SMOOTHING H-SPACES II

ERIK KJÆR PEDERSEN

0. Introduction

In [8] we showed that certain H-spaces obtained by homotopy mixing are homotopy equivalent to smooth, parallellizable manifolds. Unfortunately (as was added in proof) we needed the restriction on the fundamental group π, that D(Zπ) = 0. It is the purpose of this sequel to [8] to remove this restriction, and also to generalize the main theorem considerably. I want to thank I. Hambleton for pointing out the error in [8].

1. Notation. Statement of results

Throughout the paper space will mean topological space of the homotopy type of a connected CW complex. For a set of primes l and a nilpotent space X, X_{l} denotes the localization at l in the sense of [5]. A space X is called quasifinite if H_{∗}(X) = ⊕H_{i}(X; Z) is a finitely generated abelian group. If H_{∗}(X) is a Z_{l}-module X is called l-locally quasifinite if H_{∗}(X) is finitely generated as a Z_{l}-module.

To state our main theorem we need a couple of definitions. Let S^{i} denote the i-sphere.

1.1. Definition. A nilpotent space X admits a special 1-torus if, up to homotopy, there is a diagram of orientable fibrations

\xymatrix{S^{1} \ar[d] \ar[r] & S^{3} \ar[d] \ar[r] & S^{2} \ar[d] \\ S^{1} \ar[r] \ar[d] & X \ar[r] \ar[d] & B \ar[d] \\ ∗ \ar[r] & A \ar[r] & A}

such that

(a) A is quasifinite, B is stably reducible.

(b) Localized at 0 the diagram is homotopy equivalent to

\xymatrix{S_{0}^{1} \ar[d] \ar[r] & S_{0}^{3} \ar[d] \ar[r] & S_{0}^{2} \ar[d] \\ S_{0}^{1} \ar[r] \ar[d] & A_{0} × S_{0}^{3} \ar[r] \ar[d]^{pr_{1}} & A_{0} × S_{0}^{2} \ar[d]^{pr_{1}} \\ ∗ \ar[r] & A_{0} \ar[r] & A_{0}}

We remark that since X is nilpotent and the fibrations are orientable A and B are nilpotent so localization makes sense.

1.2. Example. Given a bundle S^{3} → X → A with A quasifinite stably reducible and the bundle stably trivial then X admits a special 1-torus by dividing out the subgroup S^{1} ⊂ S^{3} (see [8, lemma 3.6] ). All compact Lie groups other than SO(3)^{k} × T^{l} have subgroups isomorphic to S^{3} as is seen by classification and all these Lie groups admit special 1-tori. This will be further discussed in section 5.

We need a p-local version of definition 1.1. Let X be a nilpotent space, p a prime.

1.3. Definition. X admits a p-local special 1-torus if, up to homotopy, there is a diagram of orientable fibrations

\xymatrix{S_{p}^{1} \ar[d] \ar[r] & S_{p}^{3} \ar[d] \ar[r] & S_{p}^{2} \ar[d] \\ S_{p}^{1} \ar[r] \ar[d] & X \ar[r] \ar[d] & B \ar[d] \\ ∗ \ar[r] & A \ar[r] & A}

such that

(a) A is p-locally quasifinite and B is p-locally stably reducible, i. e. there are integers n and i and a map S_{p}^{n+1} → Σ^{i}B inducing isomorphism in homology in dimensions ≥ n + i.

(b) as in definition 1.1

It is clear that if X admits a special 1-torus then X_{p} admits a p-local special 1-torus. We prove the following

1.4. Theorem. Let X be a quasifinite H-space. Assume for every prime p that X_{p} is homotopy equivalent to a product C(p) × D(p) and C(p) admits a p-local special 1-torus. Then X is homotopy equivalent to a smooth, stably parallellizable manifold.

1.5. Remark. If H_{3}(X) ⊃ Z the condition of the theorem is trivially satisfied for all but finitely many primes. This is because for all but finitely many primes X_{p} is homotopy equivalent to a product of localized spheres which must include the 3-sphere. We also note that in view of example 1.2 (see section 5) this theorem is stronger than theorem 1.1 of [8].

2. Surgery

We use the special 1-tori for the following

2.1. Proposition. Let X be a quasifinite, nilpotent Poincaré complex admitting a special 1-torus. Then X is homotopy equivalent to a stably parallellizable smooth manifold.

Remark. This result only needs condition (a) of definition 1.1 Condition (b) is needed to ensure that the property of having a special 1-torus is a generic property.

Proof of proposition 2.1. In the diagram of orientable fibrations

\xymatrix{S^{1} \ar[d] \ar[r] & S^{3} \ar[d] \ar[r] & S^{2} \ar[d] \\ S^{1} \ar[r] \ar[d] & X \ar[r] \ar[d] & B \ar[d] \\ ∗ \ar[r] & A \ar[r] & A}

A is nilpotent and quasifinite, hence by [7] A is finitely dominated. It follows that X and B are finitely dominated [6]. Also A and B are Poincaré Duality spaces since X is [3]. It follows from [10] that X has 0 finiteness obstruction. Considering (B, X) a Poincaré Duality pair, we may use the stable reduction of B and a standard transversality procedure to produce a surgery problem

(M, ∂M) ⟶^{ϕ} (B, X) ϕ̂: ν_{M} → ε

where ε is the trivial bundle. Let σ(B) be the finiteness obstruction of B. Consider the exact sequence

… → H^{n+1}(Z_{2}; K̃_{0}(Zπ)) → L_{n}^{h}(π) ⟶^{δ} L_{n}^{p} →

where π = π_{1}(B) = π_{1}(X). The class of σ(B), {σ(B)}, is an element of H^{n+1}(Z_{2}, K̃_{0}(Zπ)). It follows from [9], that the surgery obstruction of ∂M → X is δ{σ(B)}. However, since A is a P. D, space of dimension n − 3 we have σ(A) = (−1)^{n−3}σ(A)^{∗} and by [10], σ(B) = 2σ(A) = σ(A)+(−1)^{n+1}σ(A)^{∗} and hence {σ(B)} = 0 in H^{n+1}(Z_{2}; K̃_{0}(Zπ)) and we are done.

3. Reducibility of H-spaces.

Browder and Spanier have shown that a finite H-space is stably reducible [2]. This is one of the steps in the attempt to prove X is a manifold, since it implies that the Spivak normal fibre space is trivial. We need to generalize the results of Browder and Spanier to a p-local situation. This is mostly straightforward. We shall nevertheless indicate the line of argument in this section. The aim of this section is to prove:

3.1. Theorem. Let D be a p-locally quasi finite H-space. Then D is p-locally stably reducible.

We need a p-local edition of S-duality.

3.2. Proposition. Let X be a simply connected p-locally quasifinite space. Then X admits a p-local CW-structure, i. e. X is homotopy equivalent to a space Y with a filtration ∗ = K_{0} ⊂ K_{1} ⊂ … ⊂ K_{n} = Y such that K_{i} is the mapping cone of some map f_{i} : S_{p}^{n(i)} → K_{i−1}, n(i) a nondecreasing function of i.

Proof. By the Hurewicz theorem we may find a finite wedge of local spheres and a map f ⋁ S_{p}^{k} → X such that H_{∗}(f) is onto in dimensions ≤ k, k ≤ 2. Using the relative Hurewicz theorem we inductively attach local cells to make H_{∗}(f) an isomorphism in higher dimensions. Since X is p-locally quasifinite and finitely generated Z_{p}-module have free resolutions of length one, we eventually obtain a homotopy equivalence.

Let X and Y be p-locally quasifinite spaces. A p-local S-duality map is a map X ∧Y → S_{p}^{n} so that slant product

f^{∗}(i)/− : H̃_{∗}(X) → H̃^{n−∗}(Y)

is an isomorphism. Here i is the generator of H^{n}(S_{p}^{n}).

Given a p-locally finite space X we note that the suspension ΣX is a simply connected p-locally quasifinite space and thus admits a p-local CW structure by Proposition 3.2. We may now go through exercises F1-7 page 463 in Spanier [11] to prove existence and stable uniqueness of a p-local S-dual with the usual functorial properties. We need the concept to complete the

Proof of Theorem 3.1. H^{∗}(D; Q) and H^{∗}(D; Z/pZ) are Hopf algebras and we may argue as in the finite case [1] that D satisfies Poincaré Duality with Z_{p} coefficients. We now only need to produce a map D → S_{p}^{n} inducing isomorphisms in dimensions ≥ n. Then we may use Hopf algebra arguments (as in the finite case [2]) to prove that the composite D^{+} ∧ D^{+} = (D × D)^{+} → D → S_{p}^{n} is a p-local S-duality map, so D^{+} is selfdual and the dual of D^{+} → S_{p}^{n} will be a stable reduction.

Localized at 0 D is a product of odd dimensional spheres so if we let l be the set of primes different from p and form the homotopy pullback

\xymatrix{Y \ar[d] \ar[r] & ∏ S_{l}^{2n_{i}+1} \ar[d] \\ D \ar[r] & ∏ S_{0}^{2n_{i}+1}}

then Y is quasifinite, nilpotent and satisfies Poincaré Duality at all primes hence [5] and [7] is a finitely dominated Poincaré Duality space. By Wall [12] Y has the homotopy type of K ∪ e^{n} where K is n − 1-dimensional and we may thus produce a map Y → K ∪ e^{n} → S^{n} by collapsing K to a point. Localizing at p we obtain D ≅ K_{p} → S_{p}^{n} with the required property and we are done

4. Proof of main Theorem.

The proof will consist of two lemmas.

4.1. Lemma. If X is a quasifinite H-space and X_{p} = C(p) × D(p) where C(p) admits a p-local special 1-torus, then X_{p} does.

Proof. Crossing the special 1-torus diagram

\xymatrix{C(p) \ar[r] \ar[d] & B \ar[d] \\ A \ar[r] & A}

with D(p) reduces the lemma to showing B × D(p) is p-locally stably reducible. Now D(p) is a retract of a p-locally quasifinite H-space and is thus itself a p-locally quasifinite H-space and thus p-locally stably reducible by theorem 3.1

4.2. Lemma. If X is a quasifinite H-space such that each X_{p} admits a p-local special 1-torus, then X admits a special 1-torus.

Proof. At all but finitely many primes X is a product of spheres, so we may consider X a homotopy pullback

\xymatrix{ &  & X \ar[dll] \ar[dl] \ar[dr] \ar[drr] \\ X_{l} \ar[drr] & X_{p_{1}} \ar[dr] &  & _{· · · · · ·} \ar[dl] & X_{p_{k}} \ar[dll] \\  &  & X_{0}}

where X_{l} is a product of odd dimensional spheres and X_{p_{i}} also admit special 1-tori. Mixing the special 1-tori in the obvious way we obtain \overline{X} which admits a special 1-torus and such that \overline{X}_{l} ≅ X_{l} and \overline{X}_{p_{i}} ≅ X_{p_{i}}; in other words \overline{X} is in the genus of X. We now argue as in [8, proposition 3.2] to show that admitting a special 1-torus is a generic property for an H-space. The key step is the result of Zabrodsky that one obtains the whole genus of an H-space by mixings defined by diagonal matrices and the observation in [8] that one of these diagonal entries may be assumed to be 1.

5. Examples.

In this section we show that compact Lie groups other than SO(3)^{k} × T^{l} do admit special 1-tori. This implies that our theorem 1.4 is indeed stronger than theorem 1.1 of [8].

5.1. Proposition. Let G be a compact connected Lie group which is not isomorphic to SO(3)^{k} × T^{l}. Then G has a subgroup isomorphic to S^{3}.

Proof. We use classification of compact Lie groups. Any compact connected Lie group is a quotient of H × T^{l} by a discrete central subgroup A. Here H is a simply connected compact Lie group. Furthermore H is a product of groups in a list, see [4, p. 346]. If we can find an S^{3} subgroup of H that intersects A trivially we are done. There are two cases. First assume H = (S^{3})^{k}. Then A can not contain the center of (S^{3})^{k} since if it does G will be isomorphic to SO(3)^{k} × T^{l}. This being the case it is easy to find a subgroup isomorphic to S^{3} not intersecting A. If H is not a product of S^{3}’s it has a simple factor different from S^{3} and we will be done if we can find a subgroup isomorphic to S^{3} in this factor, intersecting the center trivially. We do this by checking the list. We have S^{3} = SU(2) ⊂ SU(n) (n ≤ 3) intersecting the center trivially since the central element of SU(n) are the diagonal matrices with the same n’th root of unity as entry. Similarly S^{3} = SP (1) ⊂ SP (n) (n ≥ 2) and S^{3} = SU(2) ⊂ SO(4) ⊂ SO(n), (n ≥ 5) do not contain −I which is the only central element ≠ I. Furthermore E_{8} ⊃ E_{7} ⊃ E_{6} ⊃ SU(6) ⊃ S^{3} and since the center of E_{6} has order 3, S^{3} must intersect it trivially and E_{6} must intersect the center of E_{7} (cyclic of order 2) trivially. Finally F_{4} ⊃ SP (3) and G_{2} ⊃ SU(2) and these groups have trivial center. We are done.

5.2. Remark. It would be nice to have a conceptual proof of Proposition 5.1. Working in the Lie algebra it is not hard to find a subgroup isomorphic to SO(3) or S^{3} but it is crucial for us to be in the latter case.

5.3. Proposition. Let G be a compact Lie group with S^{3} as a subgroup G ⊃ S^{3} Then

\xymatrix{S^{1} \ar[d] \ar[r] & S^{3} \ar[r] \ar[d] & S^{2} \ar[d] \\ S^{1} \ar[r] \ar[d] & G \ar[r] \ar[d] & G/S^{1} \ar[d] \\ ∗ \ar[r] & G/S^{3} \ar[r] & G/S^{3}}

is a special 1-torus in G.

Proof. Lemma 3.4 of [8] shows that G/S^{1} is stably parallellizable. It follows from [8, lemma 3.3] that H^{3}(G; Q) → H^{3}(S^{3}; Q) is onto. Let G_{0} → K(Q, 3) = S_{0}^{3} represent an element in H^{3}(G; Q) hitting the generator of H^{3}(S^{3}; Q) then one sees by a spectral sequence argument that G_{0} → (G/S^{3})_{0} × S_{0}^{3} is a homology equivalence hence a homotopy equivalence and we are done.

Final Remarks. In case D(Zπ) = 0 we could replace the concept special 1-torus by the concept 1-torus (see [8]). Since admitting a 1-torus is a weaker condition than admitting a special 1-torus, it is not entirely a loss to have both concepts.

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Matematisk Institut, Odense Universitet, Danmark