . .,.f(q,q2)to be identity maps, we know t*
*hat
f(q1,1), f(q1,0), . .,.f(q1,q2)are likewise identity maps. Considering f(q*
*1,q1),
we see that it has the form of a morphism in D, while f(q1,q)is an isomorphism
(because Nq2 ! Nq1 creates only components of augmentation m), and thus by
an adjustment in the isomorphism Nq2 ~= qj=1Tj(Nq2), we can choose f(q1,q)
to be the identity map.
32 GREGORY ARONE AND KATHRYN LESH
After iterating this procedure q + 1 times, we have a commuting ladder
N0 f1! N1 f2! . . .fq! Nq
? ? ?
~=?y ~=?y ~=?y
f(1,j) q f(2,j) f(q,j) q
qj=1Tj(N0) ! j=1Tj(N1) ! . . .! j=1Tj(Nq).
where the top row is the original chain (7.1), and the bottom row is a coproduc*
*t of
chains
(7.2) Tj(N0) f(1,j)!Tj(N1)f(2,j)!.f.(.q,j)!Tj(Nq).
For 1 j q, we have Tj(N0) = . .=.Tj(Nj1) and Tj(Nj) = . .=.Tj(Nq), and
the maps f(1,j), . .,.f(j1,j)and f(j+1,j), . .,.f(q,j)are identity maps, while*
* f(j,j)is
a morphism in D. For j = 1, 0, we have Tj(N0) = . .=.Tj(Nq), and all of f(1,j),
. .,.f(q,j)are identity maps.
Recall that our goal was to show that the chain N0 ! N1 ! . .!.Nq is isomor
phic to a chain in the image of ,q : Fm1 C x (Free(Fm1 Cm ))qx Sm ! q(F0mC).
From the proof of Proposition 7.7, we know that Tj(Nj1) ! Tj(Nj) is isomorphic
to an object in the image of ', so we replace Tj(Nj1), if necessary, with an o*
*bject
that is actually in the image of Free(Fm1 Cm ) under ~. Then as a preimage (up
to isomorphism) for N0 ! N1 ! . .!.Nq in Fm1 C x (Free(Fm1 Cm ))qx Sm
under ,q, we take the object that in the Fm1 Ccoordinate is T1(Nq), in the
(Free(Fm1 Cm ))qcoordinate is T1(Nq), . .,.Tq(Nq), and in the Sm coordinate
is T0(Nq). (Here we abuse notation by regarding T0(Nq), which has the form
[0; m, . .,.m], as identified with a set of the appropriate cardinality.)
To conclude the proof, we consider a morphism between chains in q(F0mC):
0 f0 f0q
N00f1!N01 2!. ..! N0q
? ? ?
(7.3) ?y ?y ?y
N0 f1!N1  f2!. ..fq!Nq.
Because the vertical morphisms create no components of augmentation m, the
decompositions of N0qand Nq must be compatible under N0q! Nq, and so (7.3)
must actually be isomorphic to the coproduct of morphisms between chains
Tj(N00)! Tj(N01)! . ..! Tj(N0q)
?? ? ?
y ?y ?y
Tj(N0) ! Tj(N1) ! . ..! Tj(Nq),
no matter what choices are made at the various stages of the construction. The
proposition follows.
The proof of Theorem 7.1 now follows easily.
Proof of Theorem 7.1.We argue by induction on m. Clearly F0C ~= C = K0C.
Assume for the inductive hypothesis that there is an augmented weak equivalence
Fm1 C ' Km1 C. It follows from Lemma 7.3, Proposition 7.4, and Proposition 7.8
that there is a chain of monoidal functors inducing a weak equivalence of spa*
*ces
Fm C '  Bar(Fm1 C, Free(Fm1 Cm ),Sm.)
FILTERED SPECTRA 33
By Proposition 2.5, it follows that there is a chain of monoidal functors, indu*
*cing
a equivalence of spaces
Fm C ' Fm1 C Free(Fm1CmS)m.
The weak equivalence Fm1 C ' Km1 C of the inductive hypothesis restricts to a
weak equivalence Fm1 Cm ' Km1 Cm . It follows that
Fm1 C Free(Fm1CmS)m' Km1 C Free(Km1CmS)m= Km C.
So, we have obtained a chain of weak equivalences, via monoidal functors,
Fm C ' Km C.
8.Orbit categories
In this section we make additional assumptions, over and above those made in
Sections 4 and 5, that allow us to relate the subquotients of our filtration to*
* certain
classifying spaces of collections of subgroups, a grouptheoretic construction.*
* This
relates the current work to the program of [11] and [12] and allows us to estab*
*lish
that our construction gives the symmetric powers of spheres when applied to fin*
*ite
pointed sets. We continue the notation of the previous sections: B is a monogen*
*ic
category satisfying Axiom 5.1 (Injectivity Axiom), and C is a category of admis*
*sible
isomorphisms in B. Recall that Gn denotes the group of automorphisms of gn in C.
Definition 8.1. Let N = n + n1 + . .+.nt. The identification gN = gn gn1
. . .gnt defines an inclusion of groups
Gn1x . .x.Gnt,! GN ,
and we define a standard subgroup of GN to be a subgroup conjugate to the image*
* of
such an inclusion. We say that the standard subgroup Gn1x. .x.Gnthas filtration
m if n1, . .,.nt m.
Recall that a "collection" H of subgroups of a group G is a set of subgroups *
*that
is closed under conjugation [5]. The classifying space BH of the collection is *
*the
nerve of the category whose objects are orbits G=H for H 2 H and whose maps
are Gequivariant maps. For example, for a fixed n, the set of standard subgrou*
*ps
of Gn of filtration m is a collection, which we denote Hm (Gn), and we denote i*
*ts
classifying space BHm (Gn).
The additional assumptions for this section are detailed below as Axioms 8.5
and 8.7, and the goal of the section is the following theorem and its corollari*
*es,
which relate Construction 3.8 to grouptheoretic constructions. The main techni*
*cal
tool in the proofs is Proposition 8.9.
Theorem 8.2. Let C be a monogenic category`satisfying Axioms 5.1, 8.5, and 8.7.
Then BKm C is monoidally equivalent to nBHm (Gn).
Corollary 8.3. Given a category C as in Theorem 8.2, let Rm be the collection
Hm1 (Gm ), and let ffl : 1 (BRm )+ ! S be induced by the map BRm ! *. Then
0 1
1 (BRm )+ ! Am1
B ? C
Am ' hocolimB@ ffl?y CA.
S
34 GREGORY ARONE AND KATHRYN LESH
Corollary 8.4. Let A0 ! A1 ! . .b.e the sequence of spectra resulting from the
application of Construction 3.8 to the category of finite sets and isomorphisms.
Then Am ' Spm(S).
To prove these results, we need to identify the set of morphisms
[n; n1, . .,.nt] ! [r; r1, . .,.ru]
in F1 C as a certain set of equivariant maps in an orbit category of the groups*
* GN ,
where N = n + n1 + . .+.nt. The first additional assumption that we need on the
underlying category B has to do with the morphism sets.
Axiom 8.5 (Fixed Point Axiom). The pointed category B satisfies the property
that given two objects U, V 2 B, the only element of Morph B(U, V ) that is fix*
*ed
under the action of AutC(V ) is the null map, U ! 0 ! V .
The axiom clearly holds for the category of finitedimensional vector spaces *
*over
the real or complex numbers and isometric maps. It does not quite hold for the
category of finite pointed sets, because if V is the set with one nonbasepoint
element, then AutC(V ) is the trivial group, so any morphism U ! V is invariant
under the automorphisms of V . However, the axiom does hold if V is a set with
more than one element, and the main conclusions that we wish to draw do hold for
the category of finite sets. (See the proof of Corollary 8.4.) Similarly, for a*
* general
ring R, the category of free Rmodules may not satisfy Axiom 8.5 if, for exampl*
*e,
GL 1(R) is the trivial group. However, the axiom is satisfied if R is an integ*
*ral
domain in which 2 6= 0.
We require one last property of the categories B and C, which we assume for t*
*he
remainder of this section, along with Axioms 5.1 and 8.5. In order to set it ou*
*t, we
need the following definitions.
Definition 8.6. Let i : U ,! V V 0be a representative cofibration for a subobj*
*ect
U of V V 0in B.
(1) We say that the subobject U is invariant under AutC(V ) if for all oe 2
AutC(V ), there exists o 2 AutC(U) such that i O o = (oe idV)0O i.
(2) We say that U contains V if there exists an object V 00, a Cadmissible
isomorphism U ! V V 00, and a cofibration V 00,! V 0such that i factors
as a composite U ~=V V 00,! V V 0.
(3) We say that U is contained in V 0if i factors as a composite of a cofibr*
*ation
U ,! V 0with the standard inclusion V 0,! V V 0.
Axiom 8.7 (Invariance Axiom). The category B satisfies the property that if
i : U ,! V V 0is a subobject that is invariant under AutC(V ), then either U
contains V or U is contained in V 0.
The categories of finite pointed sets and finitedimensional real and complex
vector spaces satisfy the Invariance Axiom. The case of free Rmodules deserves*
* its
own lemma. (Our thanks to Bill Dwyer for a helpful conversation on this subject*
*.)
Lemma 8.8. If R is an integral domain with 2 6= 0, then the category of finitel*
*y
generated free modules over R satisfies Axiom 8.7.
Proof.Let U be a subobject of V V 0that is invariant under the action of GL (*
*V ).
There is an isomorphism i : U U0 ! V V 0, where U, U0, V, V 0are free Rmodul*
*es.
FILTERED SPECTRA 35
We assume that i(U) is invariant under GL (V ) and is not contained in V 0, and*
* we
prove that i(U) contains V .
We first assert that if r 2 R is nonzero and (rv, 0) 2 i(U), then (v, 0) 2 i(*
*U).
For suppose that (rv, 0) = i(u, 0), and suppose that (v, 0) has preimage (a, b)*
* under
the isomorphism i. Since i(ra, rb) = (rv, 0) = i(u, 0), we have (ra, rb) = (u, *
*0) and
thus rb = 0. Since r 6= 0 and V is a free module over the domain R, this tells *
*us
that b = 0, and it follows that (v, 0) = i(a, 0) 2 i(U).
Our next step is to prove that i(U) contains an element of the form (v, 0) wi*
*th
v 6= 0. Since i(U) is not contained in V 0, it contains an element of the form *
*(v, v0)
with v 6= 0. By invariance under GL (V ), we know i(U) contains (v, v0), and
therefore also (2v, 0). Then (v, 0) 2 V \ i(U) by the previous argument, since *
*2 is
nonzero in R.
Since V is a finitelygenerated free Rmodule, we can choose a basis for V and
write v = (r1, . .,.rn) where r1, . .,.rn 2 R. Since v 6= 0, suppose that r1 6=*
* 0. By
invariance of i(U) under GL (V ), we also know (r1, r2+ r1, . .,.rn) 2 V \ i(U)*
*, and
so (0, r1, 0, . .,.0) 2 V \ i(U). But r1 6= 0, so we find that (0, 1, 0 . .,.0)*
* 2 V \ i(U).
Since all permutations of the standard basis elements of V are in GL (V ), we f*
*ind
that all of the standard basis elements of V are in i(U), and we conclude that *
*i(U)
contains all of V , as required.
This completes the new assumptions that we need to make on the categories
B and C. We need more notation in order to describe how the category F1 C is
supposed to correspond to an orbit category. Given an object [n; n1, . .,.nt] *
*of
F1 C, let N = n + n1 + . .+.nt, let Gno = Gn1 x . .x.Gnt, and let O1 C be
the category whose objects are orbit spaces G=Gno for some [n; n1, . .,.nt], wi*
*th
morphisms being GN equivariant maps. That is, if r, r1, . .,.ru are integers *
*with
n + n1 + . .+.nt = r + r1 + . .+.ru, and G = Gn+n1+...+nt= Gr+r1+...+ru, then
G=Gno and G=Gro are two objects of O1 C and
Morph O1 C(G=Gno, G=Gro) = mapG (G=Gno, G=Gro).
Similarly, let Om C be the full subcategory of O1 C whose objects are orbit spa*
*ces
G=Gno where ni m for all 1 i t.
Recall from Remark 4.11 that
Morph Fm C([n; n1, . .,.nt], [r; r1, . .,.ru]) mapG (G=Gno, G=Gro).
It follows that for every m, including m = 1, there is a natural functor Fm C !
Om C sending [n; n1, . .,.nt] to G=Gno and inducing the above inclusion on mor
phisms. Further, O1 C is a symmetric monoidal category, with product given by
inducing up the product of two orbits, as in [11], and the functors Fm C ! Om C
are monoidal.
Proposition 8.9. Let B be a monogenic category that satisfies Axioms 5.1, 8.5
and 8.7, and let C be a subcategory of admissible isomorphisms. The functor F1 *
*C !
O1 C sending [n; n1, . .,.nt] to GN =Gno induces a bijection on morphism sets.
Proof.We only need to show that the map on morphism sets is surjective. Suppose
we have objects [n; n1, . .,.nt] and [r; r1, . .,.ru] with n+n1+. .+.nt= r+r1+.*
* .+.
ru, and let G = Gn+n1+...+nt. We show that an element of map G(G=Gno, G=Gro)
comes from a morphism
[n; n1, . .,.nt] ! [r; r1, . .,.ru]
36 GREGORY ARONE AND KATHRYN LESH
in F1 C.
Choose an element in mapG (G=Gno, G=Gro) ~=(G=Gro)Gno; suppose it is repre
sented by an isomorphism in G, say
f : gn gn1 . . .gnt ! gr gr1 . . .gru.
(Throughout this proof "isomorphism" means "Cadmissible isomorphism.") The
element f 2 G conjugates Gno into Gro, and so has the property that if _ 2 Gno,
then there exists OE 2 Gro such that the following diagram commutes:
gn gn1 . . .gntf!gr gr1 . . .gru
? ?
(8.1) _ ?y OE?y
gn gn1 . . .gntf!gr gr1 . . .gru.
On the other hand, a morphism [n; n1, . .,.nt] ! [r; r1, . .,.ru] is an equiv*
*alence
class of isomorphisms
gn gn1 . . .gnt ! gr gr1 . . .gru
Q
under the action of Aut(gri). We need to do three things:
(1) find an appropriate function ff : t_! u_;
(2) establish that f is compatible with ff, i.e., for each j 2 u_, the restr*
*iction
f i2ff1(j)gnifactors through the inclusion grj,! gr gr1 . . .gru;
(3) establish that fgn factors as gn ~=gr ge1 . . .geu ,! gr gr1 . . .gru.
We begin by constructing an appropriate ff. Consider f1 : gr gr1 . . .gru !
gn gn1 . . .gnt, and fix j 2 u_. The restriction of f1 to grj! gn gn1 . . .gnt
is a cofibration, and we define f1 (grj) to be the corresponding subobject of *
*gn
gn1 . . .gnt. Note that the subobject f1 (grj) is not dependent on the speci*
*fic
representative f, since any other representative is of the form OE O f for OE 2*
* Gro,
and the indeterminacy is taken care of in the equivalence relation of cofibrati*
*ons
that defines a subobject. Furthermore, because fQconjugates Gno into Gro, we see
that f1 (grj) is invariant under the action of Aut(gni), as expressed by (8.*
*1).
We now consider i = 1, . .,.t one element at a time, and decide whether or not
each one will be in ff1(j). Beginning with i = 1, we know that f1 (grj) is in*
*variant
under Aut(gn1). We apply Axiom 8.7, and one of two things can happen.
o The first possibility is that f10(grj) contains gn1. In this case, ther*
*e exists
an isomorphism h : grj~=gn1 gn1 and a direct sum of of the identity map
on gn1 with another cofibration
0 n n n n
gn1 gn1 ,! g 1 (g g 2 . . .g t)
such that f1 grjfactors through the sum. We define ff(1) = j, accepting
1 as an element of ff1(j). 0
To iterate, we need to show that the subobject defined by gn1 ,! gn
gn2Q . . .gnt is still invariant under the action of any element _ 2
t ni
i=2Aut (g ). Let OE be the element corresponding to _ that is given
FILTERED SPECTRA 37
by (8.1), say with component OEj : grj ! grj. We have a commuting dia
gram
grj h!~=gn1 gn01!gn1 (gn gn2 . . .gnt)
? ?
OEj?y~= _?y~=
grj h!~=gn1 gn01!gn1 (gn gn2 . . .gnt).
Invariance will now follow once we show that hOEjh10is the coproduct of
the identity map on gn1 with an isomorphism on gn1. This last follows
from a calculation using Lemma 5.2; the input for the lemma is provided
by going clockwise around the diagram instead and using the fact that if
the composite of a map with a cofibration is null, then the original map*
* is
null as well.
o The second possibility is that f1 (grj) is contained in gn gn2 . . .*
*gnt.
This gives us a cofibration grj,! gn gn2 . .Q.gnt through which f1 grj
factors, and again, this is invariant under ti=2Aut(gni). Observe that*
* this
implies that the composite
grj,! gr gr1 . . .gru ! gn gn1 . . .gnt ! gn1
is null. In this case, we reject 1 as an element of ff1(j).
We iterate this procedure t times, each time accepting or discarding i as an
elementPof ff1(j). After we have completed the t iterations, we define ej = rj*
* 
i2ff1(j)ni, and we have an isomorphism ij : grj ~=gni1 . . .gnik gej and
a cofibration gej ,! gn such that f1 grjis the composition of ij with the dir*
*ect
sum of the natural inclusions on gni1 . . .gnikand a cofibration gej,! gn. Note
that we do not yet know that ff is a welldefined function, since it is not obv*
*ious
that the sets ff1(j) are disjoint for different j, nor that each i 2 t_is in o*
*ne of the
sets ff1(j). We will check these below.
Our next step is to prove that f1 gr factors through gn, for which we need
Axiom 8.5. We assert that for each i, the composite morphism gr ! gn gn1
. . .gnt ! gni is invariant under Aut(gni), and therefore is null by Axiom 8.5.
This follows from diagram (8.1), since no OE 2 Gro has any effect on gr, and th*
*us
any element _ 2 Gno must fix the actual morphism gr ! gn gn1 . . .gnt, and
not just the subobject it defines. Therefore, by applying Corollary 5.3 we find*
* that
f1 gr factors through gn ,! gn gn1 . . .gnt.
The result of the preceding constructions is that we have a commuting diagram
~=
gr gr1 . . .gru! gr j2u_ i2ff1(j)gni gej
?? ?
yf1 ?y~=
gn [gn1 . . .gnt] j2u_gej gr j2u_ i2ff1(j)gni .
The top row is an isomorphism, since for each j, the map grj! ( i2ff1(j)gni) g*
*ej
is an isomorphism by construction. The right vertical map is an isomorphism
because it simply reshuffles and reassociates the terms. Thus the bottom row is
also an isomorphism, and, by construction, it is the coproduct of two maps (with
the association shown), each of which is therefore an isomorphism by Condition *
*(4)
in Definition 4.2. We are immediately entitled to conclude that fgn factors as
38 GREGORY ARONE AND KATHRYN LESH
gn ~=gr ge1 . . .geu ,! gr gr1 . . .gru, the third condition we needed in
order for f to represent a morphism in F1 C.
Finally, to check that ff is indeed a welldefined function and that f is com*
*patible
with ff, we consider the other summand of the bottom row of the diagram,
n n n n
(8.2) i2ff1(1)g i. . . i2ff1(u)g i! g 1 . . .g t,
which is also an isomorphism by Condition (4)in Definition 4.2. Notice that this
map is a coproduct of natural inclusions of summands, though it is in principle
possible for some of the summands to appear on the left more than once, while
others might not appear at all. In fact, in order to know that ff is a welldef*
*ined
function t_! u_, we need to prove precisely this does not happen, that is, we n*
*eed to
know that ff1(1), . .,.ff1(u) is actually a partition of t_. We assert that t*
*he union
of ff1(1), . .,.ff1(u) is t_, which is to say that every niappears on the lef*
*t of (8.2),
because otherwise the isomorphism (8.2)would factor through a proper subobject
of gn1 . . .gnt, which would have lower dimension, violating the assumption
of invariance of dimension in B. The proof is concluded by noting that the total
dimension on each side of the isomorphism must be the same, and so each ni can
only appear once on the left side, establishing that the sets ff1(j) are disjo*
*int for
different values of j.
The proofs of Theorem 8.2 and Corollaries 8.3 and 8.4 follow easily.
Proof of Theorem 8.2.The functor F1 C ! O1 C is a monomorphism on objects,
and every object of O1 C is isomorphic to one of the form G=Gno. By Proposi
tion 8.9, the image of F1 C ! O1 C is a full subcategory of O1 C.
Proof of Corollary 8.3.By Theorem 3.9, we have a diagram
0 1
1 (BKm1 Cm )+ ! Am1
B ? C
Am ' hocolimB@ ffl?y CA,
S
so we must show that BKm1 Cm ' BRm . However, BRm = BHm1 (Gm ), and
by Proposition 8.9, BHm1 (Gm ) ' BFm1 Cm . Finally, Theorem 7.1 says that
BFm1 Cm ' BKm1 Cm .
Proof of Corollary 8.4.Strictly speaking, Axiom 8.5 does not hold in the case t*
*hat
B is finite pointed sets; however, it still is true that Fm C and Om C are conn*
*ected
by a chain of monoidal functors, each inducing an equivalence of classifying sp*
*aces.
One way to see this is to recall that Fm C is connected in this way to the cate*
*gory
of "finite sets with multiplicity at most m" (see Example 4.10), and it is easy*
* to
see directly that the latter category is connected in the appropriate way to Om*
* C.
Let F 0( m ) denote the collection of all nontransitive subgroups of m . (T*
*he
collection F 0( m ) is actually a "family" of subgroups, meaning that it is clo*
*sed
under taking subgroups as well as under conjugation, but that does not concern *
*us
here.) The corollary follows from [12] once we know that BF 0( m ) ' BHm1 ( m *
*),
which is true because to every nontransitive subgroup of m , we can associate *
*the
unique minimal standard subgroup of filtration m  1 that contains it, and the
association is compatible with conjugation.
FILTERED SPECTRA 39
9. Relationship with the poset of directsum decompositions
For the remainder of the paper, we focus our attention on the example of fini*
*te
dimensional unitary vector spaces, and its relationship with the example of fin*
*ite
sets, and we use Am specifically to refer to the result of applying our constru*
*ction
to the former. Our goal in this section is to provide another description of t*
*he
filtration subquotients Am =Am1 in the unitary case, in preparation for relati*
*ng
our filtration to the calculus of functors. In particular, we prove Theorem 9.*
*5,
which expresses the subquotients of our filtration of bu in terms of certain sp*
*aces
of directsum decompositions, and Theorem 9.7, which codifies their behavior at*
* a
prime p.
We need some definitions. Recall that by a subgroup of a Lie group we al
ways mean a closed subgroup, and that the standard subgroups of U(m) are those
conjugate to U(m1) x . .x.U(ms). (See Definition 8.1.)
Definition 9.1. Let H be a subgroup of U(m) for m 1.
(1) We say H is bad if the action of H on Cm breaks Cm into a sum Cm ~=fl k
where fl is irreducible and nontrivial. That is, H is bad if it acts iso*
*typically
with nontrivial irreducibles.
(2) If H is not bad, then we call H good. Q
(3) IfPH is a standard subgroup conjugate to one of the form si=1U(mi) whe*
*re
mi= m and s > 1, then we call H complete.
Example 9.2. The trivial subgroup {e} U(m) is good. The center S1 =
Z(U(m)) is bad. The whole group U(m) is bad.
We will work with the following collections of subgroups of U(m):
Gm = {all good subgroups of U(m)}
Rm = {proper standard subgroups of U(m)}
Lm = {complete subgroups of U(m)}.
We note that Lm , regarded as a poset under inclusion, can be identified with t*
*he
poset of proper directsum decompositions of Cm by taking a subgroup to the dir*
*ect
sum decomposition given by its irreducible subspaces. Let Gm,ntrvand Rm,ntrv
be the collections consisting of nontrivial elements of Gm and Rm , respective*
*ly.
Note that there are inclusions Lm ,! Rm ,! Gm . There are also maps back,
Gm,ntrv! Rm,ntrv! Lm which, although not maps of posets, turn out to be useful
in the proofs of Propositions 9.8 and 9.9, where they are described in more det*
*ail.
Remark 9.3. The poset of proper directsum decompositions of Cm , was studied
in [1]. The space denoted Lm  in this paper is homeomorphic to the space deno*
*ted
"Lm, in [1], page 458, while the space Lm  introduced below is homeomorphic *
*to
the space denoted Lm (or LCm) in [1].
Given a collection H of subgroups of a group G, let EH denote the universal
space of the collection. The space EH can be constructed as the nerve of the
category whose objects are pairs (G=H, x) with H 2 H and x 2 G=H, and whose
morphisms (G=H, x) ! (G=H0, x0) are Gequivariant maps G=H ! G=H0 taking
x to x0. The space EH is characterized by the following two properties: (1) EHK
is contractible for all K 2 H, and (2) all the isotropy groups of EH are elemen*
*ts
of H. (See [2], Lemma 2.3.) The classifying space BH of the collection H is the
40 GREGORY ARONE AND KATHRYN LESH
orbit space of EH under the action of G. Recall from [5], 2.12 that there is an
equivariant map EH ! H that is a nonequivariant homotopy equivalence.2 (We
will call such a map an equivariant weak equivalence for short.) In particular,*
* there
is an equivariant weak equivalence ELm ' Lm .
Given a space X, let X denote its unreduced suspension. If X is a pointed
Gspace, let X"hGdenote the based homotopy orbits of the action of G on X:
namely X"hG= (EG xG X)=(BG x *). We let U(m) act on S2m by the onepoint
compactification of of its action on Cm . Thus the action of U(m) fixes the bas*
*epoint
of S2m , which is the point at infinity, as well as fixing the origin. Our main*
* technical
goal in this section is to prove the following theorem.
Theorem 9.4.
BRm ' (ELm ^ S2m )"hU(m)' (Lm  ^ S2m )"hU(m).
The proof of the theorem is deferred to later in the section, following the p*
*roof of
Corollary 9.14. Let us first note some consequences of the theorem for our filt*
*ration
of bu.
Theorem 9.5. There is an equivalence of spectra
Am =Am1 ' 1 (Lm  ^ S2m )"hU(m),
where Lm is the category of proper directsum decompositions of Cm .
Proof.Follows immediately from Corollary 8.3 and Theorem 9.4.
The complexes Lm  were first introduced in [1] because they play an import*
*ant
role in Weiss's orthogonal calculus. They have many interesting properties, som*
*e of
which we will discuss in the next section, where we pursue further the relation*
*ship
of our present work with the calculus of functors. For starters, we have the fo*
*llowing
proposition.
Proposition 9.6. The complex Lm  is rationally trivial for m > 1. For a prime
p, the complex Lm  is trivial at p unless m is a power of p. In particular, *
*Lm 
is contractible unless m is a power of a prime.
Proof.This is implicit in [1]. In fact, two independent proofs of this result *
*are
given there, but the statement is never made explicit. Let Sadm be the onepoint
compactification of the adjoint representation of U(m). Perhaps the quickest way
to get the results of the proposition is to start with the first sentence of [1*
*], Theorem
4(a) and the first two sentences of [1], Theorem 4(b). When unraveled, these say
that
map Lm  , 1 Sadm
is contractible rationally if m 2, is contractible unless m is a power of a p*
*rime,
and is all ptorsion when m is a positive power of p.
Since Lm  is a finite complex, it follows by SpanierWhitehead duality that
1 Lm  is rationally acyclic for m 2, and is acyclic at p unless m is a po*
*wer
of p. Therefore Lm  has the same properties. Since Lm  is connected, Lm *
* is
simply connected, and acyclicity implies contractibility.
Another way to prove the result is to use [1], Theorem 11, and then apply the
theory of homology approximations.
____________
2The result is only stated for finite groups, but the same argument works fo*
*r compact Lie
groups.
FILTERED SPECTRA 41
As a consequence, we have the following theorem.
Theorem 9.7. Let p be a prime. The successive quotients Am =Am1 are trivial at
p except when m = pk for some k 1, and are rationally trivial for m 2.
We need a series of results to prepare the ground for proving Theorem 9.4.
The collections Gm , Rm , and Lm have actions of U(m) by conjugation, and the
inclusions
Lm ,! Rm ,! Gm .
are U(m)equivariant. The following two propositions give the foundation for the
relationship between Gm , Rm and Lm that we need. Their proofs are very similar
but not particularly enlightening, and are given at the end of the section.
Proposition 9.8. The map
ELm ,! ERm,ntrv
is a U(m)equivalence.
Proposition 9.9. The map
ERm ,! EGm
is a U(m)equivalence.
Corollary 9.10. The map
ERm,ntrv! EGm,ntrv
is a U(m)equivalence.
Proof.We assert that the singular set of ERm is ERm,ntrv, which is easily veri*
*fied
by checking the isotropy groups of the chains that form the simplices of ERm .
Likewise, the singular set of EGm is EGm,ntrv. The corollary now follows from
Proposition 9.9 and [2], Lemma 2.4.
In contemplating the desired equivalence of Theorem 9.4, we see that Proposi
tion 9.8 gives us an equivalence
(9.1) (ELm ^ S2m )"hU(m)' (ERm,ntrv^ S2m )"hU(m).
Thus we tackle the righthand side of this equivalence and seek to relate it to*
* BRm .
In diagram (9.2)below, we will eventually be proving that the upper righthand
corner and the lower lefthand corner are contractible (easy, and Proposition 9*
*.13,
respectively).
Proposition 9.11. The following diagram is a homotopy pushout diagram:
(ERm,ntrv^ S2m )"hU(m)! (ERm ^ S2m )"hU(m)
? ?
(9.2) ?y ?y
(ERm,ntrv^ S2m )U(m) ! (ERm ^ S2m )U(m)
Proof.As observed in the proof of Corollary 9.10, the singular set of ERm is
ERm,ntrv. Therefore (ERm,ntrv^S2m ) contains the singular set of (ERm ^S2m ),
and the cofiber of the map
(ERm,ntrv^ S2m ) ! (ERm ^ S2m )
has a free action of U(m) (in the based sense). It follows that orbits and homo*
*topy
orbits on this cofiber are homotopy equivalent. We conclude that the induced map
42 GREGORY ARONE AND KATHRYN LESH
between the horizontal cofibers in (9.2)is a homotopy equivalence, which establ*
*ishes
the proposition.
Our next step is to show that the orbit space ERm,ntrv^ S2m U(m) in the
lower lefthand corner of (9.2)is contractible. First we look at orbit spaces f*
*or S2m
alone.
Lemma 9.12. If H 2 Rm,ntrv, then (S2m )H ' *.
Proof.First we note that (S2m )U(m) ' *. Then consider H = U(m1) x . .x.
U(ms) 2 Rm,ntrv, where at least one mi> 0. Then
(S2m )H ~=(S2m1)U(m1)^ . .^.(S2ms)U(ms)^ S2(mm1...ms),
and this establishes the result.
Now we are in a position to prove that the lower lefthand corner of (9.2)is
contractible.
Proposition 9.13.
ERm,ntrv^ S2m U(m)' *.
Proof.We argue by direct analogy to [2], Lemma 7.11 and Lemma 7.12. Suppose
that X is a pointed U(m)space whose isotropy groups are in Rm,ntrv[ {U(m)}.
We prove that (X ^ S2m )U(m) is contractible by inducting over the cells of X. *
*The
statement is certainly true if X = *. It is sufficient to show that if (Y ^ S2m*
* )U(m)
is contractible and X is obtained from Y by adding a single U(m)cell of the fo*
*rm
(U(m)=H) x( [i], @ [i]), where H = U(m1)x. .x.U(ms) with at least one mi> 0,
then (X ^ S2m )U(m) is contractible. Then we have a cofiber sequence
(Y ^ S2m )U(m) ! (X ^ S2m )U(m) ! Si^ (S2m )H ,
and the first and third terms are contractible.
In our goal of proving Theorem 9.4, we are now able to relate the right side
of the equivalence in (9.1)to (ERm ^ S2m )U(m), a space more clearly related to
BRm , which is, after all, an orbit space.
Corollary 9.14. (ERm,ntrv^ S2m )"hU(m)' (ERm ^ S2m )U(m).
Proof.Applying Proposition 9.13 to (9.2)tells us that the lower lefthand corne*
*r of
(9.2)is contractible. On the other hand, Rm contains the trivial subgroup, so E*
*Rm
is contractible, which makes the upper righthand corner of (9.2)contractible.
We are now in a position to give the proof of Theorem 9.4.
Proof of Theorem 9.4.We must establish that
BRm ' (ELm ^ S2m )"hU(m),
and then the other equivalence of the theorem follows from the equivariant weak
equivalence ELm ! Lm . We know from (9.1)that
(ELm ^ S2m )"hU(m)' (ERm,ntrv^ S2m )"hU(m).
By Corollary 9.14,
(ERm,ntrv^ S2m )"hU(m)' (ERm ^ S2m )U(m).
FILTERED SPECTRA 43
We must pick out BRm from (ERm ^ S2m )U(m).
The main calculation is to prove that the inclusion S0 ! S2m induces a U(m)
equivalence
(9.3) (ERm ^ S0) ! (ERm ^ S2m ).
We argue as in [2], Lemma 7.12. By Proposition 9.9, we know that ERm ! EGm
is a U(m)equivalence, so it suffices to prove that
(EGm ^ S0) ! (EGm ^ S2m )
is a U(m)equivalence. For this, we check the fixedpoint sets
(EGm ^ S0)H ! (EGm ^ S2m )H
for all subgroups H U(m). There are two cases. If FixH(Cm ) = 0, then (S2m )H*
* =
S0, because H also fixes the point at infinity. Thus for these subgroups H, the*
* map
on fixedpoint sets will be an equivalence because of what happens on the spher*
*es.
On the other hand, if H acts on Cm with FixH(Cm ) 6= 0, then H must be a good
subgroup, and so (EGm )H ' *. In this case, the map of fixedpoint sets is a map
between contractible spaces.
Because (9.3)is a U(m)equivalence, there is an equivalence of orbit spaces
(ERm ^ S0)U(m) ! (ERm ^ S2m )U(m).
But (ERm ^ S0)U(m) is BRm , which finishes the proof.
We close the section with the proofs of Propositions 9.8 and 9.9. We will need
the following characterizations of "standard" and "complete" in those proofs.
Lemma 9.15.
(1) A subgroup H U(m) is standard iff for every Hirreducible subspace
V Cm , either U(V ) H or U(V ) \ H = {e}.
(2) A subgroup H U(m) is complete iff for every Hirreducible subspace
V Cm , we have U(V ) H.
Proof of Proposition 9.8.We need to check that the map is a homotopy equivalence
on the fixedpoint sets:
(ELm )H ! (ERm,ntrv)H .
By Lemma 2.4 of [2], it is sufficient to check this for H 2 Rm,ntrv(because Lm
Rm,ntrv). The characteristic property of the universal space for a collection m*
*eans
that if H 2 Rm,ntrv, then (ERm,ntrv)H is contractible, so we must show that
(ELm )H ' *. Let H # Lm denote the poset of subgroups {K 2 Lm H K}. By
[5], Lemma 2.14, we know that (ELm )H is homotopy equivalent to H # Lm .
Given H 2 Rm,ntrv, we define ^H2 Lm (the "completion of H") as follows. Let
V = FixH(Cm ), and note that H U(V ?). Because H is nontrivial, we know that
V 6= Cm . Let ^H= H x U(V ) 2 (H # Lm ). Thus ^H U(V ?) x U(V ).
We assert that there is a right adjoint to the inclusion functor between the *
*posets
(H # Lm ) \ (Lm # ^H) ,!(H # Lm ).
The retraction is given by taking K 2 (H # Lm ) to K \ ^H, which we claim is an
element of (Lm # H^). Certainly K \ ^H H^, and the only question is whether
K \ ^His complete, i.e., an object of Lm .
44 GREGORY ARONE AND KATHRYN LESH
Note that the isotypical summands of ^Hare the same as those of H, and the on*
*ly
difference is that if H is not complete, then ^H, which is complete, turns FixH*
*(Cm )
into a single nontrivial irreducible summand of ^H. To check that K \H^ is comp*
*lete,
we use Lemma 9.15 (2). Suppose that W is a (K \ ^H)irreducible subspace. Note
that since H K \ ^H ^H, then W is contained in an ^Hirreducible, which means
that either W is actually a nontrivial Hirreducible or W FixH(Cm ).
o If W is a nontrivial Hirreducible, then U(W ) H since H is standard,
and thus U(W ) K \ ^H.
o If W FixH(Cm ), then U(W ) H^ by definition of H^. Further, since
K \ ^H K, we must have W W 0for an irreducible Kmodule W 0,
and so U(W ) U(W 0) K because K is complete. Thus again we have
U(W ) K \ ^H.
Thus K \ ^His complete.
But now we note that (H # Lm ) \ (Lm # ^H) has a terminal object, namely ^H.
Thus (ELm )H ' H # Lm  ' (H # Lm ) \ (Lm # ^H) ' *, which is what we needed
to establish.
Remark 9.16. It is necessary to have H nontrivial in the proof of Proposition 9*
*.8.
If H happens to be trivial, then the proof has H = {e}, ^H= U(m), (H # Lm ) =
Lm , and (Lm # ^H) = Lm . The problem is that (Lm # ^H) = Lm no longer has ^H
as a terminal object, because ^H= U(m) =2Lm .
Proof of Proposition 9.9.We need to check that the map is a homotopy equivalence
on the fixedpoint sets:
(ERm )H ! (EGm )H .
As in the preceding proof, it is sufficient to check that if H 2 Gm , then (ERm*
* )H ' *,
i.e., that H # Rm  ' *.
Given H 2 Gm , a "good" subgroup, we define ^H2 Rm as follows. Either H is
trivial, or the subgroup H acts on Cm with more than one isotypical summand by
definition, so the isotypical summands of H give a decomposition Cm ~=V0 V1
. . .Vk, where V0Qis the summand with a trivial action of H, and either V0 6= 0*
* or
k > 1. Let ^H= ki=1U(Vi). If V0 6= 0, then certainly ^His a proper subgroup of
U(m), and if V0 = 0, then k > 1, and so again ^His a proper subgroup of U(m). In
both cases, ^His in Rm .
We assert that are adjoint functors between the posets
(H # Rm ) \ (Rm # ^H) ,!(H # Rm ),
where the retraction is given by K 7! K \ ^H. What is at issue is whether K \ ^H
is in Rm , i.e., whether K \ ^His standard. The input is that K is standard and
H K.
We will use Lemma 9.15 (1). Suppose that V is a (K \ ^H)irreducible subspace
of Cm . Because K \ ^H H^, the subspace V is contained in an H^irreducible
subspace W , and because ^His standard, either U(W ) ^Hor U(W ) \ ^H= {e}.
By a similar argument, either U(W ) K or U(W ) \ K = {e}. Putting these
together, we find that either U(W ) (K \ ^H) or U(W ) \ (K \ ^H) = {e}, and so
K \ ^His standard.
FILTERED SPECTRA 45
But ^H, being standard, is an element of Rm , and thus (H # Rm ) \ (Rm # ^H)
has ^Has a terminal object. Therefore (H # Rm ) \ (Rm # ^H) is contractible, *
*and
so is H # Rm . Thus (ERm )H ' * for H 2 Gm , which finishes the proof of the
proposition.
10.Partition posets, Tits buildings and duality, revisited
Our main goal in this section is to prove the following theorem, whose proof
is found at the end of the section. This is a shifted selfduality result for *
*the
complexes Lpk of directsum decompositions introduced in Section 9. In view
of the description in [2], 1.16, we can see that Corollary 10.2 is analogous to*
* [2],
Theorem 1.17.
Theorem 10.1. Let W be a spectrum with an action of U(pk). There is a U(pk)
equivariant map that is a mod p equivalence
i j
Sk+1 ^ Lpk ^ W ' Sk ^ map Lpk , W ^ Sadpk,
where Sadpkis the onepoint compactification of the adjoint representation of U*
*(pk).
Corollary 10.2. There is a mod p equivalence of homotopy orbit spectra
i j
Sk+1 ^ (Lpk ^ W )"hU(pk)' Sk ^ map Lpk , W ^ Sadpk" .
hU(pk)
We will also need the following result in Section 11, and we give its proof a*
*t the
very end of the section. The group k is described below in (10.1).
Proposition 10.3. The homotopy orbit spectra of Corollary 10.2 are wedge sum
mands of S1 ^ W"h k(after pcompletion).
The proof of Theorem 10.1, which is the penultimate item in the section, is
closely parallel to Sections 8 and 9 of [2]. In particular, we approximate both
sides of the desired equivalence using an appropriate (symplectic) Tits building
(Proposition 10.6 and Corollary 10.9), and then show that the Tits building in
question has a selfduality property (Proposition 10.10) that gives a bridge be*
*tween
the two.
We begin by discussing the grouptheoretic aspects. A crucial ingredient of [*
*2]
is the subgroup k pk, which is characterized uniquely up to conjugacy by
being a transitive elementary abelian psubgroup of pk. The subgroup k can be
thought of as (Fp)k acting on itself by translation to give a transitive subgro*
*up of
the permutation group of pk objects. The normalizer N k can be identified with
the affine group Affk(Fp), and the Weyl group(N k)= k is GL k(Fp), so we have
a short exact sequence
1 ! k ! Affk(Fp) ! GL k(Fp) ! 1.
The analogous subgroup of the unitary group is k U(pk). It is described
in detail in [15] (see also [1], p. 464), but we give a brief description here*
*. The
subgroup k is a central extension of (Z=p)2k by the circle group S1 that is the
center of U(pk):
(10.1) 1 ! S1 ! k ! (Z=p)2k ! 1.
The realization by matrices of the quotient k=S1 ~= (Z=p)2k uses permutation
matrices for one of the copies of (Z=p)k, via the inclusion k ~=(Z=p)k ,! pk *
*,!
46 GREGORY ARONE AND KATHRYN LESH
U(pk), while the other copy of (Z=p)k is realized by certain diagonal matrices.*
* The
subgroup k is irreducible, corresponding to the fact that k is transitive. Le*
*t Ak
be the normalizer of k in U(pk). There is a short exact sequence
1 ! k ! Ak ! Sp2k(Fp) ! 1,
where Sp2k(Fp) is the symplectic group, which is therefore the Weyl group of k
in U(pk).
The subgroup k also has a uniqueness property analogous to that of k. We
follow the terminology of [1] and call a subgroup of U(m) a "projective element*
*ary
abelian psubgroup" if it is the extension of an elementary abelian psubgroup *
*of
U(m) by S1 = Z(U(m)). Then it is a standard fact from projective representation
theory that k is, up to conjugacy, the unique irreducible projective elementary
abelian psubgroup of U(pk) (e.g., see [22]). In fact, just as m has no transi*
*tive el
ementary abelian psubgroups unless m = pk for some k, so U(m) has no irreducib*
*le
projective elementary abelian psubgroups unless m = pk.
The next step is to introduce the symplectic Tits building. Because [ k, k] =
Z=p S1 = Z( k), the commutator defines a skewsymmetric bilinear map [, ] :
k x k ! Z=p . Moreover, since the connected component of the identity is the
center, it follows that the commutator passes to a skewsymmetric bilinear form*
* on
ss0( k) ~= k=S1 ~=F2kp.
The form is clearly nondegenerate, and therefore is a symplectic form.
Recall that a subspace V of a symplectic space is called coisotropic if V ? *
* V .
Let us say that a subgroup of k is "coisotropic" if it is the inverse image of*
* a
coisotropic subspace of F2kpunder the projection k ! F2kp. Let TSp2k denote the
collection of proper coisotropic subgroups of k. We define the Tits building f*
*or the
symplectic group Sp2k(Fp) to be  TSp2k , which turns out to have the homotopy
type of a wedge of spheres of dimension k.
To relate the symplectic Tits building to the constructions of Section 9, we *
*prove
the following lemma.
Lemma 10.4. A proper coisotropic subgroup of k is a good subgroup of U(pk), in
the sense of Definition 9.1.
Proof.Let H be a proper coisotropic subgroup of k, and suppose that H iskbad.
Then the restriction to H of the standard representation of U(pk) on Cp is of *
*the
form V Cs, where V is a nontrivial irreducible representation of H and Cs is *
*the
trivial sdimensional representation. It follows that dim(V )  pk, so dim(V ) *
*= pl
for some l k. Moreover, since H acts faithfully on V Cs, it follows that H
acts faithfully on V , and so H is an irreducible subgroup of U(pl). Since H is
also a projective elementary abelian psubgroup, it follows that H ~= l. On the
other hand, the assumption that H is a proper coisotropic group of k implies t*
*hat
ss0(H)? is a nontrivial subgroup of ss0(H). But then H cannot be isomorphic to
l, because ss0( l)? = {0}, and we reach a contradiction.
Recall that Gpk,ntrvdenotes the collection of all nontrivial good subgroups of
U(pk). The inclusion of collections TSp 2k,! Gpk,ntrvis equivariant with respe*
*ct
to the group inclusion Ak ,! U(pk), and it passes to a based map of unreduced
suspensions E TSp2k ! EGpk,ntrv. Inducing up, we get a U(pk)equivariant map
U(pk)+ ^Ak E TSp2k ! EGpk,ntrv. The following proposition is a special case of
FILTERED SPECTRA 47
[1], Theorem 11, namely the case P = Z(U(pk)) = S1, which acts trivially on both
U(pk)+ ^Ak E TSp2k and EGpk,ntrv. Although the definition of "good subgroup" in
this current work is not exactly the same as that used in [1] (where it is requ*
*ired
that the center, S1, be contained in a good subgroup), an argument with fixed
point sets shows that the space E(Fpk) of [1] is U(pk)equivariantly equivalent*
* to
EGpk,ntrv.
Proposition 10.5. The above U(pk)equivariant map is a mod p homology equiv
alence
U(pk)+ ^Ak (E TSp2k) ! (EGpk,ntrv) .
We will not reproduce the proof here, but we remark that an essential input
is that all irreducible projective elementary abelian psubgroups of U(pk) are *
*con
jugate to k, just as it is essential for [2] that all transitive elementary ab*
*elian
psubgroups of pk are conjugate to k.
Thus we arrive at the following mod p approximation result for the complex
Lpk of directsum decompositions.
Proposition 10.6. There is a chain of U(pk)equivariant maps that are mod p
equivalences
U(pk)+ ^Ak  TSp2k 'p Lpk .
Proof.There is a commuting square
U(pk)+ ^Ak (E TSp2k) 'p!(EGpk,ntrv)
? ?
(10.2) '?y '?y
U(pk)+ ^Ak  TSp2k ! Gpk,ntrv ,
where the top map is the mod p homology equivalence of Proposition 10.5, and
where the vertical maps are induced by E TSp2k !  TSp2k and EGpk,ntrv!
Gpk,ntrv, which are weak equivalences of spaces and equivariant with respect *
*to Ak
and U(pk), respectively, by [2], 2.10.
The proposition follows, once we establish that Lpk ! Gpk,ntrv is a weak
equivalence. By Proposition 9.8, the inclusion ELpk ! ERpk,ntrvis a U(pk)
equivalence, giving us a weak equivalence Lpk ! Rpk,ntrv. By Corollary 9.1*
*0,
ERpk,ntrv! EGpk,ntrvis a U(pk)equivalence, which gives us a weak equivalence
Rpk,ntrv ! Gpk,ntrv, finishing the proof.
Remark 10.7. Proposition 10.6 is implicit in [1], Corollary 12, in which there *
*are,
however, a couple of misprints. The most serious one is that in the description*
* of
the map at the end of [1], Corollary 12, the wrong adjoint sphere is indicated *
*as the
target of the map. It has to be the adjoint sphere of Ak (which is just S1 with*
* the
trivial action), rather than the adjoint sphere of U(pk). We will correct the t*
*ypo
in Corollary 10.9.
Proposition 10.6 has the following immediate consequence.
Corollary 10.8. Let W be a spectrum with an action of U(pk). There is a chain
of mod p equivalences
( TSp2k ^ W")hAk' Lpk ^ W"hU(pk)
48 GREGORY ARONE AND KATHRYN LESH
We also obtain the following dual result, which is a correction of [1], Corol*
*lary
12.
Corollary 10.9. Let W be a spectrum with an action of U(pk). Let Sadpkbe the
onepoint compactification of the adjoint representation of U(pk). There is a c*
*hain
of U(pk)equivariant maps that are mod p equivalences
i j
(10.3) map Lpk , W ^ Sadpk' U(pk)+ ^Ak map  TSp2k , W ^ S1.
Proof.Regarding the lefthand side of (10.3), it follows immediately from Propo
sition 10.6 that there is a mod p equivalence via equivariant maps
i j i j
map Lpk , W ^ Sadpk' map U(pk)+ ^Ak  TSp2k , W ^ Sadpk.
To transform the righthand side of (10.3), we need the Wirthm"uller isomor
phism, a standard fact in equivariant SpanierWhitehead duality. (See, for exam
ple, [13].) Let G be a compact Lie group with a closed subgroup H, and let E be*
* a
spectrum with an action of H. Let SG and SH be the onepoint compactifications
of the adjoint representations of G and H, respectively. We give G+ ^ (E ^ SH )*
* a
left action of GxH by letting letting H act diagonally on G+ ^(E ^SH ) and lett*
*ing
G act on the right of G+ by the inverse. This action passes to a left action of*
* G on
G+ ^H (E ^ SH ). Similarly, we define a left action of G x H on map (G+ , E ^ S*
*G )
as follows: H acts diagonally on E ^ SG and by conjugation on a map f, and G
acts by precomposition by right multiplication. Thus we obtain an action of G on
mapH (G+ , E ^ SG ). Then the Wirthm"uller isomorphism says that there is a cha*
*in
of Gequivariant maps that are nonequivariant equivalences
G+ ^H (E ^ SH ) ' mapH (G+ , E ^ SG ).
We apply the Wirthm"uller isomorphism with G = U(pk), H = Ak, and E =
map ( TSp2k , W.) Note that Ak is a 1dimensional group with trivial adjoint
representation, so the onepoint compactification of the adjoint representation*
* of
Ak is the circle with trivial action. Thus we get
1
U(pk)+ ^Ak map( TSp2k , W ) ^ S
i j
' mapAk U(pk)+ , map( TSp2k , W ) ^ Sadpk.
Finally, standard arguments in equivariant topology show that there is a U(pk*
*)
equivariant equivalence
i j
map U(pk)+ ^Ak  TSp2k , W ^ Sadpk
i j
' mapAk U(pk)+ , map( TSp2k , W ^ Sadpk),
We have now established approximation results for the space of directsum de
compositions in terms of the symplectic Tits building. The other ingredient tha*
*t we
need is a selfduality result for the symplectic Tits building. We follow close*
*ly the
discussion in [2], Section 8, where a similar result is established for the Tit*
*s building
for the general linear group.  TSp2k has a natural structure of a kdimensio*
*nal
simplicial complex, with kdimensional simplices identified with Sp2k(Fp)=B, wh*
*ere
FILTERED SPECTRA 49
B is the Borel subgroup of upper triangular symplectic matrices. It follows that
there is a map of spectra
ff : Sk ^ 1  TSp2k ! 1 (Sp2k(Fp)=B)+
obtained by desuspending k times the map that collapses the (k  1)skeleton to*
* a
point. Dualizing, we obtain the map
D(ff) : map ((Sp2k(Fp)=B)+ , S)! Sk ^ map( TSp2k , S).
Moreover, since Sp2k(Fp) is a finite group, there is an Sp2k(Fp)equivariant we*
*ak
equivalence fi : 1 (Sp2k(Fp)=B)+ ! map((Sp2k(Fp)=B)+ , S).
Proposition 10.10. The composition
D(ff) O fi O ff : Sk ^ 1  TSp2k ! Sk ^ map( TSp2k , S)
is an Akequivariant map inducing an isomorphism in mod p homology.
Proof.The proposition is analogous to [2], Theorem 8.2, where the same result
is proved for the general linear group, and we sketch an entirely2analogous pro*
*of.
First, both spectra2are homotopy equivalent to a wedge of pk copies of the sph*
*ere
spectrum, pk being the rank of the symplectic Steinberg representation. On the
unique nontrivial mod p homology group, the map induces the homomorphism
S : Hk( TSp2k ; Fp) ! Fp[Sp2k(Fp)=B] ! Hk( TSp2k ; Fp).
This homomorphism is analogous to the homomorphism of the same name at the
bottom of [2], page 249.
We need to prove that S is an isomorphism, and the proof proceeds exactly as
in [2], Lemma 8.1, the only difference being that the Weyl group of the symplec*
*tic
group is a reflection group of type Ck, rather than Ak. Therefore, we have k
idempotents e1, . .,.ek, and the Iwahori relations are as follows:
eiej= ejei for2 i  j,
eiei+1ei= ei+1eiei+1 for1 i < k  1,
ek1ekek1ek= ekek1ekek1.
For i = 0, . .,.k 1, we define ~Eiby ~Ei= ~eki~eki+1.~.e.k1~ek~ek1.~.e.k*
*i, and
w is defined as
w = ~Ek1~Ek2.E.~.0u.
(E~k1~Ek2. .~.E0is the longest word in the idempotents ~ei.) The rest of the *
*proof
of [2], Lemma 8.1 remains unchanged.
We can now give the proofs of Theorem 10.1 and Proposition 10.3.
Proof of Theorem 10.1.We assert that the following is a chain of mod p equiva
lences.
Sk+1 ^ Lpk ^ W' Sk ^ Lpk ^ W ^ S1
' Sk ^ U(pk)+ ^Ak  TSp2k ^ W ^ S1
' Sk ^ U(pk)+ ^Ak map( TSp2k , W ^ S1)
i j
' Sk ^ map Lpk , W ^ Sadpk.
The second equivalence follows from Proposition 10.6, the third follows from Pr*
*opo
sition 10.10, and the fourth follows from Corollary 10.9.
50 GREGORY ARONE AND KATHRYN LESH
Proof of Proposition 10.3.The proposition follows from the relationship between
Tits building and the Steinberg idempotent. Let ffl 2 Fp[Sp2k(Fp)] be the sym
plectic Steinberg idempotent. Such an idempotent can be used to split off a wed*
*ge
summand fflE from the pcompletion of any spectrum E with an action of Sp2k(Fp).
A homology calculation using the fact that the Tits building is spherical and t*
*he
Steinberg representation is projective shows that
fflE 'p Sk ^ ( TSp2k ^ E)"hSp2k(Fp).
(Cf. the comments following the proof of Theorem 1.17 of [2], page 252.) So,
Sk ^ ( TSp2k ^ E)"hSp2k(Fp)is a wedge summand of the pcompletion of E. In
particular, this holds for E = W"h kwhere W is a spectrum with U(pk) action, and
then we apply Corollary 10.8.
11.Relationship with calculus of functors
In this section we show that there is a relationship between, on the one hand,
sequences of spectra that one obtains from Construction 3.8 in the cases of fin*
*ite
sets and finitedimensional complex vector spaces, and on the other hand, certa*
*in
Taylor towers that arise from the calculus of functors. We prove Theorem 11.2,
which says that the subquotients of our construction show up as layers in Taylor
towers, and Theorem 11.3, which analyzes our filtration in the unitary case with
regard to vkperiodic homotopy.
The general references for Goodwillie's homotopy calculus and Weiss's orthogo
nal calculus are [6, 21]. We need, in particular, the Goodwillie tower of the i*
*dentity
and the Weiss tower of BU (), for which one can look at [3, 1]. For a general
discussion of the interplay between Goodwillie's calculus and periodic homotopy,
including a brief discussion of the relationship of the Goodwillie tower of the*
* identity
to the symmetric power filtration, there is Kuhn's recent survey article [10].
We very briefly recall the relevant facts. The theory of Taylor towers associ*
*ates
to a suitable functor F a tower of fibrations of functors PnF that converges, u*
*nder
favorable circumstances, to F . The homotopy fiber of the map PnF ! Pn1F ,
which we denote DnF , is the "nth layer" of F . It is closely related to the "*
*nth
derivative" of F .
An important functor to which one may profitably apply Goodwillie's theory
is the identity functor from spaces to spaces. Recall from [3] that the Goodwil*
*lie
tower of the identity functor evaluated at S1 can be written in the following f*
*orm,
up to pcompletion:
(11.1)
DpkId(S1)
??
y
. ..! PpkId(S1) ! Ppk1Id(S1)! . ..! P1Id(S1) = Q(S1)
Here k
DpkId(S1) ' 1 map (S1 ^ Ppk , 1 Sp )"h pk
is the homotopy fiber of the map Ppk(S1) ! Ppk1(S1), and Ppk is the poset of
proper, nontrivial partitions of the set {1, . .,.pk}. The tower converges to S*
*1 up
to pcompletion.
FILTERED SPECTRA 51
Similarly, one can apply M. Weiss's theory of orthogonal calculus to the func*
*tor
V 7! BU (V ) from finitedimensional complex vector spaces to topological spaces
and obtain a Taylor tower for it. This tower was studied in [1], and if we take
V = C and complete at p, the tower has the form
(11.2)
DpkBU (C)
??
y
. ..! PpkBU (C) ! Ppk1BU (C) ! . ..! P0BU (C) = BU,
where k
DpkBU (C) ' 1 map (Lpk , 1 Sadpk^ S2p )"hU(pk).
The tower converges to BU(1) (up to pcompletion).
Recall the notation
DIk = DpkId(S1)
DU k = DpkBU (C).
We want to relate these spaces to our constructions. Our present work applied to
the category of finite sets and the category of finitedimensional complex vect*
*or
spaces produces the sequences of spectra
S = Sp1(S) ! Sp2(S) ! . .!.Sp1 (S) = H Z
bu = A0 ! A1 ! A2 ! . .!.A1 = H Z,
where the quotients Spm(S)= Spm1(S) and Am =Am1 are contractible unless m is
a power of p. (The first assertion is well known, and the second is Theorem 9.7*
*.)
Moreover, if m = pk with k > 0, then the homology of the above subquotients
is all ptorsion. Therefore after pcompletion we use the usual notation L(k) =
k pk1
k Spp (S)= Sp (S) for the desuspensions of quotients of symmetric powers
of the sphere spectrum, which are the subquotients of our filtration applied to
finite sets. Similarly, after pcompletion we write T (k) = (k+1)Apk=Apk1 f*
*or
the desuspensions of the subquotients of our filtration applied to finitedimen*
*sional
complex vector spaces and unitary isomorphisms. Before stating the main result *
*of
the section, we record the following theorem,kwhich follows from Theorem 9.5 and
Proposition 10.3 upon taking W = 1 S2p .
Theorem 11.1. The spectrum T (k) is a wedge summand of the suspension spec
trum 1 (S2pk)"h k.
The main result of this section is the following theorem, which says that the
subquotients of our filtration show up as the layers in Taylor towers.
Theorem 11.2. For all k 0, there are homotopy equivalences
Bk1 DIk' 1 L(k)
Bk1 DU k' 1 T (k).
Proof.The first equivalence of the theorem is a consequence of [2], Theorem 1.1*
*8.
According to this theorem, there is an equivalence of infinite loop spaces
k pk1
B2k1DI k' 1 Spp (S)= Sp (S).
52 GREGORY ARONE AND KATHRYN LESH
Looping k times, we find that
k pk1
Bk1 DIk' 1 k Spp (S)= Sp (S)
' 1 L(k),
as desired.
For the second equivalence of the theorem, recall that
i kj
Bk1 DU k' Bk1 1 map Lpk , 1 Sadpk^ S2p" ,
hU(pk)
and, by Theorem 9.5,
i k j
Apk=Apk1' 1 (Lpk ^ S2p )"hU(pk).
Since T (k) = (k+1)Apk=Apk1,
i k j
1 T (k) ' kQ (Lpk ^ S2p )"hU(pk).
The second part of the theorem now followskreadily from Theorem 10.1. Indeed,
applying that theorem to W = 1 S2p gives the equivalence
i k j k
Sk ^ map Lpk , 1 S2p ^ Sadpk' Sk ^ 1 Lpk ^ S2p ^ S1.
Taking homotopy orbits and applying 1 , we obtain
i k j i k j
Bk1 1 map Lpk , 1 S2p ^ Sadpk" ' kQ (Lpk ^ S2p )"hU(pk),
hU(pk)
which is the desired result.
i k j
The cohomology of map Lpk , 1 S2p ^ Sadpk" was described in [1].
hU(pk)
One consequence of Theorem 11.2 and its proof, together with [1], Theorem 4(b),
is that H *(T (k)) is free over Ak1, where Ak1 is the (finite) subalgebra of *
*the
Steenrod algebra generated by fi, P 1, . .,.P pk1(Sq1, . .,.Sq2k at the prime *
*2).
Standard arguments, as in [3], then imply the following theorem.
Theorem 11.3. Let p be a prime and let K(n) denote the nth Morava Ktheory.
Then K(n)*(Apk=Apk1) = 0 for n < k, and the sequence bu = A0 ! A1 ! Ap !
Ap2! . .!.A1 ' H Z terminates at Apk in vkperiodic homotopy.
For instance, in the case k = 0 (rational homotopy), the sequence terminates
at A1 and, rationally, A1 ' H Z. Since A1=A0 = 1 CP 1 by Theorem 3.9, our
sequence 1A1=A0 ! A0 ! A1 is rationally equivalent to the rational cofibration
sequence
1 CP 1 ! bu ! H Z.
12.Conjectures
In this section, we make some conjectures about the more detailed relationship
between our construction and Taylor towers. Roughly speaking, we conjecture that
certain "chain complexes" of spectra arising from our construction are "acyclic*
*,"
and that the contracting homotopy is closely related to Taylor towers. In the f*
*inite
set case, the acyclicity is actually a theorem of Kuhn at the prime 2 and Kuhn
Priddy at odd primes [7, 9]. Informally speaking, our conjectures say, "Analogu*
*es
of the theorems of Kuhn and KuhnPriddy hold in the unitary situation, and the
structure maps of the Taylor towers deloop to contracting homotopies in both the
FILTERED SPECTRA 53
case of finite sets and the case of finitedimensional complex vector spaces." *
*More
precisely, we conjecture that certain specific maps based on stable JamesHopf *
*maps
are deloopings of the structure maps in the Taylor towers (Conjecture 12.1) and
that those same maps provide contracting homotopies for the chain complexes of
spectra in question (Conjecture 12.5).
We recall Kuhn's language on homological algebra of spectra [8]. A sequence of
spectra
E X0 X1 X2 . . .
is called a "complex over E" if it is constructed by splicing together cofiber *
*se
quences Em+1 ! Xm ! Em , with E0 = E. It is called a "resolution" if the
differentials have sections once 1 is applied. It is called "projective" if t*
*he spec
tra Xm are summands of suspension spectra, and it is called a "minimal" project*
*ive
resolution if it is a wedge summand of any other projective resolution of E.
We will be using the Goodwillie tower of (11.1)and the Weiss tower of (11.2).
In what follows, we refer to the two types of towers collectively as "Taylor to*
*wers"
when necessary; we also assume that all spaces and spectra have been completed
at the prime p.
To state our conjectures, we want to loop the Goodwillie tower (11.1)once
and loop the Weiss tower (11.2)twice. The looped Goodwillie tower starts with
Q(S1) ' Q(S0) at the bottom and converges to S1 ' Z. The twicelooped
Weiss tower starts with 2BU ' Z x BU, by Bott periodicity, and it converges to
2BU (C) ' 2CP 1 ' Z. In both the Goodwillie tower and the Weiss tower, we
have fibration sequences of the type
fflDk ! fflPpk ! fflPpk1! ffl1Dk,
where ffl is 1 or 2. In both towers we have structure maps given by the composi*
*tion
of the inclusion of the fiber with the next kinvariant,
DIk ! PpkId(S1) ! DIk+1
(12.1)
2DU k ! 2PpkBU (C) ! DUk+1 .
It is well known that these structure maps, although maps between infinite lo*
*op
spaces, are not infinite loop maps. Our first conjecture is that they are (k +*
* 1)
fold loop maps (Conjecture 12.1), and we will hazard a guess about the actual
delooping. Our conjectural models for the delooping depend on the spaces Bk DIk
and Bk1 DU kbeing retracts of certain other spaces. To begin with DIk, it is an
immediate consequence of [2], Corollary 9.6, that Bk DIk is a homotopy retract
of the space Q(Spk)"h kwhere k ~=(Z=p)k is the transitive elementary abelian
psubgroup of pk. We use the notation
k r k
Bk DIki!Q(Sp )"h k!B DIk
for the inclusion and retraction maps. Similarly, it is an immediate consequenc*
*e of
Theorems 11.1 andk11.2 that after pcompletion the space Bk1 DU kis a homotopy
retract of Q(S2p )"h k. We abuse notation and write
k r k1
Bk1 DU ki!Q(S2p )"h k!B DU k
for the inclusion and the retraction maps for the unitary case as well.
We need one more comment before stating the first conjecture. The group p o
k pk+1contains k+1, and the group p o k U(pk+1) contains k+1. In
54 GREGORY ARONE AND KATHRYN LESH
the following conjecture, we use the term "stable JamesHopf map" to refer to
the adjoint of the map 1 QX ! 1 (X^p)"h pthat one obtains from the Snaith
splitting.
Conjecture 12.1. The structure maps described in (12.1)have (k + 1)fold de
loopings. Moreover, the deloopings are given by the two compositions
k jp pk+1 tr pk+1 r k+1
Fk : Bk DIki!Q(Sp )"h k! Q(S )"h po k!Q(S )"h k+1!B DIk+1
k jp 2pk+1 tr 2pk+1 r k
gk : Bk1 DU ki!Q(S2p )"h k! Q(S )"h po k!Q(S )"h k+1!B DU k+1
where tr stands for the transfer map, i and r stand for the inclusion and retra*
*ction
maps discussed above, and jp stands for the stable JamesHopf map.
Let fk = Fk, and let g1 be the double loops on the first kinvariant in the
Weiss tower (11.2). Assuming Conjecture 12.1, we may associate the following
diagrams with the looped Taylor towers, where the few spaces that are not alrea*
*dy
pcomplete need to be pcompleted. The maps in the diagram are those defined in
Conjecture 12.1, and what is conjectural is that they are deloopings of the str*
*ucture
maps in the Taylor towers.
Diagram 12.2.
Z jI! DI0 f0! DI 1 f1! B DI2 f2!. . .
Z jU!Z x BU g1! DU0g0! DU 1 g1!B DU 2 g2!. . .
Here we have DI0= QS0, DI1 = QB p and DU0 = QCP 1. The map f0 :
QS0 ! QB p is the stable JamesHopf map (also called the KahnPriddy map).
In keeping with our exponential indexing, we think of the space BU = D0BU (C) as
DU 1 on the grounds that p1 = 0. This quirk of grading does not occur in the
case of the Goodwillie tower, because the Goodwillie tower of the identity does*
* not
have a degree zero term. We conjecture that the map g1 : Z x BU ! QCP 1,
which is the double loops on the first kinvariant in the Weiss tower, is in fa*
*ct
closely related to the BeckerGottlieb transfer 3, as considered in [4]. The ma*
*p jI
is obtained from the suspension map S1 ! QS1 by looping down once. Similarly,
the map jU is obtained from the inclusion BU(1) ! BU by looping down twice.
Note that in the case of the Goodwillie tower, we did not use the full strength*
* of
Conjecture 12.1. The number of deloopings that we need to string together the k
invariants is actually one less than provided by the conjecture, so we use fk r*
*ather
than Fk.
On the other hand, instead of looking at the filtration sequences in the Tayl*
*or
towers, we can consider the cofiber sequences of pcompleted spectra
Sppk1(S)! Sppk(S) ! kL(k)
Apk1 ! Apk ! k+1T (k).
By splicing together the cofiber sequences for different values of k, the pcom*
*pletion
of the symmetric power filtration and its buanalogue can be written in the fol*
*lowing
chain complex form.
____________
3The referee suggests that one can also get a presumably relevant map by con*
*sidering the
composite U ! QS1(S0) ! Q( CP1 )+, where the first map is the equivariant Jhom*
*omorphism
and the second map is obtained from the Segaltom Dieck splitting.
FILTERED SPECTRA 55
Diagram 12.3.
H Z fflI L(0) ff0 L(1) ff1 L(2) ff2. . .
H Z fflUbu fi1T (0)fi0T (1)fi1 T (2) fi2. . .
Recall that L(0) = S, L(1) = 1 B p and T (0) = 1 CP 1, where the latter
follows from either Construction 3.8 itself or from Corollary 8.3.
Theorem 11.2 says that 1 applied to the objects in Diagram 12.3 gives the
objects found in Diagram 12.2, but the maps are pointing in opposite directions*
*. In
view of this, we may combine Diagrams 12.2 and 12.3 into one as follows.
Diagram 12.4.
jI f0 f1 f2
Z !ffl DI0!DI1 ! B DI2 ! . . .
I ff0 ff1 ff2
jU g1 g0 g1 g2
Z !fflZ x BU ! DU0! DU 1 ! B DU 2 ! . . .
U fi1 fi0 fi1 fi2
Strictly speaking, the maps pointing to the left are infinite loops of the ma*
*ps of
the same names in Diagram 12.3, but for reasons of typesetting, we omit 1 from
the notation. Thus, the maps pointing to the left are infinite loop maps, and s*
*ince
they have nullhomotopic composites, they constitute chain complex differential*
*s.
The maps pointing to the right are not infinite loop maps, and we do not know if
their composites are nullhomotopic.
It is reasonable to ask if Diagram 12.4 represents a chain complex with a con
tracting homotopy, and this is our main conjecture.
Conjecture 12.5.
(1) For all k 1, the following maps are weak homotopy equivalences.
(a) jI O fflI + ff0 O f0 : DI0! DI0
(b) fk1 O ffk1 + ffk O fk : Bk1 DIk! Bk1 DIk
(2) Similarly, the following maps are weak equivalences.
(a) jU O fflU + fi1 O g1 : Z x BU ! Z x BU
(b) g1 O fi1 + fi0 O g0 : DU0 ! DU0
(c) gk1 O fik1 + fik O gk : Bk1 DU k! Bk1 DU k
Finally, the compositions fflI O jI and fflU O jU give the identity map on Z.
It is in fact easy to check that fflIOjI gives the identity map on Z. The fir*
*st inter
esting instance of Conjecture 12.5 in the homotopy case is (1a). It is equivale*
*nt to
the assertion that the composed map Q0(S0) jp!QB p tr!Q0(S0) is a homotopy
equivalence after pcompletion, which is the KahnPriddy theorem.
Likewise, it is easy to check that fflU O jU is the identity on Z, and the fi*
*rst
interesting instance of Conjecture 12.5 in the unitary case is (2a). It is equi*
*valent
to the assertion that the composed map BU g1!QCP 1 tr!BU is an equivalence.
Strong evidence for this assertion is given by a theorem of Segal that says tha*
*t the
space BU is a retract of QCP 1 [18]. See also [4], where Segal's theorem is rep*
*roved
and generalized using the BeckerGottlieb transfer BU ! QCP 1. This transfer
map is probably more directly pertinent to our constructions than Segal's proof.
(Or perhaps, as the referee suggests, the equivariant Jhomomorphisms is even
56 GREGORY ARONE AND KATHRYN LESH
more relevant here.) To prove (2a), one only needs to show that our map g1 is
close enough to the BeckerGottlieb transfer.
Conjecture 12.5 taken together with Theorem 11.1 would imply that the two
chain complexes of spectra in Diagram 12.3 are, in Kuhn's language, projective
resolutions of H Z and of bu<0> (the fiber of the map bu ! H Z), respectively. *
*In
the case of H Z, this consequence is the main theorem of [7] and [9]. In fact,*
* it
appears likely that [7] and [9] contain a proof of Conjecture 12.5 (1), or come*
* close.
One needs to check that certain maps considered there are equivalent to our maps
fk, and a superficial inspection of all the constructions involved suggests tha*
*t this
is the case. Thus the interesting case is Conjecture 12.5 (2).
It is also shown in [8] that the projective resolution of H Z that one obtain*
*s from
the symmetric power filtration is minimal. We conjecture that the same is true
of the resolution that Conjecture 12.5 would provide for bu<0>. If true, this w*
*ould
answer one of the open questions proposed at the end of [8].
Conjecture 12.6. The second chain complex in Diagram 12.3 gives rise to a min
imal projective resolution of bu<0>.
It is conceivable that one can prove Conjecture 12.5 (2) by carefully transfe*
*rring
the methods of Kuhn and Priddy to the unitary context, since all of the crucial
ingredients of their proof have analogues in the present context: stable James
Hopf maps, transfer maps and a suitable family of Steinberg idempotents seem to
be arranged in a similar constellation. Possibly this is the "right" way to pr*
*ove
the conjecture. However, we also wonder if the presence of the Taylor towers_
an ingredient not available at the time that [7, 9] were written_can be profita*
*bly
incorporated into the proof, thereby making it simpler or at least in some way *
*more
"conceptual." For example, a more conceptual proof might be one that does not
rely on explicit homology calculations as heavily as Kuhn's original proof.
It is intriguing to ask whether the special phenomena occurring in the cases
of finite sets and complex vector spaces hold in other cases, but if they do, t*
*hen
it may not be in an obvious way. For example, consider the category of finite
dimensional real vector spaces. Our construction associates with it a sequence *
*of
spectra interpolating between bo and H Z, and Theorems 9.5, 9.7 and 11.3 will
have appropriate analogues, although the statements are likely to be a little l*
*ess
clean than in the complex case. Moreover, there is a Weiss tower associated with
the functor V 7! BO(V ), which appears to be related to our filtration. But the*
*re
may be no realcase analogue of Theorem 11.2 to establish a precise relationship
between our filtration and the Weiss tower, and therefore no analogue for our m*
*ain
conjecture.
One final remark. Conjecture 12.5 implies that the homotopy spectral sequence
associated with the symmetric power filtration, as well as that associated to o*
*ur
filtration of bu, collapse at E2. (Of course, in the case of the symmetric pow*
*er
filtration, the collapsing at E2 is a theorem.) Taken together with Conjecture *
*12.1,
this would imply that an analogous collapsing result holds for the homotopy spe*
*ctral
sequences associated with the Taylor towers (11.1)and (11.2). In the case of the
Goodwillie tower, this was first pointed out by N. Kuhn in [10].
FILTERED SPECTRA 57
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University Mathematics Bulletin 57 (2002), 18.
Kerchof Hall, U. of Virginia, P.O. Box 400137, Charlottesville VA 22904 USA
Email address: zga2m@virginia.edu
Department of Mathematics, Union College, Schenectady NY 12308 USA
Email address: leshk@union.edu