THE FINITENESS OBSTRUCTION FOR LOOP SPACES
D. Notbohm
Abstract. For finitely dominated spaces, Wall constructed a finiteness ob*
*struction,
which decides whether a space is equivalent to a finite CW -complex or no*
*t. It was
conjectured that this finiteness obstruction always vanishes for quasi fi*
*nite H-spaces,
that are H-spaces whose homology looks like the homology of a finite CW -*
*complex.
In this paper we prove this conjecture for loop spaces. In particular, th*
*is shows that
every quasi finite loop space is actually homotopy equivalent to a finite*
* CW -complex.
1. Introduction. L
A topological space is called quasi-finite, if the direct sum nHn(X; Z) o*
*f all
integral homology groups is a finitely generated abelian group. Then you can ask
whether the space X is (weakly) homotopy equivalent to a finite CW -complex. For
finitely dominated spaces, Wall constructed an invariant which decides this pro*
*blem
[29] [30]. A space X is finitely dominated, if there exist a finite CW -comple*
*x K
and maps f : K -! X and g : X -! K such that the composition fg is homotopic
to the identity idX on X. One can show that finitely dominated spaces are alwa*
*ys
homotopy equivalent to CW -complexes and that each component is path connected
(e.g. see [17, Section 2]). Examples of finitely dominated spaces are given by
quasi-finite nilpotent spaces. Actually, for nilpotent space, both conditions, *
*finitely
dominated and quasi-finite are equivalent [15]. A space X is called nilpotent *
*if
X is homotopy equivalent to a CW -complex and if ss1(X) acts nilpotent on the
homotopy groups ss*(X). In particular, ss1(X) is a nilpotent group. The space X
is called simple if, in addition, the action is trivial. In this case, ss1(X) i*
*s abelian.
For a finitely dominated space X, the finiteness obstruction "!(X) of Wall i*
*s an
element in the reduced projective class group K"0(Z[ss1(X)]) of the integral gr*
*oup
ring. And "!(X) vanishes if and only if X is homotopy equivalent to a finite CW*
* -
complex [29] [30]. It is interesting to investigate the question how topologic*
*al
properties of spaces are reflected by properties of the finiteness obstruction.*
* For
instance, it is not known, but conjectured, that the finiteness obstruction for*
* quasi-
finite loop spaces or for H-spaces always vanishes. In this paper we will prove*
* this
conjecture for loop spaces.
A loop space L consists of a triple (L; BL; e), where L and BL are of the ho*
*mo-
topy type of a CW -complex, BL pointed, and where e : BL -! L is a homotopy
equivalence between the loop space of BL and L. Examples of loop spaces are giv*
*en
by topological groups and their classifying spaces. Loop spaces inherit proper*
*ties
from the space L; e.g. L is called quasi-finite, if the space L is so. Loop s*
*paces
______________
1991 Mathematics Subject Classification. 57Q12, 55R35, 55R10.
Key words and phrases. finiteness obstruction, Wall obstruction, loop space,*
* p-compact group.
Typeset by AM S-T*
*EX
1
2
are always nilpotent spaces. In fact, they are simple. Thus every quasi-finite *
*loop
space is finitely dominated and the finiteness obstruction "!(L) is defined.
Theorem 1.1. Let L be a quasi finite loop space. Then 0 = "!(L) 2 "K0(Z[ss1]) *
*and
L is homotopy equivalent to a finite CW -complex.
For a quasi-finite nilpotent space X, Mislin proved results of this type if *
*ss1(X)
is infinite [16] or if ss1(X) is cyclic of prime power order [15].
The proof of our result is a consequence of the next two statements. For eve*
*ry
fibration F -! E -! B, the fiber transport along loops in B establishes a homom*
*or-
phism ss1(B) -! [F; F ] from the fundamental group of B into the set of homotopy
classes of self maps of the fiber F [31, p. 98 ff]. The fibration is called ori*
*ented if
this homomorphism is trivial.
Theorem 1.2. Let L be a quasi-finite loop space, such that ss1(L) is finite. *
*Let
r := dimQ H4(BL; Q). Then there exist a semi simple connected compact Lie group
G and an oriented fibration
G -! L -! E
such that the following holds:
(1) For the universal cover G" of G, we have G"~= SU(2)r.
(2) The space E is simple and quasi-finite.
Theorem 1.3. Let X be a finitely dominated space, such that ss1(X) is finite. *
*If,
up to homotopy, there exists an oriented fibration
G -! X -! B
such that G is a connected compact Lie group and B a finitely dominated space,
then, the finiteness obstruction "!(X) vanishes and X is homotopy equivalent to*
* a
finite complex.
For principal G-fibrations X -! B Theorem 1.3 is proved in [22]. In the gene*
*ral
case, the statement is known and a simple consequence of some vanishing results*
* of
L"uck for the algebraic transfer in algebraic K-theory associated to a fibratio*
*n [14]
(see Section 2). We couldn't find a reference for it, but believe that it is wo*
*rth to
be stated.
Proof of Theorem 1.1. If ss(L) is infinite, we can apply the above mentioned re*
*sult of
Mislin [16]. If ss(L) is finite, the statement follows from Theorem 1.2 and The*
*orem
1.3, since every quasi finite nilpotent space is finitely dominated [16].
Theorem 1.2 is a corollary or a weak version of the next statement.
Theorem 1.4. Let L be a quasi-finite loop space, such that ss1(L) is finite. *
*Let
r := dimQ H4(BL; Q). Then there exist a semi simple connected compact Lie group
G, quasi-finite loop spaces M and N and a fibration
E -! BM -! BN ;
such that the following holds:
(1) For the universal cover G" of G, we have G"~= SU(2)r.
(2) The spaces G and M are homotopy equivalent as well as L and N.
(3) The space E is simple and quasi-finite.
3
The proof of Theorem 1.4 goes as follows. Since completion turns the quasi-
finite loop space L into a p-compact group(L^p; BL^p; e^p) we can use the theory
of p-compact groups to construct the above fibration at each prime (see Section
3). Localized at 0, the existence of the fibration is basically a consequence o*
*f the
fact that the rationalization BL0 is equivalent to a product of rational Eilenb*
*erg-
MacLane spaces. Using the arithmetic square we glue all these data together and
get the fibration claimed in Theorem 1.4.
In Section 2, we prove Theorem 1.2. The other sections are devoted to the pr*
*oof
of Theorem 1.4. In Section 3, we prove the p-completed version of Theorem 1.4.
In Section 4 we recall some material about the arithmetic square and the genus
of a space. Section 5 contains the construction of the fibration of Theorem 1.*
*4.
The analysis of the genus of those compact Lie groups appearing in Theorem 1.4 *
*is
worked out in section 6 and will complete the proof of Theorem 1.4.
We will switch between the p-adic completion of Bousfield and Kan [2] and the
p-profinite completion of Sullivan [28]. But, for spaces with mod-p homology of
finite type, both constructions coincide [2].
It is a pleasure thank Wolfgang L"uck for telling me that a result like Theo*
*rem
1.2 implies the vanishing of the finiteness obstruction for quasi finite loop s*
*paces
and for several helpful discussions on this subject. I am also grateful to the*
* SFB
478 "Geometrische Strukturen in der Mathematik" at M"unster for its hospitality
when part of this work was done.
2. Proof of Theorem 1.3.
For a fibration F -i!E -p!B with finitely dominated total space, base and fi*
*ber,
Ehrlich [11] constructed a geometric transfer
p!: K0(Z[ss1(B)] -! K0(Z[ss1(E)]
for the projective class groups of the integral group rings of the fundamental *
*groups
(see also [25]). The map p! also gives a transfer for the reduced projective c*
*lass
groups. Ehrlich also proved a formula relating the finiteness obstruction of E *
*and
B [12], namely
"!(E) = p*("!(B)) + O(B) . i*("!F ) 2 "K0(Z[ss1(E)]) ;
where O(B) denotes the Euler characteristic of B and where i* : K"0(Z[ss1(F )] *
*-!
K"0(Z[ss1(E)] is the map induced by induction.
In [13] [14], for fibrations of the above type, L"uck constructed an algebra*
*ic trans-
fer, showed that both coincide and proved the following vanishing result.
2.1 Theorem. (L"uck [14, 9.1]) Let G -! E -! B be an oriented fibration, such
that G is a connected compact Lie group. If ss1(B) is finite, then the algebra*
*ic
transfer
p* : K0(Z[ss1(B)] -! K0(Z[ss1(E)]
vanishes.
Proof of Theorem 1.3. The compact Lie group G is connected and homotopy equiv-
alent to a finite CW -complex. Hence "!(G) = 0. Moreover, ss1(B) is finite si*
*nce
ss1(X) is. Thus, the statement is a simple consequence of Theorem 2.1 and the
above formula.
4
3. Particular subgroups of p-compact groups.
A p-compact group X is a loop space X = (X; BX; e) such that BX is p-
complete and pointed and such that X is Fp-finite, i.e. H*(X; Fp) is finite in *
*each
degree and vanishes in almost all degrees. As turned out [6] [7] [20] p-compact
groups behave very much like compact Lie groups. In particular, most of the
classical notions for compact Lie groups are available in this context; e.g. t*
*here
exist subgroups, maximal tori and Weyl groups with the same properties and we
can speak of centralizers of p-compact toral subgroups. A p-compact toral group
P is a p-compact group P such that ss0(P ) ~=ss1(BP ) is a finite p-group and s*
*uch
that the universal cover of BP is equivalent to an Eilenberg-MacLane space of t*
*he
form K(Z^pn; 2). For details and further notions we refer the reader to the ab*
*ove
mentioned references and/or to the survey articles [4],[19] and [24].
In this section we want to prove the following statement, which is a p-compl*
*eted
version of Theorem 1.4.
Proposition 3.1. Let X be a connected p-compact group, such that ss1(X) is fin*
*ite.
Let r := dimQ H*(BX; Z^p) Q. Then there exists a connected compact Lie group
G and a map
BG^p-f!BX
such that the following holds:
(1) For the universal cover G" of G, we have G" ~=SU(2)r. If p is odd, we c*
*an
choose G = SU(2)r.
(2) The induced map H4(BX; Z^p)Q -! H4(BG^p; Z^p)Q is an isomorphism.
(3) The homotopy fiber X=G^pof f is simple and Fp-finite.
Before we start with the proof, we draw one corollary. For a space Y we deno*
*te
by Y the n-th Postnikov section and by Y (n) the n-connected cover of Y . H*
*ence
these spaces fit into a fibration
Y (n) -! Y -! Y :
3.2 Corollary. The composition BG^p-! BX -! BX<4> induces an equivalence
(BG^p)0 -'!BX<4>0.
Proof. Since ss1(G) and ss1(X) are finite and since G" ~= SU(2)r, both spaces,
(BG^p)0 and BX<4>0, are Eilenberg-MacLane spaces with non vanishing 4-th ho-
motopy group. The universal coefficient theorem shows that H4(BX; Z^p) Q is
the dual of H4(BX; Z^p)Q and hence, that H4(BG^p; Z^p)Q ~=H4(BX; Z^p)Q.
Moreover, since both spaces are rationally 3-connected, the Hurewicz map induces
isomorphisms ss4(BX0) ~=ss4(BX) Q ~=H4(BX; Z^p) Q. The first isomorphism
follows from [2; V, 4.1]. Hence, the composition induces an isomorphism between
the homotopy groups and is therefore an equivalence.
Proof of Proposition 3.1. First we assume that X is simply connected, hence 2-
connected, and simple. In this context, simple means that the representation a*
*e :
WX -! Gl(LX Q) is irreducible. This representation is given by the action of *
*the
Weyl group WX on the maximal torus TX of X respectively on the p-adic lattice
LX := H2(BTX ; Z^p).
If X is 3-connected, then H4(BX; Z^p) = 0 and the statement is obviously tru*
*e. If
ss3(X) 6= 0, then ss3(X) Q ~=Q^pkis isomorphic to the dual of the Q^p-vector s*
*pace
5
H4(BX; Z^p) Q ~=(H4(BTX ; Z^p) Q)WX . The last isomorphism follows from [6].
Every element q 2 (H4(BTX ; Z^p) Q)WX can be interpreted as a WX -invariant
quadratic form q : LX Q -! Q^p. Since ae is irreducible, every WX -invariant
quadratic form is definite and two such quadratic forms are linearly dependent.
Thus, all the groups are isomorphic to Q^p, generated by a definite WX -invaria*
*nt
quadratic form qX on LX Q. This implies that the pseudo reflection group WX *
*is
a honest reflection group, i.e. generated by elements of order 2 fixing a hyper*
*plane
of LX Q.
Let s 2 WX be any reflection, and let T := ((TX )s)e be the component of the*
* unit
of the fixed-point set of the action of s on TX . Then, the centralizer C := CX*
* (T )
X of T is a connected subgroup of X of maximal rank whose Weyl group WC ~= Z=2
is generated by s. All this follows from [6] [7] [20]. The p-compact group C ha*
*s a
finite covering , which splits into a product of a torus and a simply connected*
* p-
compact group C0, and WC0 ~=WC ~= Z=2 is of order two [20]. Since SU(2)^pis the
only simply connected p-compact group with Weyl group isomorphic to Z=2 [9], we
have C0 ~=SU(2)^pas p-compact group. This establishes a map BSU(2)^p-! BX
respectively a homomorphism SU(2)^p-! X of p-compact groups. This might not
be a monomorphism, but the kernel K SU(2)^pof this homomorphism is a finite
p-group and a central subgroup of SU(2)^p, actually of SU(2) in the honest sens*
*e of
compact Lie groups [23]. The kernel can be divided out [23] and, doing this, yi*
*elds a
monomorphism (SU(2)=K)^p-! X respectively a map B(SU(2)=K)^p-! BX. The
last claim follows from [6]. The center Z(SU(2) ~=Z=2 is a 2-group. Thus, for o*
*dd
primes, K = 0 and for p = 2 the quotient G := SU(2)=K is a connected compact
Lie group isomorphic to SU(2) or SO(3). In both cases, there exists a connected
compact Lie group G and a fibration X=G^p-! BG^p-! BX satisfying condition
(1) of the statement. The homotopy fiber X=G^pis simple since the homotopy fiber
of a map between simply connected spaces is. Moreover, X=G^pis Fp-finite since
G -! X is a monomorphism [6]. This establishes part (3).
Now we consider, the induced homomorphism
Q^p~= H4(BX; Z^p) Q -! H4(BG;^p) Q ~=Q^p:
By construction, the WX -invariant quadratic form qX is mapped onto a WG -
invariant quadratic form q0 2 H4(BG;^p) Q. If q0 = 0, the map BG^p-! BX
induces the trivial map in reduced rational cohomology [1] and is therefore null
homotopic [18]. This contradicts the fact that X=G^pis Fp-finite and finishes *
*the
proof of the statement for simple simply connected p-compact groups.
Now let X be a connected p-compact group, such that ss1(X) is finite. Then t*
*he
universal cover X" is again a p-compact group~and establishes a map BX" -! BX
which induces an isomorphism H*(BX; Z^p) Q -=!H*(BX"; Z^p) Q [20]. By [8],
X" ~=Q iXi splits into a finite product of simply connected simple p-compact gr*
*oups
Xi. For each i, we choose a connected compact Lie group Gi and a monomorphism
Gi^p-! Xi satisfying the statement. Taking the product gives a homomorphism
Y Y
G0^p:= Gi^p-! Xi ~=X" -! X ;
i i
which might not be a monomorphism. But, analogously as above, the kernel K is a
central subgroup of G0 (in the Lie group sense) and, by dividing out the kernel*
*, the
6
homomorphism can be made into a monomorphism G^p:= (G0=K)^p-! X. Again,
the center of G0 is a 2-group. Hence, for odd primes, G ~=SU(2)r. Moreover, G is
a connected compact Lie group satisfying part (1) of the statement.
By construction, all the maps establish a commutative diagram
BG0^p ----! BX"
?? ?
y ?y
BG^p ----! BX
where the vertical arrows induce isomorphisms in rational cohomology. The top
horizontal map satisfies part (2) and so does the bottom arrow. By the same
argument as above the homotopy fiber X=G^p of the bottom horizontal arrow is
simple and Fp-finite.
4. The arithmetic square and genus sets.
Most of the materialQin this section is taken from [28] and [32]. For a spac*
*e X
we define X^ := pX^p to be the product of the p-adic completions in the sense
of Bousfield and Kan [2] taken over all primes. For spaces with mod-p homology
of finite type, this is equivalent to the profinite completion of Sullivan [28]*
* [2].
Sullivan also constructed a formal completion for spaces [28]. For a space X, *
*the
p-formal completion X[pis defined to be the homotopy colimit hocolimiXi^pwhere
Xi ranges over all finite subcomplexes of X. The formal completion X[ is given
by hocolimiXi^. Both completions are coaugmented functors on the homotopy
category of topological spaces. That is they come with a map X -! X[prespective*
*ly
X -! X^p. By X[0we denote the formal completion of the rationalization of X and
by X^0the rationalization of the profinite completion of X.
4.1 Theorem. ( [Sullivan]) Let X be a connected simple CW -complex of finite
type. Then
(1) X[0' X^0.
(2) The diagram
X ----! X^
?? ?
y ?y
X0 ----! X[0' X^0
is a fiber square up to homotopy.
Proof. Actually, in [28] this is only proved for simply connected spaces, but a*
*ll
proofs also work in the case of simple spaces (see also [2]).
4.2 Remark: For a simple space X, we can identify X^0and X[0via the above
equivalence. The homotopy groups ss*(X^0) are modules over Q Z^ and therefore
topological groups. For simple spaces X and Y , a map f : (X^ )0 -! (Y ^)0 is
called ss* -continuous if ss*(f) is continuous. In the finitely generated case*
* this is
equivalent to ss*(f) being Q Z^ -linear. If f is induced from a map X0 -! Y0 by
formal completion or from X^ -! Y ^ by localization then f is always ss*-contin*
*uous.
If X^0 ' Y0^ we denote by HEC^0(X; Y ) the set of homotopy classes of ss*-
continuous homotopy equivalences f : (X^ )0 -! (Y ^)0. If X = Y , then we define
HEC^0(X) := HEC^0(X; X). The elements of these groups are called gluing maps.
7
For any element ff 2 CHE^0(X; Y ) we define W := W (X; ff; Y ) to be the hom*
*o-
topy pull back of
X0 -! (X^ )0 -ff!(Y ^)0- Y ^ :
4.3 Proposition. Let X and Y are simple spaces of finite type and let ff 2
HEC^0(X; Y ). Then, for W := W (X; ff; Y ), the following holds:
(1) W ^' Y ^ and W0 ' X0.
(2) W is a simple space of finite type.
Proof. Part (1) is obvious. By [32; 3.7], the abelian groups ssn(W ) are finit*
*ely
generated for all n and, since ssn(W ) -! ssn(W ^) ~= ssn(X^ ) is a monomorphis*
*m,
W is simple. This proves the second part of the statement.
If X = Y , the above construction gives a map HEC^0(X) -! G^0(X) where the
genus G^0(X) of X is the set of all homotopy equivalence classes of nilpotent s*
*paces
Y such that Y ^ ' X^ and Y0 ' X0. In [32; 3.8] is proved that the above assignm*
*ent
identifies the genus set of X with a particular double coset of HEC^0(X).
5. Construction of a particular fibration.
Let L be a a quasi-finite connected loop space with finite fundamental group.
Then, the rational cohomology H*(BL; Q) of the classifying space BL of L is a
finitely generated polynomial algebra with generators x1; :::xn. The generators*
* are
chosen as images of non divisible integral classes, also denoted by xi 2 H*(BL;*
* Z).
These integral classes define a map
Y
BL -e!K := K(Z; |xi|) =: K<4> x K(4)
i
where K(Z; |xi|) is an Eilenberg-MacLane space, where |xi| denotes the degree of
the class xi (which in our case is always even) and where K<4>, as the 4-th Pos*
*tnikov
section, is the product of all factors of degree less than 5 and K(4), as the 4*
*-th
connected cover, the product of all other factors. Rationally, this map establi*
*shes
equivalences BL0 -! K0 ' K<4>0 x K(4)0 and BL^0 ' BL[0-! K[0' K^0 '
K<4>^0x K(4)^0.
Let r := dimQ H*(BL; Q) = dimQ^pH*(BL; Z^p) Q. Completed at every prime,
L becomes a p-compact group. By Proposition 3.1 and Corollary 3.2, there exist a
semi simple connected compact Lie group G, such that "G~= SU(2)r, and, for every
prime p a map BG^p-! BL^psuch that the composition BG^p-! BL^p-! BL<4>^p
induces an equivalence after localizing at 0. Putting all these maps together w*
*e get
a map f : BG^ -! BL^ such that BG^0-! BL<4>^0is an equivalence.
Since BG0, BL<4>0 and K<4>0 are rational Eilenberg-MacLane spaces with iso-
morphic homotopy groups, all these spaces are homotopy equivalent. Thus, there
exists a map s : BG0 -! BL0, given by a section of BL0 -! K<4>0 such that
BG0 -! BL0 -! K(4)0 is null homotopic. We have a diagram
BG[0 ---s-! BL[0 ---e-!' K<4>[0x K(4)[0 --pr1--!BL<4>[0
? ? ?
j?y j?y j?y
BG^0 ---f-! BL^0 ---e-!'K<4>^0x K(4)^0 --pr1--!BL<4>^0 ;
8
where the vertical arrows are the obvious homotopy equivalences, where pr1 deno*
*tes
the projection onto the first factor and where the composition of the horizontal
arrows is a homotopy equivalence. The compositions es =: (s1; s2) and ef =:
(f1; f2) can be written as two components. Confusing notations, e, f and s as w*
*ell
as fi and si also denote the maps between the uncompleted or unlocalised spaces
if they exist.
We want to change the gluing map j : BL[0-! BL^0in such a way that it lifts
to the gluing map ff := (pr1ef)-1 pr1es : BG[0-! BG^0. This map is a homotopy
equivalence and ss*-continuous since all homotopy groups of BG and BL are finit*
*ely
generated and since all maps are QZ^ -linear. As a product of Eilenberg-MacLane
spaces, BL^0is an H-space. The product of the two maps
fi1 := (j|K<4>[0; f2ff(s1)-1 ) : K<4>[0-! K<4>^0x K(4)^0
and
fi2 := j|K(4)[0: K(4)[0-! K<4>^0x K(4)^0
defines a map
fi := fi1 . fi2 : BL[0' K<4>[0x K(4)[0-! K<4>^0x K(4)^0' BL^0;
which, by construction, is ss*-continuous and fits into a homotopy commutative
diagram
(L0=G0)^ ----! BG[0 ---s-! BL[0
? ? ?
fl?y ff?y fi?y
(L^ =G^ )0 ----! BG^0 --f--! BL^0 :
Here L0=G0 and L^ =G^ denote the homotopy fibers of the map s : BG0 -! BL0
and f : BG^ -! BG^ . The maps ff and fi are homotopy equivalences and so is fl.
Since f and s have left inverse, the map fl is also ss*-continuous. The homoto*
*py
groups of L0=G0 are finite dimensional rational vector spaces and the homotopy
groups of L^ =G^ finitely generated Z^ -modules.
These considerations lead to the following proposition.
5.1 Proposition. Let L be a connected quasi-finite loop space such that ss1(L)
is finite. Let r := dimQ H4(BL; Q). Then, there exist a semi simple connected
compact Lie group G, quasi-finite connected loop spaces M and N and a fibration
E -! BM -! BN
such that the following holds:
(1) E is quasi-finite and simple.
(2) "G~=SU(2)r.
(3) BM is in the genus of BG
(4) BN is in the genus of BL.
Proof. Using the arithmetic square, the above diagram establishes a fibration E*
* -!
BM -! BN. The last three properties are true by construction. The first property
follows from Proposition 4.3. The loop spaces M := BM and N := BN are in
the genus of G respectively of L and give therefore rise to quasi-finite loop s*
*paces.
This finishes the proof.
Next we want to analyze the loop fi : L[0-! L^0of the gluing map fi.
9
5.2 Lemma. fi ' j : L[0-! L^0.
Proof. Using the H-space structure of L^0' K<4>^0xK(4)^0, we can again describe
the map fi as a product of two maps, We just have to multiply the loops of fi1 *
*and
fi2. Therefore, we only have to show that fi2 = f2ff(s1)-1 ) : K<4>[0-! K(4)^0
is null homotopic. But this map fits into a commutative diagram
G[0 --ff--! G^0 --q-- G"^0 ---- G"^
? ? ? ?
'?y f2 ?y f2 ?y (f2)q?y
K0[0 --fi2--!K00^0 ________K00^0---- K00^
where K0 := K<4> and K00:= K(4) are again Eilenberg-MacLane spaces. Since
the first two arrows in the row are homotopy equivalences and since the map from
the last column to the third column is the coaugmentation of the localization a*
*t 0,
we only have to show thatQf2q : G"^ -! K00^ is null homotopic. The Eilenberg-
MacLane space K(4)^ ' lK(Z^ ; 2l) splits into a finite product such that l > 2
for all l. Since H*(BG"; Z^ is generated by classes of degree 4, each coordin*
*ate
of the map BG"^ -! K(4)^ is given by a decomposable cohomology class xl in
H*(BG"^; Z^ ). The next lemma finishes the proof.
5.3 Lemma. Let L be a quasi-finite loop space and let g : BL^ -! K(Z^ ; r)
denote a decomposable cohomology class in H*(BL^ ; Z^ ). Then, the map g :
L^ ' BL^ -! K(Z^ ; r - 1) is null homotopic.
Proof. The adjoint of g factors through BL^ -! BL -g! K(Z^ ; r). Since
BL is a co-H-space, all cup products vanish. Hence, the composition is null
homotopic as well as g.
6. Proof of Theorem 1.4.
We start with analyzing the genus of the connected compact Lie group G.
6.1 Proposition. Let G be a semi simple connected compact Lie group, such that
G" ~=SU(2)r. Then, the genus of G is rigid. That is every element of the genus *
*of
G is actually homotopy equivalent to G.
Proof. First we prove the statement for G = "G~= SU(2)r. Since
G^0' K((Q Z^ )r; 3) is an Eilenberg-MacLane space, the gluing map for M is
given by a matrix A 2 Gl(r; Q Z^ ) and therefore splits into a product RC with
R 2 Gl(r; Q) and C 2 Gl(r; Z^ ) [32]. Since G ~= SU(2)r, R can be realized by a
self map G0 -! G0 and C by a self map G^ -! G^ . This implies that M ' G [32].
Now let G be a quotient of "G, and let M 2 genus(G). Then, ss1(M) ~=ss1(G) =:
E is a finite elementary abelian 2-group. The universal cover M" is contained *
*in
the genus of G" ' SU(2)r. Hence, M" ' "G and M fits into a fibration G" -! M -!
BE. The 2-adic completion establishes an injection ss3(G") -! ss3(G"^2) and t*
*urns
the fibration into a principal fibration. Since principal fibration with conne*
*cted
structure groups are simple and since self maps of G" are classified by means of
homotopy groups, the above fibration is oriented and is classified by a map BE *
*-!
BSHE(G"), where SHE(G") denotes the set of all homotopy equivalence classes
10
of G" homotopic to the identity. The left action of G" on itself establishes a*
* map
BG" -! BSHE(G") [27].
Now, we consider the diagram
BG" ----! BSHE(G")
?? ?
y ?y
BG"^2 ----! BSHE(G")^2 --'--! BSHE(G"^2)
Since "Gis a topological group, SHE(G"^2) is 2-complete and the map SHE(G")F2 -!
SHE(G"F2) is a homotopy equivalence [2]. Since completion and passing to classi-
fying spaces commutes for nilpotent connected loop spaces [2], the second arrow*
* in
the bottom line is a homotopy equivalence. The homotopy fibers of both columns
have uniquely 2-divisible homotopy groups. Thus, obstruction theory shows that
for every map BE -! BG"^2or BE -! BSHE(G"^2) there exist a lift to BG" respec-
tively to BSHE(G"), unique up to homotopy. A small diagram chase shows that, up
to homotopy, the classifying map BE -! BSHE(G") has a unique lift BE -! BG".
Hence, G" -! M -! BE is a principal fibration. By [10], the classifying map is
induced from a homomorphism ae : E -! G". In fact, this map is a monomorphism,
since the homotopy fiber M of Bae is quasi-finite [26]. All homomorphisms E -! *
*G"
are central and therefore, M ' "G=E is a compact Lie group.
Finally we have to show that G"=E ' G. Using shift maps and permutation
of factors, one can show, analogously as for the equivalence SO(4) = SU(2) xZ=2
SU(2) ' SO(3) x SU(2), that G"=E and G are homotopy equivalent to SO(3)s x
SU`(2)r-s where s equals the rank of E.
Now we are able to finish the proof of Theorem 1.4.
Proof of Theorem 1.4. The fibration L -! E -! BM of Proposition 5.1 has almost
all the properties we stated in Theorem 1.4. It is only left to show that N ' L
and that M ' G. The first assertion follows from Lemma 5.2, and the second from
Proposition 6.1. This finishes the proof of Theorem 1.4
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Mathematisches Institut, Bunssenstr. 3-5, 37073 G"ottingen, Germany.
e-mail : notbohm at cfgauss.uni-math.gwdg.de