THE REALIZABILITY OF LOCAL LOOP SPACES AS MANIFOLDS

TILMAN BAUER AND ERIK KJÆR PEDERSEN

Abstract. As an extension of earlier work, we show that every P -local loop space, where P is a set of primes, is homotopy equivalent to the P -localization of a compact, smooth, parallelizable manifold. A similar result is also proved for P -complete loop spaces.

1. Introduction

In [BKNP02], it was shown that every quasifinite loop space is homotopy equivalent to a parallelizable, compact, smooth manifold. In this paper, we prove a stronger local version of this result.

Let R be a commutative ring. We call a nilpotent space X R-finite if H_{∗}(X; R) is totally finitely generated as an R-module, and R-local if [Y, X] = 0 for every H_{∗}(−; R)-acyclic space Y. For any such R, there exists a localization functor L_{R} : Top → Top. For a set of primes P, call a nilpotent space X P -local (resp. P - complete) if it is local with respect to R = Z_{(P)} (resp. R = Z/P := ∏_{p∈P} Z/p.) We abbreviate the corresponding localization functors by L_{(P)} and L_{P}. A nilpotent space X is P -complete if and only if X ≃ ∏_{p∈P} L_{p}X. By the universal coefficient theorem, a finite loop space is Z_{(P)}-finite and Z/P -finite for any set of primes P.

The main result of this paper can be cast in a local and in a complete setting:

Theorem 1.1. Let P be a collection of primes, and let X be a P -local, Z_{(P)}-finite loop space. Then X is homotopy equivalent to the P -localization of a compact, smooth, parallelizable manifold.

Theorem 1.2. Let P be a collection of primes, and let X be a nilpotent P -complete space. Assume that there exist integers {d_{1}, …, d_{r}} such that for all primes p ∈ P, L_{p}X is a p-compact group with degrees d_{i}. Then X is homotopy equivalent to the P -completion of a compact, smooth, parallelizable manifold.

Setting P = the set of all primes, we recover from Thm 1.1 the main result of [BKNP02] that every quasifinite (i. e. Z-finite) loop space is homotopy equivalent to a compact, smooth, parallelizable manifold.

On the other extreme, we have as a special case of Thm. 1.2:

Corollary 1.3. Every p-compact group is the p-completion of a compact, smooth, parallelizable manifold.

A natural question to ask is whether Thm. 1.1 can be reduced to the global setting of [BKNP02] by showing that every space satisfying the conditions of Thm.

Date: March 18, 2004.

2000 Mathematics Subject Classification. 55P35, 55P15, 57R67.

Key words and phrases. local loop space, p-compact group, surgery, manifold.

1.1 can be represented as the P -localization (resp. P -completion) of a quasifinite loop space. This is in fact not the case, which is illustrated by the following theorem [Cla63]

Theorem 1.4 (Clark). Every nontrivial quasifinite loop space X with finite fundamental group has H^{3}(X; Q) ≠ 0.

On the other hand, simple p-compact groups typically have trivial third cohomology.

Theorem 1.1 can be further strengthened.

It is sufficient to assume that X is a p-compact group after completing at every prime p ∈ P. For P -local spaces, we have

Proposition 1.5. If X is a P -local space such that L_{p}X is a p-compact group for each p ∈ P, then X has a unique structure as a loop space such that all X → L_{p}X are loop maps.

2. Lifts of loop space structures

Theorem 1.2 reduces to Theorem 1.1 by the implication (2)⇒(4) in the following lemma, together with the fact that giving a loop structure on a P -complete space X is equivalent to giving (independently) loop structures on every L_{p}X, p ∈ P

Lemma 2.1. Let X be a P -complete loop space with finite mod p homology for every p ∈ P. Then the following are equivalent:

(1) X is the P -completion of a Z-finite CW complex;

(2) There exist positive integers {d_{1}, …, d_{r}} such that the p-compact group L_{p}X has those degrees for all p;

(3) There exist positive integers {d_{1}, …, d_{r}} such that as loop spaces,

L_{Q}X ≃ ∏_{i=1}^{r} K(A_{P} , 2d_{i} − 1),

where A_{P} = (∏_{p∈P} Z_{p}) ⊗ Q is the ring of finite adeles of Z_{(P)}.

(4) X is the P -completion of a Z_{(P)}-finite, P -local loop space.

Proof. (1)⇒(2): Let X = L_{P} F for a finite CW-complex F.

H^{∗}(L_{p}X; Z_{p}) ⊗ Q ≅ H^{∗}(F; Z_{p}) ⊗ Q ≅ H^{∗}(F; Q) ⊗ Q_{p}.

This shows that H^{∗}(X; Q) is an exterior algebra, and the degrees are independent of p.

(2)⇒(3): By [ABGP03, Thm. 2.1], L_{p}X ≃ L_{p} (∏_{i=1}^{r} S^{2d_{i}−1}) for p > h = max_{i}{d_{i}}. Thus

L_{Q}X ≃ ∏_{p≤h} L_{Q}X_{p} × L_{Q} (∏_{p>h} ∏_{i=1}^{r} L_{p}S^{2d_{i}−1})

≃ ∏_{p≤h} L_{Q}L_{p} ∏_{i=1}^{r} S^{2d_{i}−1} × ∏_{i=1}^{r} L_{Q} ∏_{p>h} L_{p}S^{2d_{i}−1}

≃ ∏_{i=1}^{r} L_{Q}L_{P} S^{2d_{i}−1}.

Furthermore,

π_{k}L_{Q}L_{P} S^{2d−1} = Q ⊗ ∏_{p∈P} π_{k}(S^{2d−1}) = \begin{cases}0; & k ≠ 2d − 1 \\ A_{P}; & k = 2d − 1. \end{cases}

Thus L_{Q}BX is a space with homotopy groups concentrated in even dimensions, and hence a product of Eilenberg-Mac Lane spaces

L_{Q}BX ≃ ∏_{i=1}^{r} K(A_{P} , 2d_{i}).

which proves (3).

(3)⇒(4): Define BK = ∏_{i=1}^{r} K(Q, 2d_{i}), and let BK → L_{Q}BX be the product of the maps induced by the unit ring map Q → A_{P}. Let BF be the homotopy pullback of BK → L_{Q}BX ← BX. It remains to show that K = ΩBK is Z_{(P)}- finite. Since Q and ∏_{p∈P} Z_{p} generate A_{P}, the Mayer-Vietoris sequence for the homotopy groups of a fiber product splits, and hence

π_{n}(BF) ≅ ⨁_{{i|2d_{i}=n}} Z_{(P)} ⊕ ⨁_{p∈P} Tor(Z, π_{n}(BX_{p})).

Thus π_{n}(F) is finitely generated as a Z_{(P)}-module, hence so is H^{n}(F; Z_{(P)}) for all n. There exists a d such that H^{i}(F; Z_{p}) = H^{i}(X; Z_{p}) = 0 for all p ∈ P and i ≥ d. Since H^{i}(F; Z_{p}) ≅ H^{i}(F; Z_{(p)}) ⊗_{Z_{(p)}} Z_{p} and H^{i}(F; Z_{(p)}) is finitely generated, it follows that H^{i}(F; Z_{(p)}) = 0 for those i and all p ∈ P. Thus H^{i}(F; Z_{(P)}) = 0 for those i as well.

(4)⇒(1): Let F_{P} be such a P -local, Z_{(P)}-finite space, and let Q be the complementary set of primes. Choose F_{Q} to be a Q-local, Z_{(Q)}-finite space such that L_{Q}F_{Q} ≅ L_{Q}F_{P}, and let F be the pullback of F_{Q} → L_{Q}F_{P} ← F_{P}. Since F is Z_{(P)}-finite and Z_{(Q)}-finite, it is Z-finite. □

Lemma 2.2. Let X be a Q-finite rational loop space, or the rationalization of a Z/P -finite, P -complete loop space for a set of primes P. Then it has a unique deloop, i.e. for every space Y such that ΩY ≃ X we have Y ≃ BX. The unique loop structure on X is commutative.

Proof. If X is a Q-finite rational loop space, H^{∗}(X; Q) is an exterior algebra on finitely many odd-dimensional generators. Hence for any Y such that ΩY ≃ X, H^{∗}(Y; Q) is an even polynomial algebra. Any two such rational spaces are equivalent since all k-invariants necessarily vanish. The unique deloop is a product of even-dimensional Eilenberg-Mac Lane spaces, therefore X is abelian.

A similar argument works for rationalizations of P -complete Z/P -finite loop spaces. In that case, X is homotopy equivalent to a product of K(A, odd), where A = (∏_{p∈P} Z_{p}) ⊗ Q is the group of P -adeles. (If P is finite, this is of course just ∏_{p∈P} Q_{p}.) □

Lemma 2.3. Let P be a set of primes and let X satisfy the equivalent conditions of Lemma 2.1. Let f: Y → X be a P -completion map for a P -local space Y. Then Y is Z_{(P)}-finite and carries a unique loop structure such that Y → X is a loop map.

Proof. Consider the arithmetic pullback diagram

\xymatrix{Y \ar[r]^{f} \ar[d] & L_{P} Y ≃ X \ar[d] \\ L_{Q}Y \ar[r]^{L_{Q}f} & L_{Q}L_{P} Y ≃ L_{Q}X.}

The Z_{(P)}-finiteness follows from the argument given in Prop. 2.1(3)⇒(4). As X → L_{Q}X is given as a loop map, loop structures on Y such that f is a loop map are in one-to-one correspondence with loop structures on L_{Q}Y such that L_{Q}f is a loop map.

Giving a loop structure on L_{Q}Y is equivalent to giving a Hopf algebra structure on H^{∗}(Y; Q) extending the cohomology algebra structure. Since any loop structure on L_{Q}Y is commutative, such a Hopf algebra structure is completely determined by the vector space of primitives in H^{∗}(Y; Q). Now for all p ∈ P, L_{Q}f induces an isomorphism of A-algebras

f^{∗}: H^{∗}(X; Z_{P}) ⊗ Q ⟶^{≅} H^{∗}(Y; Q) ⊗ Q_{p}.

This is an isomorphism of Hopf algebras if and only if

f^{∗}(P (H^{∗}(X; Z_{P}) ⊗ Q)) ≅ P (H^{∗}(Y; Q)) ⊗ Q_{p}.

This shows that the only possible choice for the vector space of primitives in H^{∗}(Y; Q) is (f^{∗}P (H^{∗}(X; Z_{P}) ⊗ Q)) ∩ H^{∗}(Y; Q), showing existence and uniqueness of the loop space structure. □

Proof of Prop. 1.5. We have a loop structure on L_{p}X for every p ∈ P, or equivalently on the P -completion of X. We can lift this uniquely to X by Lemma 2.3, proving the Proposition. □

3. Double 1-tori

Recall [BK72, Chapter IV] that associated to any commutative ring R and group G, there is a notion of R-completion of G compatible with R-localization of nilpotent spaces. We will denote by Z_{R} the R-localization of the integers and by Q_{R} ≅ Z_{R}⊗Q the R-localization of the rationals. Thus Z_{Z/p} = Z_{p} and Z_{Z_{(p)}} = Z_{(p)}. The localizations G_{R} carry a natural profinite topology. Similarly, for R-local spaces X, the chain complex C_{∗}(X) naturally carries a profinite topology. If A is an abelian group, we denote by H_{∗}(X; A) the homology of the chain complex C_{∗}(X) ˆ⊗A and by H^{∗}(X; A) the cohomology of the cochain complex Hom^{cont.}(C_{∗}(X); A). We abbreviate H_{∗}(X) = H_{∗}(X; Z_{R}) and H^{∗}(X) = H^{∗}(X; Z_{R}).

For an R-local finite loop space X, define the level of X to be the smallest l such that H^{2l+1}(X; Q_{R}) ≠ 0. Note that since the cohomology ring is a finite dimensional Hopf algebra, it is generated by odd degree exterior classes. Define the rank of X to be the number of exterior generators in H^{∗}(X; Q_{R}).

In [BKNP02], the pivotal tool for studying the finiteness obstruction and the surgery obstruction for a loop space X as well as for mixing homotopy types is

Definition 3.1. Let R be a ring and X be an R-finite, R-local, nilpotent, connected space. We say that X admits a double 1-torus if there is a diagram of horizontal and vertical fibrations of nilpotent spaces

[figure: L_{R}S^{1}; L_{R}S^{1}; i_{0}; ∗; L_{R}S^{1} × C; i; X; p_{0}; p; Z; C; i_{1}; Y; p_{1}; Z]

where C = Z/2 if \frac{1}{2} is not in R, and trivial otherwise, satisfying the following additional conditions:

(1) Z is R-finite,

(2) Y is stably reducible,

(3) X → Y is orientable,

(4) π_{1}(p_{0}) is an isomorphism, and

(5) Y ↠^{p_{1}} Z induces an isomorphism π_{1}(Y) × C ≅ π_{1}(Z).

Such an object was called rationally splitting in [BKNP02] if it rationally splits off a Hopf fibration. This condition needs to be relaxed for the present purposes.

Definition 3.2. A 1-torus L_{R}S^{1} → X ⟶^{p_{0}} Y is called rationally splitting of level l if p_{0} rationally has a retract of the form L_{Q}L_{R}S^{1} → L_{Q}L_{R}S^{2l+1} → L_{Q}L_{R}CP^{l}.

Thus, a rational splitting in the sense of [BKNP02] is a rational splitting of level 1.

Proposition 3.3. Given a sequence of positive integers (n_{0}, n_{1}, …, n_{k}) where k ≥ 1. Then there exists a bundle of manifolds S^{1} → M → N satisfying the following conditions:

(1) N is stably parallelizable

(2) M is parallelizable

(3) Rationally the bundle is a product of the standard bundle L_{Q}S^{1} → L_{Q}S^{2n_{0}+1} → L_{Q}CP^{n_{0}} and trivial bundles ∗ → L_{Q}S^{2n_{i}+1} ⟶^{=} L_{Q}S^{2n_{i}+1} for 1 ≤ i ≤ k.

(4) There exists an N_{0} ∈ N, depending only on the integers n_{i}, such that

L_{p}M ≃ ∏_{i=0}^{k} L_{p}S^{2n_{i}+1} for all p ≥ N_{0}.

Proof. We start out with the manifold X = CP^{n_{0}} × ∏_{i=1}^{k} S^{2n_{i}+1}. The Spivak normal fibration is rationally trivial, so a multiple of the top class of Σ^{d}X is spherical for sufficiently large d. Making the Thom collapse map S^{m} → Σ^{d}X transverse to X establishes a rational surgery problem

\xymatrix{ν_{N} \ar[r] \ar[d] & X × R^{d} \ar[d] \\ N \ar[r]^{f} & X}

with f of some non-zero degree on the top homology. Since X is an integral Poincaré duality space, the only possible surgery obstruction to turning f into a homotopy equivalence is the signature obstruction when m = dim X is divisible by 4. However, the signature of X is zero since k ≥ 1, and Hirzebruch’s signature theorem implies that the signature of N is also zero since its normal bundle is trivial (condition (1)).

Since H^{2}(N, Q) = Q there is a nonzero integral class represented by a map N → BS^{1}, classifying a bundle S^{1} → M → N. The Serre spectral sequence for this fibration immediately shows that requirement (3) is satisfied. To see (2), that M is parallelizable, we proceed as in [BKNP02], using the criterion of [Dup70, Sut76]. If the dimension of M is even, the parallelizability of a stably parallelizable manifold is determined by the vanishing of the Euler characteristic of M which is clearly 0 in this case. If the dimension is 1, 3, or 7, M is automatically parallelizable; for all other odd dimensions, M is parallelizable if and only if its Kervaire semicharacteristic κ(X) vanishes. It is defined as

κ(X) = \frac{1}{2} (∑_{i≥0} dim H_{i}(X; F_{2})) (mod 2).

In this case dim N = m = 2l is even, and an easy rational computation shows that the Euler characteristic of N is 0. Thus the total dimension of H_{∗}(N; F_{2}) is even. We may assume the Serre spectral sequence of S^{1} → M → N with Z/2-coefficients collapses, since we otherwise could have composed N → BS^{1} with the degree two map BS^{1} → BS^{1} without affecting the rational types. It thus follows from the spectral sequence that the total dimension of H_{∗}(M; F_{2}) is divisible by four, and hence the Kervaire semi-characteristic is zero.

To prove (4), let {e_{i} | 0 ≤ i ≤ k} denote a basis of the indecomposables of H^{∗}(M; Z) modulo torsion and consider the lifting problem

\xymatrix{ & ∏_{i=0}^{k} S^{2n_{i}+1} \ar[d] \\ M \ar[r]_{∏ e_{i}} \ar@{.>}[ur] & ∏_{i=0}^{k} K(Z, 2n_{i} + 1)}

where the vertical map is the Hurewicz map, whose fiber will be denoted by F. The obstructions to this lifting problem lie in H^{i+1}(M; π_{i}(F)). The homotopy groups π_{i}(F) are all finite since the dimensions 2n_{i} + 1 are odd. If N_{0} = l.c.m.{#π_{i}(F) | 1 ≤ i ≤ dim M} then a lift exists after inverting N_{0}. Being parallelizable, M is in particular orientable and hence dim M = n_{0}n_{1} ⋯ n_{k}; thus N_{0} only depends on the integers n_{i}. The lift is an HF_{p}-isomorphism for p > N_{0}, and thus induces an equivalence L_{p}M → ∏ L_{p}S^{2n_{i}+1}. □

Proposition 3.4. Let P = P_{1} ∪P_{2} be a partition of a set of primes, and let G be a simply connected P -local space such that G_{(P_{1})} is a P_{1}-local finite loop space of level l and rank bigger than one, and G_{(P_{2})} is the P_{2}-localization of the total space of a bundle S^{1} → M → N satisfying the properties of Prop. 3.3. Assume that either 2 ∉ P_{1} or (⋆)

L_{2}G/Z(L_{2}G) is not equivalent to L_{2} SO(3)^{l} × L_{2} SO(5)^{ϵ} for l ≥ 0 and ϵ = 0, 1.

Then G admits a rationally splitting double 1-torus of level l.

Proof. The argument is a variation, and simplification, of the proof of [BKNP02, Prop. 5.3, Prop. 4.3, and Prop. 4.1].

Note that by [ABGP03, Thm. 2.1] and Prop. 3.3(4), there exists an N ≫ 0 such that L_{p}G is homotopy equivalent to a product of p−complete odd-dimensional spheres for all p ≥ N, p ∈ P. Thus, by property (4) of Prop. 3.3, L_{p}G is homotopy equivalent to L_{p}M for all but finitely many p ∈ P_{1}. Hence we may assume without loss of generality that P_{1} is finite.

Let r = rk(L_{(P_{1})}G) and r′′ = max{rk_{p} Z(L_{p}G) | p ∈ P_{1}} be the maximal p-rank of the centers of the p-compact groups L_{p}G. It was proved in [BKNP02, Lemma 5.2] that r′′ < r unless G has type 3^{k}. Let T′′ = (S^{1})^{r′′} and choose a torus

T′′ → L_{p}T′′ ⟶^{i′′_{p}} L_{p}G

containing the center of L_{p}G for all p ∈ P_{1} [DW95]. Extend this to a maximal torus

T = T′ × T′′ → L_{p}T′ × L_{p}T′′ ⟶^{i_{p}=i′_{p}·i′′_{p}} L_{p}G.

Let (t_{1}, …, t_{r′}) be coordinates on T′ and (t_{r′+1}, …, t_{r}) on T′′.

For p ∈ P_{1}, the map (L_{p}G)/(L_{p}T) → L_{p}BT induced by the subgroup inclusion i_{p} induces an isomorphism

H^{2}(L_{p}BT; Q_{p}) ⟶^{∼} H^{2}((L_{p}G)/(L_{p}T); Q_{p}).

Let q_{j,p} denote the image of L_{p}Bt_{j}.

Choose maps g, h, defining the rational homotopy equivalence

g × h: G ⟶^{∼_{Q}} L_{Q}M → L_{Q}S^{2l+1} × L_{Q}Y,

where Y is a product of odd-dimensional spheres such that L_{Q}N ≃ L_{Q}CP^{l} × L_{Q}Y. Since [X, L_{Q}S^{2l+1}] ≅ H^{2l+1}(X), g also denotes a class outside the image of H^{2l+1}(N; Q) → H^{2l+1}(M; Q) ≅ H^{2l+1}(G; Q). Under the map

H^{2l+1}(G; Q) ⟶^{∼} H^{2l+2}(BL_{Q}G; Q) → H^{2l+2}(BL_{Q}L_{p}G; Q_{p})

⟶^{i_{p}^{∗}} H^{2l+2}(L_{p}BT; Q_{p}) ≅ (Q_{p}[t_{1}, …, t_{r}])_{l+1} ,

g maps to nonzero homogeneous polynomials f_{p} of degree l + 1 for all p ∈ P_{1}.

For every α = (α_{1}, …, α_{r}) ∈ Z^{r}, define S^{1} ≅ S_{α} → T by t → α_{1}t_{1} + ⋯ + α_{r}t_{r}, where t is a coordinate on S^{1}. Then S_{α} is a subgroup of T if and only if the set α is coprime; moreover, i_{p}(L_{p}S_{α}) ⊆ L_{p}G intersects the center of L_{p}G trivially if p ∤ α_{j} for some 1 ≤ j ≤ r′. For p ∈ P_{1}, denote by q_{p} ∈ H^{2}((L_{p}G)/(L_{p}S_{α}); Q_{p}) the image of the coordinate t ∈ H^{2}(BS_{α}; Q) under the map classifying the fibration

L_{p}S_{α} → L_{p}G → (L_{p}G)/(L_{p}S_{α}).

We thus have a commutative diagram

[figure: H^{∗}(L_{p}BT; Q_{p}); α; H^{∗}((L_{p}G)/(L_{p}T); Q_{p}); t_{i}; q_{i,p}; α_{i}t; α_{i}q_{p}; H^{∗}(L_{p}BS_{α}; Q_{p}); H^{∗}((L_{p}G)/(L_{p}S_{α}); Q_{p}).]

In H^{2l+2}(−; Q_{p}), we also have

[figure: f_{p}; 0; f_{p}(α_{1}t, …, α_{r}t); 0; f_{p}(α_{1}, …, α_{r})t^{l+1}; f_{p}(α_{1}, …, α_{r})q_{p}^{l+1}.]

In order for S_{α} to induce a splitting 1-torus on L_{p}G for a given p ∈ P_{1}, we need α to satisfy the following two conditions:

(1) {α_{1}, …, α_{r′}} are coprime; and

(2) f_{p}(α_{1}, …, α_{r}) ≠ 0.

We can always arrange this by choosing α_{1} = 1; since f_{p} is homogeneous of degree l + 1, f′_{p} = f_{p}(1, α_{2}, …, α_{r}) is a nonzero polynomial for all p ∈ P_{1}. Since P_{1} is finite, α_{2}, …, α_{r} can be chosen in such a way that (1, α_{2}, …, α_{r}) is not a zero of f_{p} for any p ∈ P_{1}.

Now condition (1) ensures that S_{α} meets the center of L_{p}G trivially. Condition (2) implies that q_{p}^{l+1} = 0 and thus

H^{∗}((L_{p}G)/(L_{p}S_{α}); Q_{p}) ≅ Q_{p}[q_{p}]/(q_{p}^{l+1}) ⊗ H^{∗}(L_{p}G; Q_{p})/(g)

which implies that L_{p}S_{α} → L_{p}G → (L_{p}G)/(L_{p}S_{α}) has a rational splitting of level l + 1.

Let

S ≅ S_{α} → L_{P} G = ∏_{p∈P} L_{p}G

be the map thus constructed for p ∈ P_{1}, and the fiber inclusion S^{1} → M followed by p-completion for p ∈ P_{2}. Denote by L_{p}G → L_{p}G/L_{p}S the quotient of the subgroup inclusion for p ∈ P_{1} and the p-completion of the fibration M → N for p ∈ P_{2}. Thus for every p ∈ P, L_{p}S → L_{p}G → L_{p}G/L_{p}S is a rationally splitting 1-torus. Thus,

\xymatrix{L_{p}G \ar[r] \ar[d] & L_{Q}L_{p}G \ar[d] \ar[r]^{∼}_{L_{p}(g×h)} & L_{Q}L_{p}S^{2l+1} × L_{Q}L_{p}Y \ar[d] \\ L_{p}G/L_{p}S \ar[r] & L_{Q}(L_{p}G/L_{p}S) \ar[r]^{∼}_{_{\overline{h}_{p}}} & L_{Q}L_{p}CP^{l} × L_{Q}L_{p}Y.}

The map L_{p}(g×h) induces a map L_{Q}L_{P} G ⟶^{∼} L_{Q}L_{P} S^{2l+1} ×L_{Q}L_{P} Y since it comes from a map G → L_{Q}S^{2l+1}×L_{Q}. Likewise, for p ∈ P_{2}, \overline{h}_{p} is the image of a rationally defined map. This together with the fact that P_{1} is finite implies that the diagram above can be lifted to a diagram

\xymatrix{L_{P} G \ar[r] \ar[d] & L_{Q}L_{P} G \ar[d] \ar[r]^{∼} & L_{Q}L_{p}S^{2l+1} × L_{Q}L_{P} Y \ar[d] \\ L_{P} G/L_{P} S \ar[r] & L_{Q}(L_{P} G/L_{P} S) \ar[r]^{∼} & L_{Q}L_{P} CP^{l} × L_{Q}L_{P} Y.}

We now address the problem of lifting this to G.

Consider the diagram

[figure: G; L_{Q}G; ∼; L_{Q}S^{2l+1} × L_{Q}Y; Z_{0}; ∼; L_{Q}CP^{l} × L_{Q}Y; L_{P} G; L_{Q}L_{P} G; ∼; L_{Q}L_{P} S^{2l+1} × L_{Q}L_{P} Y; L_{P} G/L_{P} S; L_{Q}(L_{P} G/L_{P} S); ∼; L_{Q}L_{P} CP^{l} × L_{Q}L_{P} Y]

where the bottom two rows are the diagram constructed before, and Z_{0} is defined as the pullback.Hence we can also fill in a map L_{Q}G → Z_{0}. Let S^{1} → G → Z be the pullback of this diagram. This constitutes a rationally splitting 1-torus of level l. The only axiom that is not quite immediate is the Z_{(P)}-finiteness of Z, but it follows in the same way as in the proof of Prop. 2.1(3)⇒(4).

Now if 2 ∈ P it was shown in [BKNP02, Lemma 5.2] that under the hypothesis (⋆), this 1-torus can be extended to a double 1-torus, and the proof is finished. □

Corollary 3.5. Let P be any set of primes, and let G be an n-dimensional P - local loop space of level l and rank bigger than one satisfying (⋆). Then there exists a finite n-dimensional CW-complex X whose P -localization (resp. P -completion) is homotopy equivalent to G and such that X admits a rationally splitting double 1-torus of level l.

Proof. Let X be constructed by mixing G with the manifold constructed in Proposition 3.3 at the complementary set of primes. By Prop. 3.4, the universal cover X̃ has a rationally splitting double 1-torus of level l, and since it does not meet the center of X at any prime p ∈ P, we obtain a splitting double 1-torus on X. The stable reducibility condition is satisfied since that is a p-local condition and it is satisfied at every prime. □

Proof of Thm. 1.1. Let X be as in Theorem 1.1, and let Z be constructed by mixing as in the corollary above. If the rank of X is 0 or 1, or 2 ∈ P and L_{2}X is of the form (⋆), then the arguments from [BKNP02] and [Ped78] yield the result. Otherwise, by the above corollary Z admits a rationally splitting double 1-torus. By [BKNP02, Prop. 3.3], Z is homotopy equivalent to a compact, parallelizable, smooth manifold. □

References

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Sonderforschungsbereich 478 Geometrische Strukturen in der Mathematik, Westfälische Wilhelms-Universität Münster, Hittorfstr. 27, D-48149 Münster, Germany

   E-mail address: tbauer@math.uni-muenster.de

Department of Mathematical Sciences, SUNY at Binghamton, Binghamton, NY, 13902-6000, USA

   E-mail address: erik@math.binghamton.edu