QUASI FINITE LOOP SPACES ARE MANIFOLDS
NITU KITCHLOO AND DIETRICH NOTBOHM
Abstract. It is an old conjecture, that finite Hspaces are homotopy equ*
*ivalent
to manifolds. Here we prove that this conjecture is true for loop spaces*
*. Actually,
we show that every quasi finite loop space is equivalent to a stably par*
*allelizable
manifold. The proof is conceptual and relies on the theory of pcompact *
*groups. On
the way we also give a complete classification of all simple 2compact g*
*roups of rank
2.
10/9/2002
1.Introduction
It is an old question in the theory of Hspaces, whether finite Hspaces are*
* equivalent
to differentiable manifolds. The first major result in this direction is due to*
* Browder
who showed in a series of papers that every simply connected finite Hspace is *
*homotopy
equivalent to a closed topological manifold and, if the dimension is not congru*
*ent to 2
mod 4, then this manifold can be taken to be smooth and stably parallelizable.
The first examples of finite Hspaces which are not compact Lie groups, were*
* con
structed using Zabrodsky's method of mixing homotopy types [11] [29]. Pedersen
analyzed Zabrodsky's method in detail and showed that, in particular, Hspaces *
*in the
genus of a compact Lie group are homotopy equivalent to stably parallelizable s*
*mooth
manifolds [23] [24].
In [5], Capell and Weinberger got further results of this type for finite H*
*spaces. They
put some extra conditions on the fundamental group; e.g that the fundamental gr*
*oup
is a finite pgroup (p odd) or infinite with at most 2torsion. Under these ass*
*umptions
they were able to show that such finite Hspaces are equivalent to topological *
*manifolds.
In this paper we will concentrate on finite loop spaces in general, and show*
* that all
finite loop spaces are homotopy equivalent to stably parallelizable smooth mani*
*folds.
A loop space is a triple (L, BL, e) where L and BL are topological spaces, w*
*ith BL
pointed, and where e : BL ! L is a homotopy equivalence. By abuse of notation
we denote this loop space also by L. Then L is an Hspace with classifying spac*
*e BL.
Properties of loop spaces are inherited from L; e.g L is called finite if the s*
*pace L is
homotopy equivalent to a finite CW complex, quasi finite if H*(L; Z) vanishes *
*in large
degrees and is a finitely generated abelian group in each degree, and simply co*
*nnected
if the space L is so. Since all components of an Hspace are homotopy equivalen*
*t, we
can restrict ourselves to connected loop spaces.
Theorem 1.1. For any connected quasi finite loop space L, there exists a stably*
* paral
lelizable, smooth, finite dimensional closed manifold M such that L and M are h*
*omo
topy equivalent.
___________
1991 Mathematics Subject Classification. Primary 55P45 Secondary 57R10, 55R35.
Key words and phrases. Hspace, finite loop space, surgery, manifold.
1
2 NITU KITCHLOO AND DIETRICH NOTBOHM
This theorem also says that every quasi finite loop space is actually finite*
*. This was
already proved in [21] by methods similar to those also used in this paper.
The proof of the theorem is based on ideas and techniques developed by Peder*
*sen in
the above mentioned work [23] [24]. He constructed a special 1torus for the H*
*spaces
X he considered. A special 1torus is a fibration of the form S1 ! X ! Y w*
*ith
special extra properties (see the next section). In particular, the map S1 ! X*
* factors
through the inclusion S1 S3. Using the fact that Y is a quasi finite, stably *
*reducible,
nilpotent Poincar'e complex he could prove that X is homotopy equivalent to a s*
*tably
parallelizable manifold.
The advantage of working with finite loop spaces comes from the fact that af*
*ter
padic completion we get pcompact groups. Hence we can treat L^palmost like a
compact Lie group. This will allow us to construct particular subgroups of L^p,*
* which
are used to construct special 1tori for L.
The theory of pcompact groups will also provide enough information to show *
*that
L=S1 is a quasi finite, stably reducible, nilpotent Poincar'e complex, which is*
* the other
main ingredient to make Pedersen proof applicable. Our proof relies on work of *
*Bauer
[2] who used ideas of Klein [13] to construct an analogue of the one point comp*
*actifi
cation of the adjoint representation of a compact Lie group for pcompact group*
*s (see
Section 3).
Unfortunately, special 1tori do not exist for all quasi finite loop spaces;*
* e.g. they do
not exist for products of SO(3)'s. The rational cohomology H*(L; Q) of a quasi *
*finite
loop space is an exterior algebra generated by odd dimensional classes. We say *
*that L
is small if H*(L; Q) is generated by classes of degree less than or equal to 3.*
* Otherwise,
we call L large. We have to treat small and large quasi finite loop spaces diff*
*erently.
For large loop spaces we follow the ideas of Pedersen and construct special 1t*
*ori. For
small quasi finite loop spaces we have a complete classification.
Theorem 1.2. Let L be a small, connected, Zfinite loop space. Then there exis*
*ts
a compact Lie group G isomorphic to a product of S3's, a central elementary abe*
*lian
subgroup E G, and a torus T such that G=E x T and L are homotopy equivalent.
Remark 1.3. Since any finite loop space L (actually any finite Hspace) is equi*
*valent
to a product L0x T where T is a torus and L0 is a finite loop space (respective*
*ly, a
finite Hspace) with finite fundamental group we may assume for both theorems t*
*hat
ß1(L) is finite. And this we will do in all what follows. We call a quasi finit*
*e loop space
semi simple if it is connected and if ß1(L) is finite. Hence we only need to pr*
*ove the
above theorems for semi simple quasifinite loop spaces.
The paper is organized as follows. In the next section we prove Theorem 1.2.*
* Section
3 is devoted to a discussion of stable reducibility and completions. In Sectio*
*n 4 we
discuss the notion of special 1tori and reduce the proof of our main theorem f*
*or large
finite loop spaces to the existence of local special 1tori. In Section 5 we cl*
*assify simple
2compact groups of rank 2. This is needed to construct completed special 1tor*
*i for
pcompact groups. This is worked out in Section 6. In Section 7 we use an arith*
*metic
square argument to construct special 1tori for finite loop spaces.
We had developed our own argument for proving stable reducibility of L=S1 but
unfortunately we couldn't apply it in our situation (see Section 8). We explai*
*n our
argument in the last section, since we think that it is interesting in it's own*
* right
and since it shows the power of Klein's homotopy theoretic version of the adjoi*
*nt
representation of a compact Lie group [13]. The techniques are similar to those*
* used
QUASI FINITE LOOP SPACES ARE MANIFOLDS 3
in [2], but much simpler and very much motivated by a geometric argument for the
stable reducibility of homogeneous spaces.
Several proofs of this paper rely on the theory of pcompact groups. For a g*
*eneral
reference we refer the reader to the survey articles [15] and [19] and the refe*
*rences given
there.
We thank Erik Pedersen for bringing this question to the attention of the fi*
*rst author,
for valuable discussions and his continuous interest in this work. We also tha*
*nk the
CRM in Barcelona for it's support when part of this work was done.
2. The case of small finite loop spaces
In this section we assume that L is a small, semi simple, connected, quasi f*
*inite loop
space. We notice that padic completion makes L into a pcompact group. That *
*is
(L^p, BL^p, e^p) is a connected pcompact group.
Theorem 1.2 is part of the following statement.
Theorem 2.1. Let L be a connected quasi finite loop space. Then the following c*
*on
ditions are equivalent:
(i) The exterior algebra H*(L; Q) is generated by 3dimensional classes; i.e. *
*L is
small.
(ii) The polynomial algebra H*(BL; Q) is generated by 4dimensional classes.
(iii) There exists a compact connected Lie group G isomorphic to a product of S*
*3's and
a central elementary abelian subgroup E G such that L ' G=E (as spaces).
Proof. In the case of a pcompact group, Theorems of this type are proved in [9]
(Theorems 0.5A and 0.5B). We will be using those results.
The equivalence of (i) and (ii) follows easily from an EilenbergMoore spect*
*ral se
quence argument.
Let us assume that (ii) holds, and let L^pdenote the associated pcompact gr*
*oup. If
p = 2 we can apply Theorem 0.5B of [9]. In our case this says that there exists*
* a con
nected compact Lie group G isomorphic to a product of S3's and a central elemen*
*tary
abelian subgroup E G such that BL^2' B(G=E)^2. Obviously, the number of S3's
is determined by the number of 4dimensional generators of H*(BL^2; Z^2) Q or*
* of
H*(BL; Q).
Now we consider the case of an odd prime p. The universal cover of eL^pof L^*
*pis a
simply connected pcompact group [16] and splitsQinto a product of simply conne*
*cted
simple pcompact groups [20] or [8], i.e. eL ~= iXi. Since H*(BLe^p; Z^p) *
*Q ~=
H*(BL^p; Z^p) Q, the number of factors is determined by the number of generat*
*ors
of H*(BL; Q) and, for each factor Xi, the algebra H*(BXi; Z^p) Q is a monogen*
*ic
polynomialQalgebra generated by a 4dimensional class. Hence WXi ~=Z=2 and WL^p*
*~=
iWXi is an elementary abelian 2group. Now Theorem 0.5A of [9] tells us that
BLe^p' BH^pwhere H is a product of S3. Since H^pis center free, BH^p' BLe^p= BL*
*^p.
The number of factors again equal the number of generators of H*(BL; Q).
By construction G ~= H, and since B(G=E)^p' BG^pfor odd primes, we have
BL^p' B(G=E)^pfor all primes. Since BL and BG=E are both rationally products
of EilenbergMacLane spaces and since they have the same rational cohomology, we
also know that the rationalizations BL0 and B(G=E)0 are equivalent. That is to *
*say
4 NITU KITCHLOO AND DIETRICH NOTBOHM
that BL is in the adic genus of BG=E and hence L in the adic genus of G=E. By [*
*21]
(Theorem 6.1), the adic genus of G=E is rigid, that is every space in the adic *
*genus of
G=E is actually equivalent to G=E. This implies condition (iii).
On the other hand if L ' G=E, then the exterior algebra H*(L; Q) is generate*
*d by
3dimensional classes. This finishes the proof.
3. Stable reducibility and completions
For a quasifinite space K we define the homological dimension hdZ(K) of K b*
*y the
degree of the largest non vanishing integral homology group. For a Poincar'e co*
*mplex
this equal the formal dimension of K which is given by the degree of the fundam*
*ental
class. We call a quasifinite Poincar'e complex K of homological dimension hdZ(*
*X) = n
stably reducible, if, for some r 2 N there exists a map Sn+r ! Sr^K = rK such*
* that
Hn+r(Sn+r; Z) ! Hn+r( rK; Z) is an isomorphism or, equivalently, if the Hurewi*
*cz
map h : ßn+r( rK) ! Hn+r(K; Z) is an epimorphism. These conditions are also
equivalent to the fact that the Spivak normal bundle is stably trivial and to t*
*he fact
that the top cell splits off stably.
Using the techniques of completions we will break the question of stable red*
*ucibility
down to local ones. First we have to recall some notions.
Let R be a commutative ring with unit. A space K is called Rfinite if H*(K*
*; R)
vanishes in large degrees and is a finitely generated Rmodule in each degree. *
*In par
ticular, quasi finiteness is nothing but Zfiniteness. For such spaces the Rho*
*mological
dimension of K, denoted by hdR(K), is given by th degree of the largest non van*
*ishing
homology group (with coefficients in the ring R). A Rfinite space K with hdR(K*
*) = n
is a Poincar'e complex if H*(K; R) and H*(K; R) satisfy the usual Poincar'e dua*
*lity
properties with respect to a fundamental class [K]R 2 Hn(K; R).
We call a Rfinite Poincar'e complex K with hdR (K) = n Rstably reducible i*
*f, for
some r 2 N, there exists a map Sn+r ! rK such that the induced map
Hn+r(Sn+r; R) ! Hn+r( rK; R) is an isomorphism or, equivalently, if the Hurewi*
*cz
map
ßn+r( rK) R ! Hn+r( rK; R) is an epimorphism.
For a space K we denote the padic completion by K^p. If K is nilpotent or p*
*good
completion induces an isomorphism in modp homology and cohomology. Hence, for
such spaces K and K^phave the same modp properties.
Lemma 3.1. Let K be a Zfinite, nilpotent, Poincar'e complex of homological dim*
*ension
n. Then the following are equivalent:
(i) K is stably reducible.
(ii) For all primes p, K is modp stably reducible.
(iii) For all primes p, the completion K^pis modp stably reducible.
Proof. Since K is a Poincar'e complex of dimension n, we know that Hn+r( rK; Fp*
*) ~=
Hn+r( rK; Z) Fp ~=Fp. We consider the exact sequence
ßn+r( rK) ! Hn+r( rK; Z) ~=Z ! Q ! 1
where Q is the cokernel of the Hurewicz map. We choose r big enough so that we *
*are
in the stable range and Q does not depend on r. Then Q = 0 if and only if Q F*
*p = 0
for all primes. That is K is stably reducible if and only if K is modp stably *
*reducible
for all p.
QUASI FINITE LOOP SPACES ARE MANIFOLDS 5
The equivalence between (ii) and (iii) follows from the fact that ß*( rK) *
*Fp ~=
ß*( rK^p) Fp.
We record another easy lemma.
Lemma 3.2. Let K be a Z(p)finite, nilpotent, Poincar'e complex of Z(p)homolog*
*ical
dimension n. Then the following are equivalent:
(i) K is Z(p)stably reducible.
(ii) K is modp stably reducible.
(iii) K^pis modp stably reducible.
Proof. The equivalence of (ii) and (iii) is exactly as in the above lemma. Hen*
*ce one
only needs to establish the equivalence of (i) and (ii). This follows rather tr*
*ivially since
the map ßn+r( rK) Z(p)! Hn+r( rK; Z(p)) is an epimorphism if and only if the
map ßn+r( rK) Fp ! Hn+r( rK; Fp) is an epimorphism.
In [2] (Theorem 1.3), Bauer showed that for a pcompact group X there exists*
* a
pcompleted sphere spectrum SX with a stable Xaction whose dimension equals the
Fphomological dimension hdFp(X). Here we have to work in the category of simpl*
*icial
spaces and have to replace the loop space X by the associated simplicial group.*
* More
over, if X is the completion of a compact Lie group, we can take for SX the one*
* point
compactification of the Lie algebra of G with the adjoint action. In particula*
*r, if G
is abelian, SG has the trivial action. He also showed that, for a pcompact sub*
*group
Y X there exists a map of spectra SX ! X ^Y SY which induces an isomorphism
in Hn(; Fp). Here, we take the modp cohomology of spectra. This result enable*
*s us
to prove the following proposition.
Proposition 3.3. Let X be a pcompact group and let T ! X be a pcompact subt*
*orus.
Then, the Fpfinite homogeneous space X=T is modp stably reducible.
Proof. Since T is the completion of an abelian compact Lie group, the sphere sp*
*ectra
ST carries the trivial T action. Therefore, we get a map SX ! X+ ^T ST ' X=T+*
* ^ST
which induces an isomorphism in modp homology in the right degree. In particul*
*ar,
this tells us that the top cell of X=T splits off stably and that X=T is modp *
*stably
reducible.
4.Special 1tori and the proof of Theorem 1.1
In [24] Pedersen introduced the concept of special 1tori for spaces, which *
*is his main
concept to get control of the surgery obstructions (see [24] Proposition 2.1). *
*We will
recall his notion, Actually, we only need the plocal version. A fibration F !*
* E ! B
is called orientable if ß1(B) acts trivially on the set [F, F ] of homotopy cla*
*sses of self
equivalences of F .
6 NITU KITCHLOO AND DIETRICH NOTBOHM
Definition 4.1. A nilpotent space K admits a plocal special 1torus if, up to *
*homotopy,
there exists a diagram of orientable fibrations
S1(p)___//_S3(p)_//_S2(p)
  
  
fflffl fflffl fflffl
S1(p)____//K______//B
  
  
fflffl fflffl fflffl
* ______//_A_____//A
such that
(i) A is Z(p)finite.
(ii) B is Z(p)finite and Z(p)stably reducible.
(iii) Localized at 0, the diagram is homotopy equivalent to
S10______//S30________//_S20
  
  
fflffl fflffl fflffl
S10____//A0 x S30__//_A0 x S20
  
  
fflffl fflffl fflffl
* _______//_A0________//_A0
where all vertical fibrations are trivial.
Using the notion of special 1tori Pedersen could prove the following result*
* (see [24]
Theorem 1.4).
Theorem 4.2. (Pedersen) Let X be a Zfinite Hspace. If for every prime p the
localization X(p)admits a special 1torus, then X is homotopy equivalent to a s*
*mooth
stably parallelizable manifold.
Obviously, there also exists a notion of a global special 1torus. For the *
*proof of
the above theorem, Pedersen first showed that, under the above assumption, X ha*
*s a
global special 1torus. Then he used this extra structure to prove that the fin*
*iteness
obstruction for X vanishes. Since Hspaces are stably reducible [4], their Spiv*
*ak nor
mal bundle is stably trivial. The existence of a special 1torus then also imp*
*lies the
vanishing of the surgery obstruction for the existence of a homotopy equivalenc*
*e to a
stably parallelizable manifold (see [24] 2.1 and 4.2).
To prove Theorem 1.1, it is therefore only left to show that any large Zfin*
*ite loop
space admits plocally a special 1torus. And this is a consequence of the fol*
*lowing
proposition.
Proposition 4.3. Let L be a large connected Zfinite loop space. Then there exi*
*sts a
f
loop space N and a fibration A ! BS3(p)! BN(p)such that A is simple, Z(p)fi*
*nite
and such that N and L are homotopy equivalent spaces. Moreover, localized at 0,*
* there
exists a left inverse s : BM0 ! BS30of f, i.e. sf0 = idBS30.
The proof of this proposition will be given in Section 7.
Corollary 4.4. For a large Zfinite loop space L and a prime p, the localizatio*
*n L(p)
admits a plocal special 1torus.
QUASI FINITE LOOP SPACES ARE MANIFOLDS 7
Proof. Let S1 S3 be the maximal torus of S3. Since the loop space N of the l*
*ast
proposition is equivalent to L we only have to prove the claim for N or equival*
*ently, we
may assume that there exist a fibration BS3(p)!BL(p)with the desired propertie*
*s.
Passing to classifying spaces and localizations, and taking homotopy fibers *
*we get a
commutative diagram of fibration sequences
A


fflffl
S1(p)____//S3(p)_//_S2(p)_//_BS1(p)i_//BS3(p)
=   = f
fflffl fflffl fflffl fflfflg fflffl
S1(p)____//L(p)___//_B____//_BS1(p)__//BL(p)
=  
fflffl fflffl= fflffl
* ______//_A_____//_A
Here B is the homotopy fiber of the composition BS1(p)!BS3(p)!BL(p). As the
homotopy fiber of maps between simply connected spaces, A and B are simple.
The three left columns of the above diagram will establish a plocal special*
* 1torus
for L(p). All rows of this 3 x 3diagram are given by principal fibrations and *
*therefore
orientable. The same holds for the two left columns. For the right column we ha*
*ve a
pull back diagram
S2(p)_____//B________//A
=  
fflffl fflffl fflffl
S2(p)___//_BS1(p)__//_BS3(p)
The bottom row is an orientable fibration. Hence, this also holds for the top r*
*ow. This
shows that the above 3 x 3diagram consists of orientable fibrations.
Since A is Z(p)finite, a Serre spectral sequence argument shows that the sa*
*me holds
for B.
Localized at 0, there exists a left inverse s : BL0 ! BS30. Since sg0 = sf0*
*i0 = i0,
this left inverse establishes rationally compatible left inverses for all verti*
*cal arrows
between the second and third row of the above large diagram. In particular this*
* shows
that, localized at 0, the vertical fibrations of the 3 x 3diagram are trivial *
*and that this
diagram satisfies the third condition of special 1tori.
To complete the proof it remains to show that B is Z(p)stably reducible. We
pass to completions. Then L^pbecomes a pcompact group. We get a fibration
B^p! BS1^p! BL^p. Since B was Z(p)finite and simple, B and B^phave isomorph*
*ic
modp homology. This shows that B^pis Fpfinite, that S1^p! L^pis a monomorphi*
*sm
of pcompact groups and that B is equivalent to the homogeneous space L^p=S1^p.*
* By
Proposition 3.3, B is Fpstably reducible and by Lemma 3.2, B is Z(p)stably re*
*ducible.
This completes the proof and shows that L(p)admits a plocal special 1torus.
Proof of Theorem 1.1: We already discussed th ecase of small quasi finite lo*
*op spaces.
Let L be a large Zfinite loop space. By Corollary 4.4 every localization L(p)a*
*dmits a
plocal special 1torus. By Theorem 4.2 this implies that L is homotopy equival*
*ent to
a smooth stably parallelizable manifold.
8 NITU KITCHLOO AND DIETRICH NOTBOHM
Remark 4.5. Unfortunately, there exists no global version of Proposition 4.3; i*
*.e. a
large quasi finite loop space N might not contain a S3 or a S1 as a subgroup. H*
*ence,
in general there exists no sequence of fibrations of the form
S1 ! L ! L=S1 ! BS1 ! BN,
where L and N are homotopy equivalent. If such a sequence were to exist, the ar*
*gument
of Section 8 would give a proof of the stable reducibility of L=S1.
5.2compact groups of rank 2
In this section we will classify all simple 2compact groups of rank 2. For *
*the con
venience of the reader and to fix notation we recall some material about pcomp*
*act
groups.
A pcompact group X is a loop space X = (X, BX, e) such that BX is pcomplete
and pointed and such that X is Fpfinite. Every pcompact group X has a maximal
torus TX , a maximal torus normalizer NX , and a Weyl group WX acting on TX . T*
*hese
loop spaces fit into a diagram
BTX H____//_BNX____//_BWX
HHH 
HHH 
H$$Hfflffl
BX
Here, BTX ' K((Z^p)n, 2) is homotopy equivalent to an EilenbergMacLane space of
degree 2. The top row is a fibration and determines the action of WX on TX , ac*
*tually
on LX := ß1(TX ) ~=(Z^p)n. We call LX the associated WX lattice and n the ran*
*k of
X. This action can also be described by a representation WX ! Gl(LX ). If X*
* is
connected, this representation is faithful and makes the finite group WX into a*
* pseudo
reflection group. And if in addition p = 2, then WX is a 2adic reflection grou*
*p. We call
X simple if X is connected and if the associated representation WX ! Gl(LX *
*Q)
is irreducible. For details and further notions we refer the reader to the surv*
*ey articles
[15] and [19] and the references mentioned there.
The following theorem might be known to the experts. But since we couldn't f*
*ind a
reference for it, we will also include a proof.
Theorem 5.1. Any simple 2compact group X of rank 2 is isomorphic to the 2adic
completion of SU(3), Spin(5) = Sp(2), SO(5) or G2.
The rest of this section is devoted to the proof of this statement. For com*
*pact
connected Lie groups we will abuse notation and denote by G the associated 2co*
*mpact
group obtained by 2completion.
Let U be a finite dimensional vector space over Q^2with an action of a finit*
*e group W
defined by a homomorphism W ! Gl(U). A W lattice L of U is a Z^2lattice L *
* U
of maximal rank fixed under the action of W ; i.e. L is a Z^2[W ]module and L *
*Q ~=U.
We say that two W lattices L and L0of U are isomorphic if L ~=L0as Z^2[W ]mod*
*ules.
A W1lattice L1 and a W2lattice L2 are called weakly isomorphic if there exist*
*s an
isomorphism W1 ~=W2 such that L1 and L2 are isomorphic as W1 lattices.
We say that two pcompact groups X and Y have the same Weyl group data if the
representations WX ! Gl(LX ) and WY ! Gl(LY ) are weakly isomorphic. Renami*
*ng
the elements of WY we always can identify WY with WX and assume that the two
lattices are actually isomorphic.
QUASI FINITE LOOP SPACES ARE MANIFOLDS 9
From the ClarkEwing list [6] we get a complete list of all irreducible refl*
*ection groups
of rank 2 defined over Q^2. These are given by the dihedral groups D6, D8 and D*
*12with
their standard representation as reflection groups. In fact, these are the only*
* dihedral
groups which can be represented as reflection groups over Q^2. The first is the*
* rational
Weyl group representation of SU(3), the second of Spin(5) or SO(5) and the last*
* of
the exceptional Lie group G2. The classification of Clark and Ewing only works *
*up to
weak equivalence.
The Lie groups Spin(5) and Sp(2) are isomorphic. Hence, the Weyl groups WSpi*
*n(5)
and WSp(2)are also isomorphic and the associated lattices LSpin(5)and LSp(2)wea*
*kly
isomorphic. In the following, we will always use the one of these two which see*
*ms to
be more appropriate.
The universal cover Xe of a pcompact group X is again a pcompact group and*
*, if
ß1(X) is finite, X and eXhave the same rational Weyl group data and eX~= eX=Z w*
*here
Z Xe is a central subgroup [16]. Simple pcompact groups have finite fundamen*
*tal
groups [16]. Therefore, Theorem 5.1 is a consequence of the following classifi*
*cation
result for simply connected simple 2compact groups.
Theorem 5.2. Let G = SU(3), Sp(2) or G2. A simply connected 2compact group X
has the same rational Weyl group data as G if and only if X and G are isomorphi*
*c as
2compact groups.
For the proof of this theorem we first have to classify all 2adic lattices *
*of the repre
sentation WG ! GL(LG Q).
Lemma 5.3. Let U := (Q^2)2 and W ! GL(U) be a reflection group.
(i) If W = D6, D12 then, up to isomorphism, there exists exactly one W lattice*
* of the
representation W ! Gl(U).
(ii) If W = D8 each W lattice of U is isomorphic either to LSO(5)or to LSpin(5*
*). And
both lattices are weakly isomorphic.
Proof. For D6 and D12 this follows from [1] (Proposition 4.3 and Theorem 6.2).
Now let W = D8. In this case we have two non isomorphic lattices LSO(5)and L*
*Spin(5).
Let L be another W lattice of U. For a large r, the lattice 2rL, the submodule*
* of all
elements divisible by 2r, is a submodule of LSO(5)~= Z^2 Z^2. We choose r mini*
*mal
with this property, i.e. 2rL LSO(5)but 2rL 6 2LSO(5). Since L Q ~=LSO(5) *
*Q,
we get a short exact sequence of Z^2[W ]modules
j
0 ! 2rL ! LSO(5) ! Q ! 0.
The minimality of r implies that Q is a finite cyclic group; i.e. Q ~=Z=2s gene*
*rated by
either æ((1, 0)) or æ((0, 1)). The dihedral group D8 is generated by the three *
*elements
oe1, oe2, ø, where oei multiplies the ith coordinate by 1 and ø exchanges the*
* two coor
dinates. Since the automorphism group of Q is abelian, the action of W on Q fac*
*tors
through the abelianization of W , ab(W ). It follows that the element oe1oe2 = *
*oe1øoe1ø acts
trivially on Q. Hence the elements (1, 0), (0, 1) 2 M are mapped onto elements *
*of order
2 in Q. Thus, either Q = 0 or Q = Z=2. In the first case, we have L ~=M = LSO(5*
*).
In the second case, D8 acts trivially on Q with æ((1, 0)) = æ((0, 1)) 6= 0 in Z*
*=2 and
consequently L ~=LSpin(5). This proves the first part of (ii).
The second part follows from the facts that LSp(2)and LSpin(5)are weakly iso*
*morphic
and that LSp(2)and LSO(5)are isomorphic.
10 NITU KITCHLOO AND DIETRICH NOTBOHM
Proof of Theorem 5.2: If X and G have the same rational Weyl group data, the
above lemma shows that they also have the same 2adic Weyl group data. We can
assume that W := WG = WX and that L := LG = LX . We also can identify the
maximal tori T := TG ~=TX .
For G = SU(3) or G2, this implies X ~=G. For SU(3) this follows from [17] an*
*d for
G2 from [27].
For Spin(5) = Sp(2) uniqueness result are only known in terms of the maximal
torus normalizer [22] [26]. We have to show that NX ~=NSp(2)as loop spaces; i*
*.e.
BNX ' BNSp(2).
Since X and Sp(2) have the same rational Weyl group data, they have isomorph*
*ic
rational cohomology. Hence, H*(X; Z^2) Q is an exterior algebra with generato*
*rs in
degree 3 and 7. If H*(X; Z^2) has 2torsion, then X and G2 have isomorphic mod*
*2
cohomology [12]. The Bockstein spectral sequence then shows that X does not have
the right rational cohomology. Therefore, X has no 2torsion, H*(X; Z^2) is an *
*exterior
algebra with generators in degree 3 and 7 and H*(BX; F2) ~=F2[x4, x8] is a poly*
*nomial
algebra generated by a class of degree 4 and one of degree 8. Since H*(BX; F2) *
*is a
finitely generated module over H*(BT ; F2), the composition
H*(BX; F2) ~=H*(BX; Z^2) F2 ! H*(BT ; Z^2)W F2 ~=H*(BSp(2); F2) ! H*(BT ; F2)
is a monomorphism. The isomorphism H*(BT ; Z^2)W F2 ~=H*(BSp(2); F2) follows
from the fact that X and Sp(2) have the same 2adic Weyl group data (Lemma 5.3).
Since the first and third term are both polynomial algebras of the same type,
H*(BX; F2) ! H*(BT ; Z^2)W F2 ~=H*(BSp(2); F2)
is an isomorphism.
Let t T denote the elements of of order 2 and H := S3 x S3 Sp(2) the obv*
*ious
subgroup. We have a chain of inclusion t T H Sp(2) and H = CSp(2)(t).
The action of D8 on t factors through the Z=2action on t given by switching the
coordinates.
Now we use Lannes' Tfunctor theory (e.g. see [25]). We get a map f : Bt ! *
*BX
which looks in mod2 cohomology like the map Bt ! BSp(2). This map is Z=2
equivariant up to homotopy. The mod2 cohomology of the classifying space BCX (*
*t) :=
map(Bt, BX)f of the centralizer CX (t) can be calculated with the help of Lanne*
*s'
T functor and H*(BCX (t); F2) ~= H*(BCSp(2)(t); F2) ~= H*(BH; F2). Moreover, *
*the
Weyl group of CX (t) is given by the elements of D8 acting trivially on t. Hen*
*ce
WCX(t)~= Z=2 x Z=2. By [9] (Theorem 0.5B), this implies that BCX (t) ' BH. We
will identify CX (t) with H. The Z=2action on t induces a Z=2action on H. S*
*ince
Bt ! BX was Z=2equivariant up to homotopy, the inclusion BCX (t) ! BX extends
to a map BY := BHhZ=2! BX. In this case, the homotopy orbit space BY happens
to be a 2compact group and has the same Weyl group as X. That is NY = NX .
Moreover, the space BY fits into a fibration
BH ! BY ! BZ=2 ,
which is classified by obstructions in H*(BZ=2; ß*(BSHE(BH))). Here, SHE(BH)
is the space of self equivalences of BH homotopic to the identity. Since SHE(BH*
*) '
(BZ=2)2 [7] and since Z=2 acts on ß2(B2(Z=2)2) ~=(Z=2)2 by switching the coordi*
*nates,
all obstruction groups vanish and the above fibration splits. This shows that *
*BY '
B(H o Z=2) := BH0 and that BNX = BNY ' BNH0 = BNSp(2). That is X and
QUASI FINITE LOOP SPACES ARE MANIFOLDS 11
Sp(2) = Spin(5) have isomorphic maximal torus normalizer and shows that X ~=
Sp(2).
Remark 5.4. The only simply connected 2compact group of rank 1 is S3. Hence,
we get the following complete list (up to isomorphism) of connected 2compact g*
*roups
of rank 2, namely S1 x S1, S1 x S3, U(2), S1 x SO(3), S3 x S3, S3 x SO(3), SO(4*
*),
SU(3), Sp(2), SO(5) and G2.
The following_corollary is needed for later purpose. For a pcompact group *
*X we
denote by X the associated center free quotient.
Corollary 5.5. For any simple connected 2compact group X of rank_2, there exis*
*ts
a homomorphism S3 ! X such that the composition S3 ! X ! X is a monomor
phism.
Proof. Because of Theorem 5.1 we only have to check this for the compact connec*
*ted
Lie groups SU(3), Sp(2), SO(5) and G2. There exists a chain of monomorphisms
S3 = SU(2) SU(3) G2. Both groups, SU(3) and G2, are 2adically center free.
This proves the claim in these two cases. Let S3 Sp(2) denote the inclusion *
*into
the first coordinate. Since the intersection of S3 and the center of Sp(2) is t*
*rivial, the
composition S3 Sp(2) ! SO(5) is also a monomorphism. This proves the claim in
the other cases.
6. Particular subgroups of pcompact groups.
In this section we will construct particular subgroups of large pcompact su*
*bgroups.
A pcompact group X is called large, if the exterior algebra H*(X; Z^p) Q has*
* a
generator of degree 5. We want to prove the following proposition.
Proposition 6.1. Let X be a large semi simple connected pcompact group. Let r *
*:=
dimQ^pH4(BX; Z^p) Q be the dimension of the Q^pvector space H4(BX; Z^p) Q.
Then there exists a compact Lie Group G and a map
f : BG^p! BX
such that the following hold:
(i) G ~=S3 x H with H semi simple and it's universal cover eH isomorphic to (S3*
*)r1.
If p is odd, we can choose G = (S3)r.
(ii) The induced map H4(BX; Z^p) Q ! H4(BG^p; Z^p) Q is an isomorphism.
(iii) The homotopy fiber X=G^pof f is simple and Fpfinite.
Proof. Comparing the statement with Proposition 3.1 of [21] there is an extra a*
*ssump
tion on the generators of H*(X; Z^p) Q and the corresponding additional outpu*
*t is
that G contains a factor S3. Actually, for odd primes, the statements of both p*
*roposi
tions are the same. Therefore we only have to prove the statement for p = 2. Ag*
*ain,
for a compact connected Lie group we denote by G the associated 2compact group.
Let X be a semi simple 2compact group, i.e. ß1(X) is a finite 2group. If H*
**(X; Z^2)
Q is not generated by classes of degree 3, i.e. the polynomial algebra H*(BX; Z*
*^2) Q
is not generated by classes of degree 4, then the Weyl group WX is non abelian*
* [9]
(Theorem 0.5B), but a honest reflection group, since WX is defined over Q^2. Th*
*at is,
WX is generated by elements of order 2 fixing a hyperplaneQof codimension 1. *
*The
universal cover Xe of X splits into a direct product Xe ~= Xi of simple, simpl*
*y con
nected pieces [8]. Since X and Xe have isomorphic Weyl groups, we can assume th*
*at
12 NITU KITCHLOO AND DIETRICH NOTBOHM
X1 has a non abelian Weyl group W1. For this piece we will construct_a monomor
phism BG1 := BS3 ! BX1, such that_the_composition BS3 ! BX1 ! BX 1 is also
a monomorphism (see below). Here, X 1denotes the center free quotient of X1. Mo*
*re
over, the map BG1 ! BX1 will induce an isomorphism in H4(; Z^p) Q.
Having done this we can proceed similarly as in [21]. For all other pieces *
*there
exists monomorphisms BGi! BXiinducing an isomorphism on H4(; Z^p) Q such
thatQGi isQisomorphic to S3 or to SO(3) (see [21]). This produces a homomorphi*
*sm
Gi! Xi~= eX! X of pcompact groups. The kernel KQof this homomorphism,
which might be nontrivial,_is a central subgroupQof_G1 x i>1Gi [18]. Since_t*
*he
center free quotient X is isomorphic to iX_iwe have a homomorphism X ! X1.
By construction the composition S3 ! X1 ! X 1 is a monomorphism. We get a
commutative diagram
Q
K ____//_S3 x i>1Gi___//_X

=  
fflffl fflffl __fflffl
K _________//_S3________//_X1
__
where the right arrow in the bottomQrow is a monomorphism. Since X 1 is center
freeQthe composition K ! S3Qx i>1Gi! S3 is trivial. Therefore, K is a sub*
*group
of i>1GiQand the map S3 x i>1Gi! X factors through a monomorphism G :=
S3 x (( i>1Gi)=K) ! X with all the desired properties.
It remains to show that, for a simple, simply connected 2compact group X wi*
*th non
abelian Weyl group, there exists a monomorphism_S3 ! X inducing a isomorphisms
in H4(; Z^p) Q such that S3 ! X ! X is also a monomorphism. Let W 0 WX be
a subgroup of0the Weyl group of X generated by two non commuting reflections of*
* WX .
Let T TXW TX denote the connected component of the fixedpoint set of the W*
*X 
action on TX , which has codimension 2. The centralizer C := CX (T ) is a conn*
*ected
2compact group, whose Weyl group WC contains W 0[16]. There exists a finite co*
*vering
of C which splits into a product Y xT where Y is a simply connected 2compact g*
*roup of
rank 2 with Weyl group isomorphic to WC . The action of W 0on the maximal torus*
* TY
of Y gives rise to an irreducible representation over Q^2. Otherwise, W 0would *
*split into
a product and the two chosen reflections would commute. Hence, the 2compact gr*
*oup
Y is simple and of rank_2. By Corollary 5.5 there exists a monomorphism S3 !*
* Y
such that S3 ! Y ! Y is a monomorphism. Putting all these homomorphisms and
groups into a diagram we get
S3 ______//Y_____//_X
=  
fflffl fflffl _fflffl_
S3 ____//_Y=K____//_X .
__
Here, K denotes the kernel_of_Y ! X . In particular, K is a central subgroup *
*of Y .
Since S3 ! Y ! Y=K ! Y is monomorphism, this_also holds for the compositio*
*n of
the first two arrows. Moreover, since_Y=K ! X is a monomorphism, the same hol*
*ds
for the composition S3 ! Y=K ! X. This proves the above claim and finishes t*
*he
proof of the proposition.
QUASI FINITE LOOP SPACES ARE MANIFOLDS 13
7. Proof of Proposition 4.3
In this section, we want to prove Proposition 4.3. The proof is based on an *
*arithmetic
square argument. First we need a statement about the existence of a particular*
* sub
loop space, a global version of Proposition 6.1.
Proposition 7.1. Let L be a large semi simple Zfinite loop space. Then there e*
*xists
a semi simple compact Lie group G, loop spaces M and N and a fibration
A ! BM ! BN
such that the following hold:
(i) A is simple and Zfinite.
(ii) G ~=S3 x H and the universal cover of H is isomorphic to a product of S3's.
(iii) The spaces G and M as well as L and N are homotopy equivalent.
(iv) H4(BN; Q) ! H4(BM; Q) is an isomorphism.
(v) There exists a commutative diagram
BM^p _____//BN^p
 
 
fflffl fflffl
BG^p _____//BL^p
where the vertical maps are equivalences. The same holds for the rationalizati*
*ons of
the classifying spaces.
Proof. This statement is a refinement of Proposition 1.4 of [21]. The proof of*
* that
statement is an arithmetic square argument which uses it's pcompleted version,*
* namely
Proposition 3.1 of [21], as input. The proof carries over word for word. We o*
*nly
have to replace that proposition by a pcompleted version of the above claim, n*
*amely
by Proposition 6.1. In particular, the bottom rowin the diagram of (v) is the *
*map
constructed in Proposition 6.1. Claim (ii), which is not part of Proposition 1.*
*4 of [21],
is a consequence of the same formula in Proposition 6.1.
Remark 7.2. The above proposition establishes an oriented fibration G ! L ! L*
*=G.
And the existence of such an oriented fibration is already sufficient to show t*
*hat the
finiteness obstruction vanishes and that every quasi finite loop space is actua*
*lly finite
(see [21]). The existence of a special tori is needed for the vanishing of the *
*appropriate
surgery obstruction.
For the proof of Proposition 4.3 we need two more lemmas.
Lemma 7.3. Let A 2 Gl(n, Z^p). Then, there exists a vector v = (v1, ..., vn) 2 *
*(Z^p)n
such that vi is a square of a padic unit for all i and such that Av is a vecto*
*r whose
components are given by elements of Z(p).
Proof. Let B := A1. We have to solve the following problem: Find a vector w 2 *
*Zn(p)
such that Bw 6= 0 has as components squares of padic units. The question wheth*
*er
a padic unit is a square, can be decided_by reducing to Z=p for p odd or to Z=*
*8 for
p = 2. In both cases the reduction B of B is an invertible matrix and induces t*
*herefore
an epimorphism on (Z=p)n =: V . In particular, if __v2 V is a vector with compo*
*nents
given by squares mod_p such that all entries are units in Z=p, there exists a v*
*ector
w 2 (Z)n such that B w = v. Hence, Bw is a vector whose components are squares *
*of
nontrivial padic units. For p = 2, the same argument works, we only have to re*
*place
Z=p by Z=8.
14 NITU KITCHLOO AND DIETRICH NOTBOHM
Proof of Proposition 4.3. Let M and N denote the loop spaces and G the Lie g*
*roup
constructed in Proposition 7.1 Since BM^p' BG^pand BM0 ' BG0 we have a pull
back diagram
BM(p) _______________//_BG^p
 
 
fflffl A fflffl
BG0 _____//_BG[p___//(BG^p)0
Here BG[pis the formal padic completion of the rationalization BG0 in the sens*
*e of
Sullivan, and (BG^p)0 the localization at 0 of BG^p. The map A is an equivalen*
*ce
between the homotopy equivalent spaces BG[pand (BG^p)0, and induces a continuous
map in homotopy. The homotopy groups ß*((BG^p)0 carry a natural topology, since
ß*(BG^p) ~=ß*(BG) Z^p(details may be found in [28]). (BG^p)0 ~=K(Q^pr, 4) is *
*a ra
tional EilenbergMacLane space. Since self maps of rational EilenbergMacLane s*
*paces
are determined by the induced maps in homotopy, and since A induces a continuous
map in homotopy, we can think of A as a matrix in Gl(n, Q^p) inducing a continu
ous self equivalence of (Q^p)n. Such matrices can be written as a product BR w*
*here
B 2 GL(n, Z^p) and R 2 Gl(n; Q). For example, this follows from the fact that *
*the
adic genus of products of S1's is rigid. Since R can be realized as a self equi*
*valence of
BG0, replacing A by B does not change the homotopy type of the pull back. Hence
we may assume that A 2 Gl(n, Z^p).
We have an analogous pull back diagram as above for the classifying space BM*
*f of
the universal cover fM of M with the same gluing map A, namely
BMf(p)___________________________//BS3^px BHe^p
 
 
fflffl fflffl
__A_//_ 3 ^
BS30x BHe0 ____//_(BS3 x BHe)[p ((BS x BHe)p)0
Here we used the fact that G ~=S3xH with H = eH=K, and eHa product of S3's. Sin*
*ce
every equivalence BS3^p! BS3^pinduces in ß4(BS3^p) multiplication by a non tri*
*vial
square unit of Z^p, Lemma 7.3 shows that there exists a map BS3 ! BS3^px BHe^p
such that the composition
1 3 [
BS3(p)!BS3^p! BS3^px BHe^p! ((BS3 x BHe)^p)0 A! (BS x BHe)p
lifts to a map BS3(p)!BS30x BHe0. Moreover, localized at 0, composition with t*
*he
projection on the first factor is an equivalence. This establishes a map BS3(p)*
*!BMf(p)
such that the completion of the composite BS3(p)!BMf(p)! BM(p)is induced by t*
*he
monomorphism S3^p! S3^px eH^p! S3^px eH=K^p= G^pof pcompact groups. This
shows that the homotopy fiber of BS3(p)!BM(p)is simple and Z(p)finite as is t*
*he
g 4
homotopy fiber of the composite f : BS3(p)!BM(p)! BN(p). Since H (BN0; Q) ~=
H4(BM0; Q), there exists a left inverse s : BN0 ! BM0 for g0. Projection onto *
*the
first factor gives a left inverse of BS30! BM0 ' BS30x BHe0. This shows that,
localized at 0, the map f : BS3(p)!BN(p)has a left inverse and finishes the pr*
*oof of
the proposition.
QUASI FINITE LOOP SPACES ARE MANIFOLDS 15
8.Stable Reducibility of abelian quotients
Let L be a finite loop space. Let T be an abelian compact Lie group, and let*
* B' :
BT ! BL be a monomorphism. We will show in this section that the Zfinite fiber*
* L=T
is stably reducible. Unfortunately, here we use a stronger assumption than we a*
*re able
to produce in our case. In general, the fibration sequence S1 ! L ! L=S1 ! B*
*S1
cannot be extended one further step to the right. This is only possible after c*
*ompletion.
We begin with the case of a Lie group L to develop our intuition.
Lemma 8.1. Let ' : T ! L be a monomorphism between compact Lie groups, where
T is abelian, then L=T is stably reducible.
Proof. Let L and T be the Lie algebras of L and T respectively. It is easy to s*
*ee that
the tangent bundle of L=T is given by L xT (L=T ), where the action of T on the*
* Lie
algebras is given by the adjoint representation. Since T is abelian, the bundle*
* L xT T
is trivial. On adding it to the tangent bundle of L=T , we get the bundle L xT*
* L.
Notice that the adjoint action of T on L extends to L and hence L xT L is trivi*
*al.
This shows that L=T has a stably trivial tangent bundle, which is equivalent to*
* being
stably reducible.
One would like to extend this argument to the case of L being a finite loop *
*space. The
theory that best preserves the analogy with compact Lie groups has been develop*
*ed by
John Klein in [13]. For a topological group G, Klein defines the dualizing Gsp*
*ectrum,
DG = Map (EG+ , S[G])G , where S[G] is the spectrum 1 G+ with the left G actio*
*n.
The residual right G action on S[G] induces the G action on DG . The spectrum D*
*G is
the appropriate notion of the Adjoint representation. This is justified by the *
*fact that
for a compact Lie group G, there is an equivalence DG = SAd, where SAd denotes *
*the
onepoint compactification of the adjoint representation of G.
The magic of the spectrum DG lies in the following two theorems of Klein [13]
Theorem 8.2. Assume that BG is a finitely dominated space. Then the following a*
*re
equivalent:
(i) BG is a Poincar'e duality space of formal dimension n.
(ii) DG has the (unequivariant) homotopy type of a sphere spectrum of dimension*
* n.
Moreover, in the above two cases, the Thom spectrum EG+ ^G DG is the Thom
spectrum of the Spivak normal bundle of BG.
Theorem 8.3. Assume 1 ! H ! G ! Q ! 1 is an extension of topological groups,
then if BH is a finitely dominated Poincar'e duality space, then there is a wea*
*k equiv
alence of spectra
DG ~=DH ^ DQ
Moreover, one may replace DH by an Hequivalent spectrum so as to make the above
equivalence Hequivariant.
We will apply the above theorems to the extension of topological groups
1 ! (L=T ) ! T ! L ! 1
In order to make sense of this extension, we must work in the modelcategory of*
* sim
plicial groups and replace the above groups by equivalent topological groups (c*
*.f [10]).
Let us record a simple lemma about simplicial groups that will be useful in *
*the
sequel.
16 NITU KITCHLOO AND DIETRICH NOTBOHM
Lemma 8.4. Let sH ! sG be an acyclic fibration of simplicial groups. Then on ta*
*king
realizations one gets an extension of topological groups
1 ! K ! H ! G ! 1
with a contractible kernel K. Moreover, there is an Hequivalence of spectra DH*
* ~= DG .
Proof. The realization of an acyclic fibration of simplicial groups is an acycl*
*ic Serre
fibration. Hence we get an extension of topological groups with a contractible *
*kernel K.
We may consider the space EH=K as a model for EG. Then the required equivalence
is induced by the (2sided) Hequivalence S[H] ! S[G] and is given by
DH = Map (EH+ , S[H])H ! Map (EH+ , S[G])H = Map (EG+ , S[G])G = DG
We are now ready to prove the main theorem of this section
Theorem 8.5. Let L be a finite loop space of formal dimension n, and let B' : B*
*T !
BL be a monomorphism, where T is an abelian compact Lie group of rank r, then t*
*he
Zfinite space L=T is stably reducible.
Proof. For a connected space X, let s X denote the simplicial Kan loop group of*
* X
(c.f [10]). The map B' induces a simplicial homomorphism
s' : s BT ! s BL
The properties of model categories allow us to factor s' through an acyclic cof*
*ibration
followed by a fibration. Consequently, we may assume that s' is a fibration. Un*
*for
tunately, s BT is a highly nonabelian model for T . This is the price one has *
*to pay
for obtaining a fibration. Fortunately however, the simplicial group s BT is re*
*lated
to an abelian simplicial model for T via adjointness. We have an acyclic fibrat*
*ion
s BT ! sT a
where sT ais a simplicial abelian model for T .
Now taking the realization of s', we obtain an extension
1 ! K ! T f! Lf ! 1 ,
where T fand Lf denote the free models of T and L, we obtain by realizing the s*
*implicial
Kan loop groups s BT and s BL respectively. The group K is clearly equivalent to
(L=T ). Using Theorem 8.3, one obtains a Kequivariant equivalence
DTf ~=DK ^ DLf
Hence K acts trivially on DLf. In fact, DLf is a sphere spectrum of dimension n*
* with a
trivial K action. One sees this as follows: by Theorem 8.2, DK is a sphere of d*
*imension
r  n. Moreover, by Lemma 8.4 DTf ~=DTa ~=DT is also a sphere of dimension r, t*
*hus
DLf is a sphere of dimension n with a trivial K action. Hence DK ~= nDTf as a
Kspectrum. It now follows from Theorem 8.2 that the Thom spectrum of the Spivak
Normal bundle of L=T is given by nEK+ ^K DTf. Thus to show that L=T is stably
reducible, it is sufficient to show that DTf is equivalent to a sphere spectrum*
* with a
trivial T faction. This follows by applying Lemma 8.4 to notice that DTf is equ*
*ivalent
to DTa which clearly has a trivial T faction.
QUASI FINITE LOOP SPACES ARE MANIFOLDS 17
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Department of Mathematics, 404 Krieger Hall, John Hopkins University, Balti
more, MD 21218, USA
Email address: nitu@math.jhu.edu
Department of Mathematics & Computer Science, University of Leicester, Unive*
*r
sity Road, Leicester, LE1 7RH, U.K.
Email address: dn8@mcs.le.ac.uk