THE FINITENESS OBSTRUCTION FOR LOOP SPACES

D. Notbohm

Abstract. For finitely dominated spaces, Wall constructed a finiteness obstruction, which decides whether a space is equivalent to a finite CW -complex or not. It was conjectured that this finiteness obstruction always vanishes for quasi finite 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, this shows that every quasi finite loop space is actually homotopy equivalent to a finite CW -complex.

1. Introduction.

A topological space is called quasi-finite, if the direct sum ⨁_{n} H_{n}(X; ℤ) of 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 problem [29] [30]. A space X is finitely dominated, if there exist a finite CW -complex K and maps f : K ⟶ X and g : X ⟶ K such that the composition fg is homotopic to the identity id_{X} on X. One can show that finitely dominated spaces are always 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 π_{1}(X) acts nilpotent on the homotopy groups π_{∗}(X). In particular, π_{1}(X) is a nilpotent group. The space X is called simple if, in addition, the action is trivial. In this case, π_{1}(X) is abelian.

For a finitely dominated space X, the finiteness obstruction ω̃(X) of Wall is an element in the reduced projective class group K̃_{0}(ℤ[π_{1}(X)]) of the integral group ring. And ω̃(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 topological properties of spaces are reflected by properties of the finiteness obstruction. For instance, it is not known, but conjectured, that the finiteness obstruction for quasifinite 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 homotopy type of a CW -complex, BL pointed, and where e : ΩBL ⟶ L is a homotopy equivalence between the loop space of BL and L. Examples of loop spaces are given by topological groups and their classifying spaces. Loop spaces inherit properties from the space L; e.g. L is called quasi-finite, if the space L is so. Loop spaces

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 57Q12, 55R35, 55R10.

Key words and phrases. finiteness obstruction, Wall obstruction, loop space, p-compact group.

are always nilpotent spaces. In fact, they are simple. Thus every quasi-finite loop space is finitely dominated and the finiteness obstruction ω̃(L) is defined.

Theorem 1.1. Let L be a quasi finite loop space. Then 0 = ω̃(L) ∈ K̃_{0}(ℤ[π_{1}]) and L is homotopy equivalent to a finite CW -complex.

For a quasi-finite nilpotent space X, Mislin proved results of this type if π_{1}(X) is infinite [16] or if π_{1}(X) is cyclic of prime power order [15].

The proof of our result is a consequence of the next two statements. For every fibration F ⟶ E ⟶ B, the fiber transport along loops in B establishes a homomorphism π_{1}(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 oriented if this homomorphism is trivial.

Theorem 1.2. Let L be a quasi-finite loop space, such that π_{1}(L) is finite. Let r := dim_{ℚ}H^{4}(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 G̃ of G, we have G̃ ≅ SU(2)^{r}.

(2) The space E is simple and quasi-finite.

Theorem 1.3. Let X be a finitely dominated space, such that π_{1}(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 ω̃(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 general case, the statement is known and a simple consequence of some vanishing results of Lück for the algebraic transfer in algebraic K-theory associated to a fibration [14] (see Section 2). We couldn’t find a reference for it, but believe that it is worth to be stated.

Proof of Theorem 1.1. If π_{1}(L) is infinite, we can apply the above mentioned result of Mislin [16]. If π(L) is finite, the statement follows from Theorem 1.2 and Theorem 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 π_{1}(L) is finite. Let r := dim_{ℚ}H^{4}(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 G̃ of G, we have G̃ ≅ 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.

The proof of Theorem 1.4 goes as follows. Since completion turns the quasifinite 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 of the fact that the rationalization BL_{0} is equivalent to a product of rational Eilenberg-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 1.3. The other sections are devoted to the proof 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 nilpotent spaces with mod-p homology of finite type, both constructions coincide [2].

It is a pleasure thank Wolfgang Lück for telling me that a result like Theorem 1.2 implies the vanishing of the finiteness obstruction for quasi finite loop spaces and for several helpful discussions on this subject. I am also grateful to the SFB 478 ”Geometrische Strukturen in der Mathematik” at Münster 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 fiber, Ehrlich [11] constructed a geometric transfer

p^{!} : K_{0}(ℤ[π_{1}(B)] ⟶ K_{0}(ℤ[π_{1}(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 class groups. Ehrlich also proved a formula relating the finiteness obstruction of E and B [12], namely

ω̃(E) = p^{!}(ω̃(B)) + χ(B) · i_{∗}(ω̃F) ∈ K̃_{0}(ℤ[π_{1}(E)]) ,

where χ(B) denotes the Euler characteristic of B and where i_{∗} : K̃_{0}(ℤ[π_{1}(F)] ⟶ K̃_{0}(ℤ[π_{1}(E)] is the map induced by induction.

In [13] [14], for fibrations of the above type, Lück constructed an algebraic transfer, showed that both coincide and proved the following vanishing result.

2.1 Theorem. (Lück [14, 9.1]) Let G ⟶ E ⟶ B be an oriented fibration, such that G is a connected compact Lie group. If π_{1}(B) is finite, then the algebraic transfer

p^{∗} : K_{0}(ℤ[π_{1}(B)] ⟶ K_{0}(ℤ[π_{1}(E)]

vanishes.

Proof of Theorem 1.3. The compact Lie group G is connected and homotopy equivalent to a finite CW -complex. Hence ω̃(G) = 0. Moreover, π_{1}(B) is finite since π_{1}(X) is. Thus, the statement is a simple consequence of Theorem 2.1 and the above formula. □

3. Particular subgroups of p–compact groups.

A p–compact group X is a loop space X = (X, BX, e) such that BX is pcomplete and pointed and such that X is 𝔽_{p}-finite, i.e. H^{∗}(X; 𝔽_{p}) 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. there 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 π_{0}(P) ≅ π_{1}(BP) is a finite p-group and such that the universal cover of BP is equivalent to an Eilenberg-MacLane space of the form K(ℤ_{p}^{∧}^{n}, 2). For details and further notions we refer the reader to the above 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-completed version of Theorem 1.4.

Proposition 3.1. Let X be a connected p–compact group, such that π_{1}(X) is finite. Let r := dim_{ℚ}H^{4}(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 G̃ of G, we have G̃ ≅ SU(2)^{r}. If p is odd, we can choose G = SU(2)^{r}.

(2) The induced map H^{4}(BX; ℤ_{p}^{∧})⊗ℚ ⟶ H^{4}(BG_{p}^{∧}; ℤ_{p}^{∧})⊗ℚ is an isomorphism.

(3) The homotopy fiber X/G_{p}^{∧} of f is simple and 𝔽_{p}-finite.

Before we start with the proof, we draw one corollary. For a space Y we denote by Y ⟨n⟩ the n-th Postnikov section and by Y (n) the n-connected cover of Y. Hence these spaces fit into a fibration

Y (n) ⟶ Y ⟶ Y ⟨n⟩ .

3.2 Corollary. The composition BG_{p}^{∧} ⟶ BX ⟶ BX⟨4⟩ induces an equivalence

(BG_{p}^{∧})_{0} ⟶^{≃} BX⟨4⟩_{0}.

Proof. Since π_{1}(G) and π_{1}(X) are finite and since G̃ ≅ SU(2)^{r}, both spaces, (BG_{p}^{∧})_{0} and BX⟨4⟩_{0}, are Eilenberg-MacLane spaces with non vanishing 4-th homotopy group. The universal coefficient theorem shows that H^{4}(BX; ℤ_{p}^{∧}) ⊗ ℚ is the dual of H_{4}(BX; ℤ_{p}^{∧})⊗ℚ and hence, that H_{4}(BG_{p}^{∧}; ℤ_{p}^{∧})⊗ℚ ≅ H_{4}(BX; ℤ_{p}^{∧})⊗ℚ. Moreover, since both spaces are rationally 3-connected, the Hurewicz map induces isomorphisms π_{4}(BX_{0}) ≅ π_{4}(BX) ⊗ ℚ ≅ H_{4}(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 2connected, and simple. In this context, simple means that the representation ρ : W_{X} ⟶ Gl(L_{X} ⊗ ℚ) is irreducible. This representation is given by the action of the Weyl group W_{X} on the maximal torus T_{X} of X respectively on the p-adic lattice L_{X} := H_{2}(BT_{X}; ℤ_{p}^{∧}).

If X is 3-connected, then H^{4}(BX; ℤ_{p}^{∧}) = 0 and the statement is obviously true. If π_{3}(X) ≠ 0, then π_{3}(X)⊗ℚ ≅ ℚ_{p}^{∧}^{k} is isomorphic to the dual of the ℚ_{p}^{∧}-vector space H^{4}(BX; ℤ_{p}^{∧}) ⊗ ℚ ≅ (H^{4}(BT_{X}; ℤ_{p}^{∧}) ⊗ ℚ)^{W_{X}}. The last isomorphism follows from [6]. Every element q ∈ (H^{4}(BT_{X}; ℤ_{p}^{∧}) ⊗ ℚ)^{W_{X}} can be interpreted as a W_{X}-invariant quadratic form q : L_{X} ⊗ ℚ ⟶ ℚ_{p}^{∧}. Since ρ is irreducible, every W_{X}-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 W_{X}-invariant quadratic form q_{X} on L_{X} ⊗ ℚ. This implies that the pseudo reflection group W_{X} is a honest reflection group, i.e. generated by elements of order 2 fixing a hyperplane of L_{X} ⊗ ℚ.

Let s ∈ W_{X} be any reflection, and let T := ((T_{X})^{s})_{e} be the component of the unit of the fixed-point set of the action of s on T_{X}. Then, the centralizer C := C_{X}(T) ⊂ X of T is a connected subgroup of X of maximal rank whose Weyl group W_{C} ≅ ℤ/2 is generated by s. All this follows from [6] [7] [20]. The p–compact group C has a finite covering , which splits into a product of a torus and a simply connected p– compact group C′, and W_{C′} ≅ W_{C} ≅ ℤ/2 is of order two [20]. Since SU(2)_{p}^{∧} is the only simply connected p–compact group with Weyl group isomorphic to ℤ/2 [9], we have C′ ≅ SU(2)_{p}^{∧} as 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)_{p}^{∧} of this homomorphism is a finite p-group and a central subgroup of SU(2)_{p}^{∧}, actually of SU(2) in the honest sense of compact Lie groups [23]. The kernel can be divided out [23] and, doing this, yields 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 odd 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_{p}^{∧} is simple since the homotopy fiber of a map between simply connected spaces is. Moreover, X/G_{p}^{∧} is 𝔽_{p}-finite since G ⟶ X is a monomorphism [6]. This establishes part (3).

Now we consider, the induced homomorphism

ℚ_{p}^{∧} ≅ H^{4}(BX; ℤ_{p}^{∧}) ⊗ ℚ ⟶ H^{4}(BG; ℤ_{p}^{∧}) ⊗ ℚ ≅ ℚ_{p}^{∧} .

By construction, the W_{X}-invariant quadratic form q_{X} is mapped onto a W_{G}- invariant quadratic form q′ ∈ H^{4}(BG;_{p}^{∧}) ⊗ ℚ. If q′ = 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_{p}^{∧} is 𝔽_{p}-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 π_{1}(X) is finite. Then the universal cover X̃ is again a p–compact group and establishes a map BX̃ ⟶ BX which induces an isomorphism H^{∗}(BX; ℤ_{p}^{∧}) ⊗ ℚ ⟶^{≅} H^{∗}(BX̃; ℤ_{p}^{∧}) ⊗ ℚ [20]. By [8], X̃ ≅ ∏_{i} X_{i} splits into a finite product of simply connected simple p–compact groups X_{i}. For each i, we choose a connected compact Lie group G_{i} and a monomorphism G_{i}_{p}^{∧} ⟶ X_{i} satisfying the statement. Taking the product gives a homomorphism

G′_{p}^{∧} := ∏_{i} G_{i}_{p}^{∧} ⟶ ∏_{i} X_{i} ≅ X̃ ⟶ X ,

which might not be a monomorphism. But, analogously as above, the kernel K is a central subgroup of G′ (in the Lie group sense) and, by dividing out the kernel, the homomorphism can be made into a monomorphism G_{p}^{∧} := (G′/K)_{p}^{∧} ⟶ X. Again, the center of G′ 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

BG′_{p}^{∧} ⟶ BX̃
↓  ↓
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 𝔽_{p}-finite. □

4. The arithmetic square and genus sets.

Most of the material in this section is taken from [28] and [32]. For a space X we define X^{∧} := ∏_{p} X_{p}^{∧} to be the product of the p–adic completions in the sense of Bousfield and Kan [2] taken over all primes. For nipotent 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_{p}^{♭} is defined to be the homotopy colimit hocolim_{i}X_{i}_{p}^{∧} where X_{i} ranges over all finite subcomplexes of X. The formal completion X^{♭} is given by hocolim_{i}X_{i}^{∧}. Both completions are coaugmented functors on the homotopy category of topological spaces. That is they come with a map X ⟶ X_{p}^{♭} respectively X ⟶ X_{p}^{∧}. By X_{0}^{♭} we denote the formal completion of the rationalization of X and by X_{0}^{∧} the 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^{∧}
↓  ↓
X_{0} ⟶ X_{0}^{♭} ≃X_{0}^{∧}

is a fiber square up to homotopy.

Proof. Actually, in [28] this is only proved for simply connected spaces, but all proofs also work in the case of simple spaces (see also [2]). □

4.2 Remark: For a simple space X, we can identify X_{0}^{∧} and X_{0}^{♭} via the above equivalence. The homotopy groups π_{∗}(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 π_{∗} -continuous if π_{∗}(f) is continuous. In the finitely generated case this is equivalent to π_{∗}(f) being ℚ ⊗ ℤ^{∧}-linear. If f is induced from a map X_{0} ⟶ Y_{0} by formal completion or from X^{∧} ⟶ Y^{∧} by localization then f is always π_{∗}-continuous.

If X_{0}^{∧} ≃ Y_{0}^{∧} we denote by HEC_{0}^{∧}(X, Y) the set of homotopy classes of π_{∗}- 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.

For any element α ∈ CHE_{0}^{∧}(X, Y) we define W := W (X, α, Y) to be the homotopy pull back of

X_{0} ⟶ (X^{∧})_{0} ⟶^{α} (Y^{∧})_{0} ⟵ Y^{∧} .

4.3 Proposition. Let X and Y are simple spaces of finite type and let α ∈ HEC_{0}^{∧}(X, Y). Then, for W := W (X, α, Y), the following holds:

(1) W^{∧} ≃ Y^{∧} and W_{0} ≃ X_{0}.

(2) W is a simple space of finite type.

Proof. Part (1) is obvious. By [32; 3.7], the abelian groups π_{n}(W) are finitely generated for all n and, since π_{n}(W) ⟶ π_{n}(W^{∧}) ≅ π_{n}(X^{∧}) is a monomorphism, W is simple. This proves the second part of the statement. □

If X = Y is a nipotent space of finite type, 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 spaces Y of finite type such that Y^{∧} ≃ X^{∧} and Y_{0} ≃ X_{0}. In [32; 3.8] is proved that the above assignment 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 x_{1}, ...x_{n}. The generators are chosen as images of non divisible integral classes, also denoted by x_{i} ∈ H^{∗}(BL; ℤ). These integral classes define a map

BL ⟶^{e} K := ∏_{i} K(ℤ, |x_{i}|) =: K⟨4⟩ × K(4)

where K(ℤ, |x_{i}|) is an Eilenberg-MacLane space, where |x_{i}| denotes the degree of the class x_{i} (which in our case is always even) and where K⟨4⟩, as the 4-th Postnikov 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 establishes equivalences BL_{0} ⟶ K_{0} ≃ K⟨4⟩_{0} × K(4)_{0} and BL_{0}^{∧} ≃ BL_{0}^{♭} ⟶ K_{0}^{♭} ≃ K_{0}^{∧} ≃ K⟨4⟩_{0}^{∧} × K(4)_{0}^{∧}.

Let r := dim_{ℚ}H^{4}(BL; ℚ) = dim_{ℚ_{p}^{∧}}H^{4}(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 G̃ ≅ SU(2)^{r}, and, for every prime p a map BG_{p}^{∧} ⟶ BL_{p}^{∧} such that the composition BG_{p}^{∧} ⟶ BL_{p}^{∧} ⟶ BL⟨4⟩_{p}^{∧} induces an equivalence after localizing at 0. Putting all these maps together we get a map f : BG^{∧} ⟶ BL^{∧} such that BG_{0}^{∧} ⟶ BL⟨4⟩_{0}^{∧} is an equivalence.

Since BG_{0}, BL⟨4⟩_{0} and K⟨4⟩_{0} are rational Eilenberg-MacLane spaces with isomorphic homotopy groups, all these spaces are homotopy equivalent. Thus, there exists a map s : BG_{0} ⟶ BL_{0}, given by a section of BL_{0} ⟶ K⟨4⟩_{0} such that BG_{0} ⟶ BL_{0} ⟶ K(4)_{0} is null homotopic. We have a diagram

BG_{0}^{♭} ⟶^{s} BL_{0}^{♭} ⟶_{≃}^{e} K⟨4⟩_{0}^{♭} × K(4)_{0}^{♭} ⟶^{pr_{1}} BL⟨4⟩_{0}^{♭}
j↓  ^{j}↓  ^{j}↓
BG_{0}^{∧} ⟶^{f} BL_{0}^{∧} ⟶_{≃}^{e} K⟨4⟩_{0}^{∧} × K(4)_{0}^{∧} ⟶^{pr_{1}} BL⟨4⟩_{0}^{∧} ,

where the vertical arrows are the obvious homotopy equivalences, where pr_{1} denotes the projection onto the first factor and where the composition of the horizontal arrows is a homotopy equivalence. The compositions es =: (s_{1}, s_{2}) and ef =: (f_{1}, f_{2}) can be written as two components. Confusing notations, e, f and s as well as f_{i} and s_{i} also denote the maps between the uncompleted or unlocalised spaces if they exist.

We want to change the gluing map j : BL_{0}^{♭} ⟶ BL_{0}^{∧} in such a way that it lifts to the gluing map α := (pr_{1}ef)^{−1}pr_{1}es : BG_{0}^{♭} ⟶ BG_{0}^{∧}. This map is a homotopy equivalence and π_{∗}-continuous since all homotopy groups of BG and BL are finitely generated and since all maps are ℚ⊗ℤ^{∧}-linear. As a product of Eilenberg-MacLane spaces, BL_{0}^{∧} is an H-space. The product of the two maps

β_{1} := (j|_{K⟨4⟩_{0}^{♭}}, f_{2}α(s_{1})^{−1}) : K⟨4⟩_{0}^{♭} ⟶ K⟨4⟩_{0}^{∧} × K(4)_{0}^{∧}

and

β_{2} := j|_{K(4)_{0}^{♭}} : K(4)_{0}^{♭} ⟶ K⟨4⟩_{0}^{∧} × K(4)_{0}^{∧}

defines a map

β := β_{1} · β_{2} : BL_{0}^{♭} ≃ K⟨4⟩_{0}^{♭} × K(4)_{0}^{♭} ⟶ K⟨4⟩_{0}^{∧} × K(4)_{0}^{∧} ≃ BL_{0}^{∧} ,

which, by construction, is π_{∗}-continuous and fits into a homotopy commutative diagram

(L_{0}/G_{0})^{∧} ⟶ BG_{0}^{♭} ⟶^{s} BL_{0}^{♭}
γ↓  ^{α}↓  ^{β}↓
(L^{∧}/G^{∧})_{0} ⟶ BG_{0}^{∧} ⟶^{f} BL_{0}^{∧} .

Here L_{0}/G_{0} and L^{∧}/G^{∧} denote the homotopy fibers of the map s : BG_{0} ⟶ BL_{0} and f : BG^{∧} ⟶ BG^{∧}. The maps α and β are homotopy equivalences and so is γ. Since f and s have left inverse, the map γ is also π_{∗}-continuous. The homotopy groups of L_{0}/G_{0} 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 π_{1}(L) is finite. Let r := dim_{ℚ}H^{4}(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) G̃ ≅ 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 := ΩBM and N := ΩBN are in the genus of G respectively of L and give therefore rise to quasi-finite loop spaces. This finishes the proof. □

Next we want to analyze the loop Ωβ : L_{0}^{♭} ⟶ L_{0}^{∧} of the gluing map β.

5.2 Lemma. Ωβ ≃ j : L_{0}^{♭} ⟶ L_{0}^{∧}.

Proof. Using the H-space structure of L_{0}^{∧} ≃ ΩK⟨4⟩_{0}^{∧}×K(4)_{0}^{∧}, we can again describe the map Ωβ as a product of two maps, We just have to multiply the loops of β_{1} and β_{2}. Therefore, we only have to show that Ωβ_{2} = Ωf_{2}α(s_{1})^{−1}) : ΩK⟨4⟩_{0}^{♭} ⟶ ΩK(4)_{0}^{∧} is null homotopic. But this map fits into a commutative diagram

G_{0}^{♭} ⟶^{Ωα} G_{0}^{∧} ⟵^{q} G̃_{0}^{∧} ⟵ G̃^{∧}
≃↓  ^{Ωf_{2}}↓  ^{Ωf_{2}}↓  ^{(Ωf_{2})q}↓
K′_{0}^{♭} ⟶^{Ωβ_{2}} K′′_{0}^{∧}  K′′_{0}^{∧} ⟵ K′′^{∧}

where K′ := ΩK⟨4⟩ and K′′ := ΩK(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 at 0, we only have to show that Ωf_{2}q : G̃^{∧} ⟶ K′′^{∧} is null homotopic. The Eilenberg-MacLane space K(4)^{∧} ≃ ∏_{l} K(ℤ^{∧}, 2l) splits into a finite product such that l > 2 for all l. Since H^{∗}(BG̃; ℤ^{∧} is generated by classes of degree 4, each coordinate of the map BG̃^{∧} ⟶ K(4)^{∧} is given by a decomposable cohomology class x_{l} in H^{∗}(BG̃^{∧}; 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 Ωg : L^{∧} ≃ ΩBL^{∧} ⟶ K(ℤ^{∧}, r − 1) is null homotopic.

Proof. The adjoint of Ωg factors through ΣΩBL^{∧} ⟶ BL ⟶^{g} K(ℤ^{∧}, r). Since ΣΩBL is a co-H-space, all cup products vanish. Hence, the composition is null homotopic as well as Ωg. □

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 G̃ ≅ 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 = G̃ ≅ 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 ∈ Gl(r, ℚ ⊗ ℤ^{∧}) and therefore splits into a product RC with R ∈ Gl(r, ℚ) and C ∈ Gl(r, ℤ^{∧}) [32]. Since G ≅ SU(2)^{r}, R can be realized by a self map G_{0} ⟶ G_{0} and C by a self map G^{∧} ⟶ G^{∧}. This implies that M ≃ G [32].

Now let G be a quotient of G̃, and let M ∈ genus(G). Then, π_{1}(M) ≅ π_{1}(G) =: E is a finite elementary abelian 2-group. The universal cover M̃ is contained in the genus of G̃ ≃ SU(2)^{r}. Hence, M̃ ≃ G̃ and M fits into a fibration G̃ ⟶ M ⟶ BE. The 2-adic completion establishes an injection π_{3}(G̃) ⟶ π_{3}(G̃_{2}^{∧}) and turns the fibration into a principal fibration. Since principal fibration with connected structure groups are simple and since self maps of G̃ are classified by means of homotopy groups, the above fibration is oriented and is classified by a map BE ⟶ BSHE(G̃), where SHE(G̃) denotes the set of all homotopy equivalence classes of G̃ homotopic to the identity. The left action of G̃ on itself establishes a map BG̃ ⟶ BSHE(G̃) [27].

Now, we consider the diagram

BG̃ ⟶ BSHE(G̃)
↓  ↓
BG̃_{2}^{∧} ⟶ BSHE(G̃)_{2}^{∧} ⟶^{≃} BSHE(G̃_{2}^{∧})

Since G̃ is a topological group, SHE(G̃_{2}^{∧}) is 2-complete and the map SHE(G̃)𝔽_{2} ⟶ SHE(G̃𝔽_{2}) is a homotopy equivalence [2]. Since completion and passing to classifying 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 ⟶ BG̃_{2}^{∧} or BE ⟶ BSHE(G̃_{2}^{∧}) there exist a lift to BG̃ respectively to BSHE(G̃), unique up to homotopy. A small diagram chase shows that, up to homotopy, the classifying map BE ⟶ BSHE(G̃) has a unique lift BE ⟶ BG̃. Hence, G̃ ⟶ M ⟶ BE is a principal fibration. By [10], the classifying map is induced from a homomorphism ρ : E ⟶ G̃. In fact, this map is a monomorphism, since the homotopy fiber M of Bρ is quasi-finite [26]. All homomorphisms E ⟶ G̃ are central and therefore, M ≃ G̃/E is a compact Lie group.

Finally we have to show that G̃/E ≃ G. Using shift maps and permutation of factors, one can show, analogously as for the equivalence SO(4) = SU(2) ×_{ℤ/2} SU(2) ≃ SO(3) × SU(2), that G̃/E and G are homotopy equivalent to SO(3)^{s} × 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 Göttingen, Germany. e-mail : notbohm at cfgauss.uni-math.gwdg.de