Invent. math. 131, 321-344 (1998)  Inventiones

Mathematicae

Compact 3-Sasakian 7-Manifolds with Arbitrary Second Betti Number

Charles P. Boyer Krzysztof Galicki Benjamin M. Mann Elmer G. Rees

Introduction

Amongst all Riemannian geometries the class of Einstein metrics stands out as perhaps the most natural and interesting [Bes]. Even so there are still many open questions about the relation between the topology of a compact manifold and the possible existence of Einstein metrics. In dimensions bigger than four almost nothing seems to be known in general. Yet, Einstein metrics on compact manifolds are relatively rare and they usually appear as part of additional geometric structure which makes their study tractable. In recent years the first three authors [BGM1,BGM2] have studied a class of Riemannian manifolds known as 3-Sasakian manifolds which have proven to be a remarkable source of compact Einstein manifolds of positive scalar curvature. In view of this work several seemingly unrelated questions regarding the possible breakdown of finiteness and Betti number bounds come to mind:

Question 1: Do there exist 3-Sasakian manifolds with second Betti number greater than one?

Question 2: Are there singular Fano varieties with arbitrarily large Picard number?

Question 3: Are there compact Einstein manifolds of positive scalar curvature with arbitrarily large total Betti number?

Question 4: Are there compact Einstein manifolds of positive scalar curvature which do not admit metrics of nonnegative sectional curvature?

These four questions are related and it is the purpose of this paper to give an affirmative answer to all of these questions by explicit construction. The first question is of interest in light of the recent bound of LeBrun [Le,LeSa] that for quaternionic Kähler manifolds of positive scalar curvature we must have b_{2} ≤ 1. It then follows [GS] that for regular 3-Sasakian manifolds b_{2} ≤ 1 also. The methods used to prove this bound relate to question 2 through twistor geometry. Indeed the bound follows from a result of Wiśniewski [Wi] about Fano manifolds of large index. Question 2 is of interest in its own right owing to the result of Mori and Mukai [MM] which says that for smooth Fano 3-folds b_{2} ≤ 10. That an affirmative answer to question 3 implies an affirmative answer to question 4 is a well known result of Gromov [Gro] that says that if a compact manifold admits a Riemannian metric of nonnegative sectional curvature then the total Betti number is bounded by a constant depending only on dimension. In fact, more is true when there is a bound on the diameter [Gro]. As the total Betti number grows some sectional curvatures must become more and more negative. The main result of our paper is:

During the preparation of this work the first three authors were supported by an NSF grant.

Theorem A: Let k be a positive integer, and let (a, b) ∈ (ℤ^{∗})^{k} ⊕(ℤ^{∗})^{k} whose components (a^{i}, b^{i}) are pairs of relatively prime integers for i = 1, ⋯, k that satisfy the condition that if for some pair i, j a^{i} = ±a^{j} or b^{i} = ±b^{j} then we must have b^{i} ≠ ±b^{j} or a^{i} ≠ ±a^{j}, respectively. Then there exist 3-Sasakian manifolds S(a, b) of dimension 7 depending on (a, b) such that b_{2}(S(a, b)) = k. In particular, there exist simply connected Einstein 7-manifolds of positive scalar curvature with arbitrary second Betti number.

Actually we construct a family of 3-Sasakian orbifolds depending on \frac{k(k+5)}{2} integers, but there appears to be no general algorithm for determining when these orbifolds are smooth manifolds. It should also be emphasized that the existence of the manifolds S(a, b) is determined constructively, and that they are generalizations of the 3-Sasakian manifolds described in example 4.6 of [BGM1].

There are several important corollaries of Theorem A. The first follows immediately from Theorem A and Gromov’s Theorem [Gro].

Corollary B: There are infinitely many compact simply-connected Einstein 7-manifolds of positive scalar curvature, namely the S(a, b) of Theorem A, that do not admit metrics of nonnegative sectional curvature. Furthermore, for any negative real number κ there are infinitely many 3-Sasakian manifolds S(a, b) which do not admit metrics whose sectional curvatures are all greater than or equal to κ.

Actually question 4 with the condition “Einstein manifold of positive scalar curvature” replaced by “nonnegative Ricci curvature” was problem 5 of Yau’s famous problem section of the 1979-80 Princeton Seminar [Y]. This question was answered affirmatively in 1989 by Sha and Yang [SY], but to the best of the authors’ knowledge our construction gives the first examples for Einstein manifolds of positive scalar curvature.

Another result relating to the breakdown of finiteness comes from a theorem of Anderson [An] which implies there is finite number of diffeomorphism types of Einstein manifolds of positive scalar curvature with a lower bound on the injectivity radius. Thus using our examples in [BGM2,BGM3] or those of Theorem A give:

Corollary C: There are infinitely many 3-Sasakian 7-manifolds with arbitrarily small injectivity radius.

It is interesting to compare the examples of Theorem A with our previous examples [BGM2] where in particular we constructed infinitely many 3-Sasakian manifolds that are homotopy inequivalent and which admit metrics of positive sectional curvature, but which by Anderson’s theorem have arbitrarily small injectivity radii. On the other hand by Gromov’s theorem, Anderson’s theorem, and Theorem A, there are infinitely many 3-Sasakian manifolds with arbitrarily small injectivity radii which cannot admit any metric of nonnegative sectional curvature.

Our next corollary is a partial classification result. It follows immediately from Theorem A and results of [GS].

Corollary D: In dimension seven there exist 3-Sasakian manifolds with every allowable rational homology type.

It is known [BGM1,BGM2] that every 3-Sasakian manifold has two distinct homothety classes of Einstein metrics only one of which is 3-Sasakian. Furthermore, in dimension 7 both of these metrics have weak G_{2} holonomy [GS,FKMS]. Thus, we have

Corollary E: There exist 7-manifolds with arbitrary second Betti number having metrics of weak G_{2} holonomy.

In [BG] it was shown that the twistor space of any 3-Sasakian manifold has the structure of a ℚ-factorial Fano variety, and this provides the link with question 2. Thus, results of [BG] and Theorem A give:

Corollary F: There exist ℚ-factorial contact Fano 3-folds X with b_{2}(X) = l for any positive integer l.

There is a well known relationship [BGM1,BG] between 3-Sasakian geometry on the one hand and both quaternionic Kähler geometry of positive scalar curvature and Fano contact geometry on the other (Here l = k+1 for the l in Corollary F and k in Theorem A). But in general this relationship involves Riemannian metrics with orbifold singularities for both the quaternionic Kähler and Fano geometries. It is the existence of these singularities that cause the violation of finiteness. This is most easily seen in the Fano contact case where Wiśniewski’s theorem fails in the orbifold category since both the contact divisor and the anticanonical divisor are now ℚ-divisors, and the singularity index can be arbitrarily high.

It is interesting to view our results from the perspective of holonomy. In the case of an Einstein n-manifold M with positive scalar curvature and irreducible special holonomy (i.e., M is not locally symmetric and the restricted holonomy group is a proper subgroup of SO(n) given by the Berger’s classification theorem [Bes]), there are only two possibilities; M is Kähler-Einstein of positive scalar curvature, or M is quaternionic Kähler of positive scalar curvature. In both cases there is a finiteness theorem. The first is the boundedness of deformation types of Fano manifolds by Kollár, Miyaoka, and Mori [KMM], and the second is LeBrun’s Finiteness theorem [Le] mentioned above. This contrasts with 3-Sasakian manifolds whose holonomy is always the full special orthogonal group SO(4n − 1) [GS].

Another known case where finiteness fails are the homogeneous manifolds of Aloff and Wallach [AlWa]. Moreover, they admit Einstein metrics of positive scalar curvature [Wa]. However, since they are homogeneous, b_{2} is bounded (in fact, b_{2} = 1 here). In our case we have:

Corollary G: If the second Betti number b_{2}(S(a, b)) = k > 3, the 3-Sasakian manifolds S(a, b) (of Theorem A) are not homotopy equivalent to any homogeneous space.

Another question related to those mentioned above is whether there exist compact quaternionic Kähler orbifolds O of positive scalar curvature with arbitrary second Betti number. In dimension four, these spaces are compact, self-dual, Einstein orbifolds. To each 3-Sasakian manifold S(Ω) there is a naturally associated quaternionic Kähler orbifold O(Ω) [BGM1, BGM2].

Theorem H: Let O(a, b) be the compact, self-dual, Einstein orbifold associated to the 7-dimensional 3-Sasakian manifold S(a, b) given in Theorem A. Then

b_{2}(O(a, b)) = b_{2}(S(a, b)) = k.

Hence, there are compact, self-dual, Einstein orbifolds of positive scalar curvature with arbitrary second Betti number.

We begin our constructions in section one by reviewing the 3-Sasakian reduction [BGM2] of the standard sphere S^{4n−1} by a k-torus action. Next, in section two, we determine necessary and sufficient conditions for the k-torus to act freely on the zero level set of the 3-Sasakian moment map. Moreover, we determine normal forms for these toral action.

Since the k-torus is a subtorus of the maximal torus in the group Sp(n) of 3-Sasakian isometries, the reduced 3-Sasakian manifold S we construct admits an (n − k)-torus of 3-Sasakian isometries. Combining these isometries with the Sp(1) group of 3-Sasakian isometries generated by the 3-Sasakian vector fields we obtain a larger group of 3-Sasakian isometries on our quotients. In particular, in the case when n = k + 2, the resulting 3-Sasakian 7-manifolds S each have a five dimensional isometry group. Taking the quotient of S by this isometry group gives rise to a 2-dimensional quotient space with a cell structure represented by a 2-disc whose boundary is a (k + 2)-gon. This in turn gives a natural stratification of S where the fibres are tori crossed with 3-dimensional lens spaces. This stratification is described in section three.

Then, in section four we use a Leray spectral sequence to determine the rational homology of S and thus prove Theorem A. Finally, Theorem H follows from a slightly modified version of the stratification argument used to prove Theorem A.

Acknowledgements. The first three authors would like to thank the ICMS in Edinburgh for support and hospitality during their visit there in June of 1996. The third named author would like to thank the University of Edinburgh for support and hospitality during his sabbatical stay there and for partial support through an EPSRC Visiting Fellowship. We all thank Dimitri Alekseevsky for bringing to our attention results on cohomogeneity two Riemannian manifolds and Claude LeBrun for several helpful comments. Finally, the second named author would like to thank Professor Hirzebruch and the MPI in Bonn. The work on this article started during his visit there in June and July of 1995.

§1. 3-Sasakian Reductions

In this section we review the 3-Sasakian reduction procedure [BGM2] which constructs new 3-Sasakian manifolds from a given one. The key to this construction is the quaternionic quotient described in [BGM1]. With each quaternionic Kähler space O, in a canonical way, one associates three bundles: the twistor bundle Z, the Swann bundle U , and the Konishi bundle S. The twistor bundle Z carries a complex contact structure and a Kähler-Einstein metric. The Swann bundle U is a hyperkähler manifold with a certain isometric SU(2) action. Finally, the Konishi bundle S is a 3-Sasakian manifold. (In case O is a compact orbifold all the bundles exist as V -bundles.) The quaternionic Kähler reduction of [GL] has a canonical lift to three other reductions: the twistor space or contact reduction of Hitchin [Hi2], the hyperkähler reduction of [HKLR] and the 3-Sasakian reduction given in [BGM2] and reviewed in this section. In contrast to the better known quaternionic Kähler and hyperkähler quotients, it should be mentioned that this last reduction is effective in producing new compact and smooth manifolds. There is not a single known example of a hyperkähler reduction which leads to a compact manifold and in the quaternionic Kähler case such examples are extremely rare.

To begin let (S, g_{S}, ξ^{a}) be a 3-Sasakian manifold and let I_{0}(S, g_{S}) denote the connected component of the group of 3-Sasakian isometries. By the embedding theorem [BGM2], M = S × ℝ^{+} is a hyperkähler manifold with respect to the cone metric g_{M}. The group I_{0}(S, g_{S}) extends to a group I_{0}(M, g_{M}) ≅ I_{0}(S, g_{S}) of isometries on M by defining each element to act trivially on ℝ^{+}. Furthermore, it follows that these isometries I_{0}(M, g_{M}) preserve the hyperkähler structure on M. Recall from [HKLR] that any subgroup G ⊂ I_{0}(M, g_{M}) gives rise to a hyperkähler moment map μ : M⟶𝔤^{∗} ⊗ ℝ^{3}, where 𝔤 denotes the Lie algebra of G and 𝔤^{∗} is its dual. Thus, we can define a 3-Sasakian moment map

1.1  μ_{S} : S ⟶ 𝔤^{∗} ⊗ ℝ^{3}

by restriction μ_{S} = μ | S. We denote the components of μ_{S} with respect to the standard basis of ℝ^{3}, which we have identified with the purely imaginary quaternions, by μ_{S}^{a}. Recall that ordinarily moment maps determined by Abelian group actions (in particular, those associated to 1-parameter groups) are only specified up to an arbitrary constant. This is not the case for 3-Sasakian moment maps since we require that the group Sp(1) generated by the Sasakian vector fields ξ^{a} acts on the level sets of μ_{S}. In fact, there is a unique 3-Sasakian moment map μ_{S} such that the zero set μ_{S}^{−1}(0) is invariant under the group Sp(1) generated by the vector fields ξ^{a}, and this moment map is given by the simple expression

1.2  < μ_{S}^{a}, τ > = \frac{1}{2}η^{a}(X^{τ}),

where τ ∈ 𝔤 has associated vector field X^{τ} , and η^{a} denotes the 1-form dual to the vector field ξ^{a}. We now have:

Reduction Theorem 1.3 [BGM2]: Let (S, g_{S}, ξ^{a}) be a 3-Sasakian manifold with a compact Lie group G acting on S by 3-Sasakian isometries. Let μ_{S} be the corresponding 3-Sasakian moment map and assume both that 0 is a regular value of μ_{S} and that G acts freely on the submanifold μ_{S}^{−1}(0). Furthermore, let

ι : μ_{S}^{−1}(0) ⟶ S

and

π : μ_{S}^{−1}(0) ⟶ μ_{S}^{−1}(0)/G

denote the corresponding embedding and submersion. Then

(Š = μ_{S}^{−1}(0)/G, ǧ_{S}, ξ̌^{a})

is a smooth 3-Sasakian manifold of dimension 4(n−dim 𝔤)−1 with metric ǧ_{S} and Sasakian vector fields ξ̌^{a} determined uniquely by the two conditions

ι^{∗}g_{S} = π^{∗}ǧ_{S}

and

π_{∗}(ξ^{a} |μ_{S}^{−1}(0)) = ξ̌^{a}.

In the next section we apply this theorem with S = S^{4n−1} and G = T^{k} the k dimensional torus. We also need the following result concerning 3-Sasakian isometries.

Proposition 1.4 [BGM2]: Assume that the hypothesis of Theorem 1.3 holds. In addition assume that (S, g_{S}) is complete and hence compact. Let C(G) ⊂ I_{0}(S, g_{S}) denote the centralizer of G in I_{0}(S, g_{S}) and let C_{0}(G) denote the subgroup of C(G) given by the connected component of the identity. Then C_{0}(G) acts on the submanifold μ_{S}^{−1}(0) as isometries with respect to the restricted metric ι^{∗}g_{S} and the 3-Sasakian isometry group I_{0}(Š, ǧ_{S}) of the quotient (Š, ǧ_{S}) contains an isomorphic copy of C_{0}(G).

By abuse of notation we denote this isomorphic copy by C_{0}(G). In the case at hand G = T^{k}, C_{0}(T^{k}) = T^{n} the maximal torus of Sp(n). Notice that C_{0}(T^{k}) does not act effectively on the quotient. The normalizer of the maximal torus, namely, the Weyl group W(Sp(n)) will also play a role when discussing equivalence between quotients in the next section.

We end this section with an example. The diagram below describes all the homogeneous circle quotients [Bat] (see [BGM2] for details and further examples):

1.5 \begin{aligned}& & ℍ^{n} \ \{0\} & & & ⟶^{S^{1}} & & & N \\ & {ℝ^{+} \atop ↙} & & {ℂ^{∗} \atop ↘} & & & & {t(ℂ) \atop ↙} & & {t(ℝ) \atop ↘} \\ S^{4n−1} & & ↓ℍ^{∗} & & ℂℙ^{2n−1} & ⟶^{S^{1}} & Z & & ↓t(ℍ) & & S, \\ & ↘ & & ↙ & & & & ↘ & & ↙ \\ & & ℍℙ^{n−1} & & & ⟶^{S^{1}} & & & Gr_{2}(ℂ^{n}) \end{aligned}

where S = \frac{U(n)}{U(n−2)×U(1)} and Z is the complex flag manifold F_{n,2,1} = \frac{U(n)}{U(n−2)×U(1)×U(1)}.

In the next section we describe a quotient construction of a large family of inhomogeneous examples by considering k-dimensional torus actions on spheres. On the one hand these generalize the inhomogeneous examples S(p) given in [BGM2] since S(p) is obtained by considering the most general 1-torus action. On the other hand the general toral quotients are very different as they are never homogeneous, whereas S(p) is actually homogeneous for p = 1.

§2. Torus Actions on the 3-Sasakian Sphere

Let u = (u_{1}, ..., u_{n}) ∈ ℍ^{n}, be the quaternionic coordinates on the n-dimensional quaternionic vector space ℍ^{n}, equipped with the flat metric. Consider the unit sphere

S^{4n−1} = \{u ∈ ℍ^{n} | ∑_{α=1}^{n} \overline{u}_{α}u_{α} = 1\},

with its canonical metric g_{can} obtained from the flat metric by the inclusion S^{4n−1} ⊂ ℍ^{n}. This sphere has two natural 3-Sasakian structures determined by whether one treats ℍ^{n} as a right or left quaternionic vector space. We choose the left module structure on ℍ^{n} and this chooses the right 3-Sasakian vector fields ξ_{r}^{a}. The subgroup of the isometry group O(n) of the (S^{4n−1}, g_{can}) that normalizes this structure is Sp(n) · Sp(1) = (Sp(n) × Sp(1))/ℤ_{2}, where the Sp(1) is the group generated by the 3-Sasakian vector fields ξ_{r}^{a}. The group Sp(n) · Sp(1) acts on the sphere as:

Sp(n) × Sp(1) × S^{4n−1} → S^{4n−1}

2.1  ((𝔸, σ); u) ⟶ 𝔸uσ^{−1},

where 𝔸 ∈ Sp(n) is the quaternionic n × n matrix of the quaternionic representation of Sp(n), and σ ∈ Sp(1) is a unit quaternion. As the diagonal ℤ_{2} acts trivially we see that Sp(n) · Sp(1) acts on the sphere by isometries. The group Sp(n) is the subgroup of the full isometry group which commutes with the 3-Sasakian Sp(1) action, i.e., we have I_{0}(S^{4n−1}) = Sp(n). In this paper we shall consider the maximal torus T^{n} ⊂ Sp(n) and its subgroups acting on S^{4n−1}.

Every quaternionic representation of a k-torus T^{k} on ℍ^{n} can be described by a diagonal matrix of the form

2.2  ( \begin{matrix}∏_{i=1}^{k} τ_{i}^{a_{1}^{i}} & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & ∏_{i=1}^{k} τ_{i}^{a_{n}^{i}} \end{matrix} ) ,

where (τ_{1}, .., τ_{k}) ∈ S^{1} × ⋯ × S^{1} = T^{k} are the complex coordinates on T^{k}, and a_{j}^{i} ∈ ℤ. In turn this representation defines a k × n integral weight matrix

2.3  Ω = ( \begin{matrix}a_{1}^{1} & a_{2}^{1} & … & a_{k}^{1} & … & a_{n}^{1} \\ a_{1}^{2} & a_{2}^{2} & … & a_{k}^{2} & … & a_{n}^{2} \\ ⋮ & ⋮ & ... & ⋮ & … & ⋮ \\ a_{1}^{k} & a_{2}^{k} & … & a_{k}^{k} & … & a_{n}^{k} \end{matrix} ) .

Let {e_{j}}_{j=1}^{k} denote the standard basis for 𝔱_{k}^{∗} ≃ ℝ^{k}. Then the 3-Sasakian moment map [BGM2] μ_{Ω} : S^{4n−1}⟶𝔱_{k}^{∗} ⊗ ℝ^{3} defined by the action with Ω given by 2.3 is μ_{Ω} = ∑_{j} μ_{Ω}^{j}e_{j} with

2.4  μ_{Ω}^{j}(u) = ∑_{α} ū_{α}ia_{α}^{j}u_{α}.

We choose the complex structure in ℍ^{n} determined by i and write the quaternionic coordinates in terms of complex coordinates u_{α} = z_{α} + w_{α}j. The moment map then takes the form

2.4′  μ_{Ω}^{j}(z, w) = i ∑_{α} a_{α}^{j}(|z_{α}|^{2} − |w_{α}|^{2}) + 2k ∑_{α} a_{α}^{j}w̄_{α}z_{α}

where k = ij. We are interested in the zero set of this moment map. Under the appropriate conditions, 0 is a regular value of μ_{Ω}, so that N(Ω) = μ_{Ω}^{−1}(0) is a smooth compact submanifold of S^{4n−1} of codimension 3k.

Assuming that N(Ω) is a smooth submanifold, we are interested in conditions that guarantee that the T^{k}-action is free on N(Ω). Actually this condition is a bit too strong, since the action may not be effective. So what we want is a free action after quotienting to an effective action. However, by considering an appropriate notion of equivalence, we shall show that any ‘good’ action will be equivalent to an effective one. Let us assume that no row has all zeroes since this would correspond to a T^{k−1} action, and consider the \binom{n}{k} minor determinants

2.5  Δ_{α_{1}...α_{k}} = det ( \begin{matrix}a_{α_{1}}^{1} & … & a_{α_{k}}^{1} \\ ⋮ & & ⋮ \\ a_{α_{1}}^{k} & … & a_{α_{k}}^{k} \end{matrix} )

obtained by deleting n − k columns of Ω.

Definition 2.6: Let Ω ∈ M_{k×n}(ℤ) satisfy

(1) Δ_{α_{1}······α_{k}} ≠ 0, ∀ 1 ≤ α_{1} < ⋯ < α_{k} ≤ n.

Suppose that condition (1) is satisfied and let g be the kth determinantal divisor, i.e. the gcd of all the k by k minor determinants Δ_{α_{1}······α_{k}}. Then Ω is said to be admissible if in addition we have

(2) gcd(Δ_{α_{2}···α_{k+1}}, ..., Δ_{α_{1}···α̂_{s}···α_{k+1}}, ..., Δ_{α_{1}···α_{k}}) = g for all sequences 1 ≤ α_{1} < ⋯ < α_{s} < ⋯ < α_{k+1} ≤ n.

Next we discuss the notion of equivalent T^{k}-actions on S^{4n−1} and obtain a normal form for admissible weight matrices. We are free to change bases of the Lie algebra 𝔱_{k}. This can be done by the group of unimodular matrices GL(k, ℤ). Moreover, if we fix a maximal torus T^{n} of Sp(n), its normalizer, the Weyl group W(Sp(n)) ≃ Σ_{n}^{>}◁ (ℤ_{2})^{n}, preserves the 3-Sasakian structure on S^{4n−1} and intertwines the T^{k} actions. Thus, there is an induced action of GL(k, ℤ) × W(Sp(n)) on the set of weight matrices M_{k×n}(ℤ). The group GL(k, ℤ) acts on M_{k×n}(ℤ) by matrix multiplication from the left, and the Weyl group W(Sp(n)) acts by permutation and overall sign changes of the columns. Actually we want a slightly stronger notion of equivalence than that described above. If the ith row of Ω has a gcd d_{i} greater than one, then by reparameterizing the one-parameter subgroup τ′_{i} = τ_{i}^{d_{i}} we obtain τ_{i}^{a_{α}^{j}} = (τ′_{i})^{b_{α}^{i}} where gcd{b_{α}^{i}}_{α} = 1. So the action obtained by using the matrix whose ith row is divided by its gcd d_{i} is the same as the original action. This is related to the kth determinantal divisor g. We say that a matrix Ω satisfying condition (1) of definition 2.6 is in reduced form (or simply reduced) if g = 1. The following lemma says that it is sufficient to consider matrices in reduced form.

Lemma 2.7: Every weight matrix Ω satisfying condition (1) of definition 2.6 is equivalent to a matrix in reduced form.

Proof: By standard theory there are U ∈ GL(k, ℤ) and V ∈ GL(n, ℤ) such that

Ω = USV where S = ( \begin{matrix}s_{1} & 0 & … & 0 & 0 & … & 0 \\ 0 & s_{2} & … & 0 & 0 & … & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋯ & ⋮ \\ 0 & 0 & ⋯ & s_{k} & 0 & … & 0 \end{matrix} ) .

Here s_{1}, ⋯, s_{k} are the invariant factors of the matrix Ω satisfying s_{i}|s_{i+1} and s_{1}s_{2} ⋯ s_{k} = g. The matrix SV is a k by n matrix with the gcd of the ith row equal to s_{i}. Furthermore, the kth determinantal divisor of the k by n matrix V′ obtained by deleting the last n − k rows of V is one, and so is in reduced form.

Henceforth, we shall only consider matrices in reduced form.

Definition 2.8: Let A_{k×n}(ℤ) denote the subset of reduced admissible matrices in M_{k×n}(ℤ).

The subset A_{k×n}(ℤ) is invariant under the action of GL(k, ℤ) × W(Sp(n)), and the unordered set {|Δ_{s}|} of the absolute values of all k by k minor determinants are invariants of this subset. Let m = min_{s}{|Δ_{s}|}, and let Δ_{m} realize this minimum. Then by transforming Ω by an element of W(Sp(n)) if necessary, we can arrange that the first k columns of Ω has minor determinant Δ_{m}.

Recall (See e.g. [AW]) that a Hermite normal form is a normal form for k by n matrices over principal ideal domains R under multiplication by GL(k, R) from the left. We are working with the classical case R = ℤ. For any nonzero integer a, we let P (a) = {0, 1, ⋯, |a|−1} denote the set of residues modulo |a|. In our case of the GL(k, ℤ)-invariant subset A_{k×n}(ℤ) the Hermite normal form takes the form:

2.9  Ω = ( \begin{matrix}a_{1}^{1} & a_{2}^{1} & … & a_{k}^{1} & … & a_{n}^{1} \\ 0 & a_{2}^{2} & … & a_{k}^{2} & … & a_{n}^{2} \\ ⋮ & ⋮ & ... & ⋮ & … & ⋮ \\ 0 & 0 & … & a_{k}^{k} & … & a_{n}^{k} \end{matrix} )

with a_{i}^{i} > 0, and if j < i then a_{i}^{j} ∈ P (a_{i}^{i}). From the standard theory we have

Proposition 2.10: Every Ω ∈ A_{k×n}(ℤ) is equivalent by left multiplication by an element of GL(k, ℤ) to a matrix of the form 2.9.

The Hermite normal form is unique when transforming by GL(k, ℤ), but in general there may be several Hermite normal forms when transforming by the larger group GL(k, ℤ) × W(Sp(n)). For example, the weight matrices

2.11  ( \begin{matrix}1 & 0 & −1 & 2 \\ 0 & 1 & −1 & 3 \end{matrix} )  ( \begin{matrix}1 & 0 & −1 & 1 \\ 0 & 1 & −1 & 3 \end{matrix} )

are inequivalent under both GL(2, ℤ) and W(Sp(4)) separately, but are equivalent under the product.

As described above we always choose a normal form such that the absolute value of the determinant of the first k by k minor is minimal. Thus, the GL(k, ℤ) × W(Sp(n))- invariant subset A_{k×n}(ℤ) decomposes as a disjoint union of invariant subsets labelled by minima |Δ_{m}|, viz.

2.12  A_{k×n}(ℤ) = ⨆_{m=1}^{∞} A_{k×n}(m; ℤ).

As the subsets A_{k×n}(m; ℤ) are somewhat intractable for m > 1, we focus our attention on the case m = 1. In this case the Hermite normal form gives

Proposition 2.13: Let Ω lie in the invariant subset A_{k×n}(1; ℤ), then Ω is equivalent to a weight matrix of the form

Ω = ( \begin{matrix}1 & 0 & … & 0 & a_{k+1}^{1} & … & a_{n}^{1} \\ 0 & 1 & … & 0 & a_{k+1}^{2} & … & a_{n}^{2} \\ ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋯ & ⋮ \\ 0 & 0 & ⋯ & 1 & a_{k+1}^{k} & … & a_{n}^{k} \end{matrix} ) ,

where the all the entries of the last n − k columns are non vanishing. Furthermore, if Δ_{1} is unique, then this normal form is unique up to permutations and sign changes of the last n − k columns.

Our interest in admissible weight matrices is the following:

Theorem 2.14: Let Ω ∈ A_{k×n}(ℤ), that is Ω is a reduced admissible weight matrix. Then the quotient space N(Ω)/T^{k} is a smooth 3-Sasakian manifold of dimension 4(n − k) − 1.

The proof of this theorem follows directly from the Reduction Theorem 1.3 and the following lemma.

Lemma 2.15: Let Ω ∈ M_{k×n}(ℤ) satisfy condition (1) of definition 2.6, then

(i) The isotropy subgroup Γ ⊂ T^{k} of any point of N(Ω) is discrete.

(ii) 0 is a regular value of the moment map μ_{Ω}.

(iii) If in addition Ω is in reduced form and satisfies condition (2) of 2.6, then T^{k} acts freely on N(Ω).

Proof: The 3-Sasakian version of a standard fact of symplectic reduction says that the image of the differential Dμ_{Ω}(p) of the moment map at p ∈ N(Ω) is 𝔤_{p}^{⊥} ⊗ ℝ^{3} where 𝔤_{p} is the Lie algebra of the isotropy subgroup at p. So (i) and (ii) are equivalent. We prove (i). At most n − k − 1 quaternionic coordinates u_{j} can vanish on N(Ω) for if n − k of the u_{j}’s were to vanish the moment map equation would become in matrix notation

Ω_{k} · ( ū_{α_{1}}iu_{α_{1}} ; ⋮ ; ū_{α_{k}}iu_{α_{k}} ) = 0

for some indices 1 ≤ α_{1} < ⋯ < α_{k} ≤ n and k by k minor Ω_{k}. But by (1) of definition 2.6 Ω_{k} is invertible implying ū_{α_{j}}iu_{α_{j}} = 0 for j = 1, ⋯ k. Then equation 2.4′ implies that all the coordinates u_{j} vanish. But this is impossible since N(Ω) ⊂ S^{4n−1}. Thus, there are k + 1 quaternionic coordinates that do not vanish. Let {α_{j}}_{j=1}^{k+1} denote the indices of the non-vanishing coordinates. The equations that determine a fixed point of N(Ω) are

2.16  ∏_{i} τ_{i}^{a_{α}^{i}} = 1 for each α = α_{j}, j = 1, ⋯ k + 1.

This system of k + 1 equations in k unknowns τ_{i} can be linearized by taking logarithms. Writing τ_{j} = e^{2πit_{j}} 2.16 becomes

2.17  ∑_{i} a_{α}^{i}t_{i} = r_{α}

for each α = α_{j} and some integers r_{α_{j}}. Since all minor determinants are non vanishing this system has only a discrete set of solutions. Hence, the isotropy groups are discrete proving (i).

Now suppose that Ω is in reduced form and satisfies condition (2) of definition 2.6, i.e., Ω ∈ A_{k×n}(ℤ). From above at most n − k − 1 of the quaternionic coordinates can vanish, so there must be at least k + 1 non-vanishing coordinates u_{α_{j}}. The equations determining the fixed points of T^{k} are Eq. 2.16 which must hold for all α such that u_{α} ≠ 0. Again let Ω_{k} be any k by k minor of Ω and let C_{k} denote the cofactor matrix of Ω_{k}. Then inverting 2.17 gives

Δ_{k}t_{i} = (C_{k} · r)_{i},

where Δ_{k} is the determinant of Ω_{k} and r = (r_{1}, ⋯, r_{n}) ∈ ℤ^{n}. Since the matrix C_{k} is integer valued this implies

2.18  τ_{i}^{Δ_{k}} = 1

for all minor determinants Δ_{k}. But since the gcd of all minor determinants is 1, this implies τ_{i} = 1 for all i = 1, ⋯ k. This proves the lemma, and thus Theorem 2.14.

Remark 2.19: If condition (1) of definition 2.6 holds, but condition (2) fails, then the quotient N(Ω)/T^{k} will be a 3-Sasakian orbifold, but not a smooth manifold.

In general Theorem 2.14 is not yet an existence theorem, since A_{k×n}(ℤ) could be empty. Indeed, we shall show in a forthcoming work that for each k there is an upper bound on n in order that admissible weight matrices exist. However, in this work we are only concerned with the case n = k + 2 when the 3-Sasakian quotient is a 7-manifold. So we now restrict ourselves to this case. Our next result determines precisely the admissible weight matrices in the subset A_{k×k+2}(1; ℤ). Any such matrix takes the form

2.20  Ω = ( \begin{matrix}1 & 0 & … & 0 & a^{1} & b^{1} \\ 0 & 1 & … & 0 & a^{2} & b^{2} \\ ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ \\ 0 & 0 & … & 1 & a^{k} & b^{k} \end{matrix} ) .

In this case the admissibility conditions simplify. The remaining subsets A_{k×k+2}(m; ℤ) for m ≠ 1 are more difficult to treat. For low values of k it is not difficult to construct admissible weight matrices by hand, but there appears to be no general algorithm.

Proposition 2.21: Let Ω ∈ A_{k×k+2}(1; ℤ) be in the normal form 2.20, then Ω is admissible if and only if gcd(a^{j}, b^{j}) = 1 for all j = 1, ⋯, k, and if for any pairs a^{i} = ±a^{j} or b^{i} = ±b^{j} then we must have b^{i} ≠ ±b^{j} or a^{i} ≠ ±a^{j}, respectively.

Proof: To check admissibility we need to select k + 1 columns and then look at the gcd of the resulting k + 1 minor determinants. If we select the first k + 1 columns or the k + 1 columns obtained by deleting column k + 1, the gcd of the corresponding k + 1 minor determinants is identically 1. The remaining conditions are all obtained by deleting one of the first k columns and then considering the corresponding k + 1 k by k minor

determinants. So the conditions for admissibility imply that det ( \begin{matrix}a^{i} & b^{i} \\ a^{j} & b^{j} \end{matrix} ) ≠ 0 for all

1 ≤ i < j ≤ k, and that

gcd(a^{j}, b^{j}, det ( \begin{matrix}a^{1} & b^{1} \\ a^{j} & b^{j} \end{matrix} ) , ⋯, det ( \begin{matrix}a^{j} & b^{j} \\ a^{k} & b^{k} \end{matrix} )) = 1

for all j = 1, ⋯ k, where of course the term det ( \begin{matrix}a^{j} & b^{j} \\ a^{j} & b^{j} \end{matrix} ) is deleted. The gcd condition is

satisfied if and only if gcd(a^{j}, b^{j}) = 1 for all j = 1, ⋯, k, and then the remaining conditions follow easily from the non vanishing of the corresponding 2 by 2 minor determinants.

Example 2.22 The following example was first introduced in [BGM1] and it is related to Kronheimer’s construction of the ALE-spaces for Γ = ℤ_{k+1}. Consider the weight matrix

2.23  Ω(a) = ( \begin{matrix}1 & −1 & 0 & … & 0 & a_{1} \\ 0 & 1 & −1 & … & 0 & a_{2} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & … & 1 & −1 & a_{k} \end{matrix} ) ,

where a = (a_{1}, ..., a_{k}). Clearly Ω(a) ∈ A_{k×k+2}(1; ℤ) and the corresponding 3-Sasakian quotient S(Ω(a)) is 7-dimensional. Note that Ω(a) is equivalent to a matrix of the form 2.20 with a^{i} = −1 and b^{i} = a_{i} + ⋯ + a_{k}. Condition (2) of the Definition 2.6 is now automatically satisfied. Condition (1) means that a must be chosen away from a union of \frac{1}{2}k(k + 1) hyperplanes in ℝ^{k} defined by

2.24  ∀ j = 0, 1, ..., k − 1 ∀i ≤ k − j a_{i} + a_{i+1} + ⋯ + a_{i+j} ≠ 0.

The construction of S(Ω(a)) is related to the quotient construction of the gravitational instantons of Gibbons and Hawking [GH] and further analyzed by Hitchin [Hi1]. The non-Abelian case is due to Kronheimer [Kr]. We briefly describe the quotient construction [HKLR] as described by Kronheimer [Kr] for general Γ and then in more details for Γ = ℤ_{k+1}. For any discrete subgroup Γ of SU(2) Kronheimer introduces a subgroup G(Γ) ⊂ U(|Γ|) of dimension |Γ| − 1 which then gives a linear action on the model flat hyperkähler manifold, the quaternionic vector space ℍ^{|Γ|}. Now G(Γ) ⊂ U(|Γ|) and acts on ℍ^{|Γ|} by hyperkähler isometries. Hence we obtain the hyperkähler moment map

μ(Γ) : ℍ^{|Γ|} ⟶ G^{∗}(Γ) ⊗ ℝ^{3},

where G^{∗}(Γ) is the dual of the Lie algebra of G(Γ). Let T (Γ) ⊂ G(Γ) be the center of G(Γ). Now, if ξ ∈ T^{∗}(Γ) ⊗ ℝ^{3} is in the so-called “good set” then the hyperkähler quotient M(Γ, ξ) = μ(Γ)^{−1}(ξ)/G(Γ) is a 4-dimensional hyperkähler manifold. Let us review the Γ = ℤ_{k+1} case. Then G(Γ) = T^{k} ⊂ SU(k + 1) ⊂ U(k + 1) is just the maximal torus in SU(k + 1) acting on ℍ^{k+1} as 𝔸(τ) matrix multiplication from the left, where

𝔸(τ) = ( \begin{matrix}τ_{1} & 0 & 0 & … & 0 \\ 0 & \overline{τ}_{1}τ_{2} & 0 & … & 0 \\ ⋮ & ⋮ & ⋮ & & ⋮ \\ 0 & 0 & 0 & … & \overline{τ}_{k} \end{matrix} ) ,

and τ = (τ_{1}, ..., τ_{k}) ∈ T^{k}. The moment map for this action can be easily computed and one gets

μ^{j}(u) = \overline{u}_{j}iu_{j} − \overline{u}_{j+1}iu_{j+1}, j = 1, ..., k,

with μ = ∑_{j} μ^{j}e_{j}, where {e_{j}}_{j=1}^{k} denote the standard basis for 𝔱_{k}^{∗} ≃ ℝ^{k}.

Let us denote by μ_{k}^{−1}(ξ) the inverse image of some element ξ = (ξ_{1}, ..., ξ_{k}) ∈ 𝔱_{k}^{∗} ⊗ ℝ^{3}. We say that ξ is in a good set if conditions 2.24 with a_{i} replaced by ξ_{i} are satisfied. It can be easily checked (see also [BD] for more general cases of toral quotients) that T^{k} acts freely on the level set μ_{k}^{−1}(ξ) if and only if ξ is in the good set. The quotient space M(ℤ_{k+1}, ξ) is a simply-connected hyperkähler manifold diffeomorphic to the minimal resolution of Kleinian singularity ℂ^{2}/ℤ_{k+1}. The first and second integral cohomology groups are H^{1}(M(ℤ_{k+1}, ξ); ℤ) = 0 and H^{2}(M(ℤ_{k+1}, ξ); ℤ) = ⊕^{k}ℤ.

Clearly, the 3-Sasakian quotient S(Ω(a)) is a modification of the above construction and it was first discussed in [GN]. Namely, choose a = (a_{1}, ..., a_{k}) ∈ ℤ^{k} and consider a homomorphism h_{a} : T^{k} ⟶ T^{1}, given by

h_{a}(τ) = ∏_{j=1}^{k} τ_{j}^{a_{j}}.

This allows one to extend Kronheimer’s action of the T^{k} on ℍ^{k+1} to another action of the T^{k} on ℍ^{k+2} as follows:

(τ; (u, u_{k+2})) ⟶ (𝔸(τ)u, h_{a}(τ)u_{k+2}), u ∈ ℍ^{k+1}, u_{k+2} ∈ ℍ,

and the hyperkähler moment map of this action restricted to the S^{4k+7} gives the corresponding 3-Sasakian moment map and we get S(Ω(a)).

§3. A Stratification of S(Ω)

In this section we describe a stratification of the orbit space S(Ω) = N(Ω)/T^{k} in the case n = k + 2, that is dim S(Ω) = 7. To do this we consider the space Q(Ω) defined as the space of orbits of N(Ω) by the isometry group T^{n}·Sp(1). For the moment we consider general n. Let Ω ∈ A_{k×n}(ℤ) and consider the manifold N(Ω). It follows from Proposition 1.4 that the groups T^{n} × Sp(1) and T^{n−k} × Sp(1) act as isometry groups on N(Ω) and S(Ω), respectively. Let Q(Ω) denote orbit space N(Ω)/T^{n} × Sp(1). We have the following commutative diagram

3.1  \begin{aligned}& & N(Ω) \\ & ↙ & & ↘ \\ Q(Ω) & & ⟵ & & S(Ω). \end{aligned}

The southeast arrow is a principal bundle with fibres T^{k}, but the other arrows are not fibrations. The southwest arrow has generic fibres T^{n} ·Sp(1) ≃ T^{n} × S^{3}, while the west arrow has generic fibres T^{n−k} ·Sp(1) homeomorphic either to T^{n−k} × ℝℙ^{3} or T^{n−k} × S^{3} depending on Ω. The dimension of the orbit space Q(Ω) is 3(n − k) − 4, and by analyzing this diagram we shall obtain very simple stratifications of both N(Ω) and S(Ω) in the case n = k + 2, when dim Q(Ω) = 2. This stratification is related to the stratification by orbit types [Bre], but is cruder and will provide the quotient Q(Ω) with a very simple CW decomposition. Let us define the following subsets of N(Ω) : (Recall from the proof of Lemma 2.15 that in the case n = k + 2 at most one quaternionic coordinate can vanish.)

3.2

N_{0}(Ω) = {u ∈ N(Ω)| u_{α} = 0 for some α = 1, ⋯, k + 2},
N_{1}(Ω) = {u ∈ N(Ω)| for all α = 1, ⋯, k + 2, u_{α} ≠ 0 and

there is a pair (u_{α}, u_{β}) that lies on the same

complex line in ℍ},

N_{2}(Ω) = {u ∈ N(Ω)| for all α = 1, ⋯, k + 2, u_{α} ≠ 0 and

no pair (u_{α}, u_{β}) lies on the same complex line

in ℍ}.

Here the strata are labeled by the dimension of the cells in the resulting CW decomposition of Q(Ω). Clearly, N(Ω) = N_{0}(Ω) ⊔ N_{1}(Ω) ⊔ N_{2}(Ω) and N_{2}(Ω) is a dense open submanifold of N(Ω).

Lemma 3.3: Let n = k + 2 and Ω ∈ A_{k×k+2}(ℤ), and let ρ_{N} : N(Ω)⟶Q(Ω) denote the natural projection.

(i) The orbits in N_{2}(Ω) are T^{k+2} ·Sp(1) ≃ T^{k+2} × S^{3}, and ρ_{N} restricted to N_{2}(Ω) is a principal T^{k+2}·Sp(1) bundle over a two dimensional base space Q_{2}(Ω).

(ii) The orbits in N_{1}(Ω) are T^{k+1}·Sp(1) ≃ T^{k+1} × S^{3}.

(iii) The orbits in N_{0}(Ω) are T^{k}·Sp(1) ≃ T^{k} × S^{3}.

The ≃ above indicates diffeomorphism.

Proof: The action of T^{k+2} × Sp(1) on ℍ^{k+2} is given in quaternionic coordinates u_{α} by u_{α} ↦ e^{iθ_{α}}u_{α}q for α = 1, ⋯, k + 2, where 0 ≤ θ_{α} ≤ 2π and q ∈ Sp(1). This action clearly restricts to the zero set of the moment map N(Ω). The action is not effective since the element e^{iθ_{α}} = −1, q = −1 acts as the identity on ℍ^{k+2}. We do get an effective action of the factor group T^{k+2}·Sp(1).

In the description of N_{2}(Ω) and N_{1}(Ω) we can use the Sp(1) action to choose a complex line in ℂ^{2} for a chosen quaternionic coordinate u_{α}. For example writing u_{α} = z_{α} + w_{α}j, we choose u_{k+2} = z_{k+2} or equivalently w_{k+2} = 0. Then 3.2 becomes

3.4

N_{2}(Ω) = {u ∈ N(Ω)| u_{α} ≠ 0 for all α = 1, ⋯, k + 2, and z_{α}w_{α} ≠ 0 for some α},
N_{1}(Ω) = {u ∈ N(Ω)| for all α = 1, ⋯, k + 2, u_{α} ≠ 0 and z_{α}w_{α} = 0},
N_{0}(Ω) = {u ∈ N(Ω)| u_{α} = 0 for some α = 1, ⋯, k + 2}.

We show that T^{k+2}·Sp(1) acts freely on N_{2}(Ω). The subgroup of Sp(1) that stabilizes the complex line determined by w_{k+2} = 0 is an S^{1} given by multiplication from the right, and the stabilizer in the full isometry group is G = T^{k+2}·S^{1}. In this case the action of G on N_{2}(Ω) is given by

3.5  z_{α} ↦ e^{i(θ_{α}+ϕ)}z_{α},  w_{α} ↦ e^{i(θ_{α}−ϕ)}w_{α},

where e^{iϕ} parameterizes the S^{1} from the right. At any point of N_{2}(Ω) there is an α such that z_{α}w_{α} ≠ 0. So if this point is fixed 3.5 forces either the identity or e^{iϕ} = −1 and e^{iθ_{α}} = −1. But then since for all β = 1, ⋯, k + 2 either z_{β} or w_{β} is non vanishing we get that either e^{iθ_{β}} = 1 or e^{iθ_{β}} = e^{±iϕ} = −1. Thus, T^{k+2} ·Sp(1) acts freely on N_{2}(Ω), and restricting ρ_{N} to this stratum gives a principal T^{k+2}·Sp(1) bundle.

Now consider the orbits in N_{1}(Ω). Again setting w_{k+2} = 0 we have the stabilizer G = T^{k+2} ·S^{1} and Eq.3.5 gives that e^{iθ_{k+2}} = e^{−iϕ}. Now since for each α precisely one of either z_{α} or w_{α} is non vanishing, Eq.3.5 implies that for each α = 1, ⋯, k + 1 we have e^{iθ_{α}} = e^{±iϕ}. Hence, the isotropy group at every point of N_{1}(Ω) is an S^{1}, and the inclusion map ι : S^{1}⟶T^{k+2}·S^{3} has the following form: On the first factor ι has the form τ ↦ (τ^{±1}, ⋯, τ^{−1}) where the signs are determined by certain choices. On the second factor ι is the standard inclusion in the Hopf fibration. It follows that the fibres in N_{1}(Ω) are (T^{k+2}·S^{3})/S^{1} ≃ T^{k+1}·S^{3}.

Next we analyze the orbits in N_{0}(Ω). Any Ω ∈ A_{k×k+2}(ℤ) has the form 2.20 with n = k + 2. As far as the equations 2.4′ describing N(Ω) is concerned we can pass from ℤ to its field of fractions ℚ. So for the equations 2.4′ we can choose Ω to be in the form 2.20 by inverting the first k by k block by an element of GL(k, ℚ). Now as before if u_{k+2} ≠ 0, we can use the Sp(1) action to set w_{k+2} = 0. If u_{k+2} is the coordinate that vanishes do this for u_{k+1}. Then our transformed equations for the moment map take the form

3.6  \begin{aligned}|z_{j}|^{2} − |w_{j}|^{2} + f_{j}(|z_{k+1}|^{2} − |w_{k+1}|^{2}) + g_{j}|z_{k+2}|^{2} & = 0 \\ w̄_{j}z_{j} + f_{j}w̄_{k+1}z_{k+1} & = 0 \end{aligned}

for j = 1, ⋯, k and f_{j}, g_{j} ∈ ℚ − {0}. The second equation immediately implies that with the choices made on N(Ω) :

(1) If z_{α}w_{α} = 0 for some α it vanishes for all α.

From the proof of Lemma 2.15 at most one quaternionic coordinate u_{α} can vanish. So suppose that u_{α} = 0 then the circle e^{iθ_{α}} contributes an S^{1} to the isotropy subgroup of any point of N_{0}(Ω). Moreover, since (1) above holds the analysis for orbits in N_{1}(Ω) above shows that the isotropy subgroup of any point in N_{0}(Ω) is the 2-torus T^{2}, so analyzing as before shows that these orbits are T^{k}·S^{3}.

Our next lemma relates our stratification with the finer stratification by orbit types [Bre].

Lemma 3.7: Assuming the hypothesis of Lemma 3.3, the stratification N(Ω) by orbit types refines the stratification defined by 3.2 as follows:

(i) N_{2}(Ω) is precisely the stratum of principal orbits.

(ii) N_{1}(Ω) is disconnected and decomposes into components ⊔_{α=1}^{k+2}N_{1}(α, Ω), and each component consists of a single orbit type.

(iii) N_{0}(Ω) consists of k + 2 disjoint copies of the group T^{k}·S^{3}, and each component is an orbit type.

(iv) There are no exceptional orbits.

Proof: (i) and (iv) are obvious from Lemma 3.3. Consider the orbits in N_{0}(Ω). Recall that at most one quaternionic coordinate u_{α} can vanish. Suppose u_{k+2} ≠ 0 (if this is the coordinate that vanishes interchange the role of u_{k+1} and u_{k+2}). As before we analyze the moment map equations by considering a slice of the Sp(1) action yielding Eqs. 3.4. We need to consider various cases. We assume that both f_{j}, g_{j} > 0. (The other cases are handled similarly.) First, suppose that u_{k+1} = 0 and choose w_{k+2} = 0 as before, then Eqs. 3.6 imply w_{j} ≠ 0, z_{j} = 0 for j = 1, ⋯, k. Moreover, the first of Eqs. 3.6 together with the sphere constraint determine the moduli

3.8  |z_{k+2}|^{2} = \frac{1}{1 + ∑_{j} g_{j}} ,  |w_{j}|^{2} = \frac{g_{j}}{1 + ∑_{j} g_{j}} .

The only free parameters are the phases of w_{j} and these determine the T^{k} of the fibre. So there is a single orbit with fibre T^{k}·S^{3}. A similar analysis can be done at the vertices u_{j} = 0. In this case which of the coordinates z_{i}, w_{i} vanish depends on the sign of g_{i}f_{j} −f_{i}g_{j}, and one sees again that k phases are undetermined. This proves (iii).

Now consider orbits in N_{1}(Ω). Again we use the form 3.4 and set u_{k+2} = z_{k+2} ≠ 0. The isotropy subgroup G_{u} ≃ S^{1} only depends on whether w_{α} or z_{α} vanish (recall from (1) above that the product vanishes for all α). Choosing this for each α chooses a component of N_{1}(Ω), and this component is a trivial principal T^{k+1}·S^{3} bundle. Hence, on any component all fibres are equivalent. We could do a similar, albeit more tedious analysis than above, to count the number of components. However, it will follow easily from Lemma 3.11 below that there are precisely k + 2 components.

Since T^{k+2} is the center of T^{k+2}·Sp(1), diagram 3.1 and Lemmas 3.3 and 3.7 imply

Lemma 3.9: Let n = k + 2 and Ω ∈ A_{k×k+2}(ℤ), and consider the natural projection ρ_{S} : S(Ω)⟶Q(Ω). Then there is a stratification S(Ω) = S_{0}(Ω) ⊔ S_{1}(Ω) ⊔ S_{2}(Ω) such that

(i) The orbits in S_{2}(Ω) are precisely the principal orbits of type

\frac{T^{k+2}·Sp(1)}{T^{k}} = G(Ω).

Moreover, H_{∗}(G(Ω); ℚ) ≅ H_{∗}(T^{2} × Sp(1); ℚ).

(ii) The orbits in S_{1}(Ω) are of the form (G(Ω)/S^{1}) and S_{1}(Ω) decomposes into components ⊔_{α=1}^{k+2}S_{1}(α, Ω), where each component is an orbit type. Furthermore, the Weyl group W(Sp(k + 2)) acts transitively on the set of components.

(iv) S_{0}(Ω) consists of k + 2 disjoint copies of the form (G(Ω)/T^{2}) and each component is an orbit type. Again the Weyl group W(Sp(k + 2)) acts transitively on the set of components.

(v) There are no exceptional orbits.

Remark 3.10: The group G(Ω) defined in Lemma 3.9 is isomorphic to either T^{2} × Sp(1) or T^{2} × SO(3). For example, if Ω has the form of 2.20, we see that

G(Ω) ≃ \{ \begin{matrix}T^{2} × SO(3) if ∑ a^{i} and ∑ b^{i} are both odd \\ T^{2} × Sp(1) otherwise. \end{matrix}

Next we analyze the quotient space Q(Ω). The stratification given by 3.2 induces a stratification of the quotient, namely Q(Ω) = Q_{0}(Ω) ⊔ Q_{1}(Ω) ⊔ Q_{2}(Ω). We now have

Lemma 3.11: Under the hypothesis of Lemma 3.3, the following hold:

(i) The orbit space Q(Ω) is homeomorphic to the closed disc D̄^{2}, and the subset of singular orbits Q_{1}(Ω) ⊔ Q_{0}(Ω) is homeomorphic to the boundary ∂D̄^{2} ≃ S^{1}.

(ii) Q_{2}(Ω) is homeomorphic to the open disc D^{2}.

(iii) Q_{1}(Ω) is homeomorphic to the disjoint union of k + 2 copies of the open unit interval.

(iv) Q_{0}(Ω) is a set of k + 2 points.

Proof: Consider the natural projection ρ_{S} : S(Ω)⟶Q(Ω). Since, S(Ω) has finite fundamental group [BGM2], it follows from Corollary II.6.5 of Bredon [Bre] that H_{1}(Q(Ω), ℚ) = 0. Moreover, from (v) of Lemma 3.9 there are no exceptional orbits. So as in the proof of Theorem IV.8.6 of Bredon [Bre] Q(Ω) is homeomorphic to the closed disc D̄^{2}, and the subset of singular orbits Q_{1}(Ω)⊔Q_{0}(Ω) is homeomorphic to the boundary ∂D̄^{2} ≃ S^{1}. It follows from (iv) of Lemma 3.9 that Q_{0}(Ω) is a set of k+2 points. But then Q_{1}(Ω) ≃ ∂D̄^{2} −Q_{0}(Ω) is homeomorphic to k + 2 disjoint arcs of a circle. The remainder of the lemma as well as filling the gap in the proof of Lemma 3.7 now follow easily.

Lemma 3.11 implies that the stratification of Q(Ω) provides it with a fairly natural CW decomposition. This CW decomposition is depicted in the following diagram.

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§4. The Rational Homology of S(Ω)

In this section we restrict our attention to the case where n = k + 2. Thus, each S(Ω) is a 7-dimensional 3-Sasakian manifold. As mentioned in the introduction the results of [BGM2] and [GS1] imply the rational homology of any such manifold is given by

4.1  H_{∗}(S(Ω); ℚ) ≅ \{ \begin{matrix}ℚ, & if ∗ = 0, 7; \\ 0, & if ∗ = 1, 3, 4, 6; \\ ℚ^{r} & if ∗ = 2, 5; \end{matrix}

for some integer r. Hence, the only rational homology invariant for these manifolds is the Betti number b_{2}(S(Ω)) = b_{5}(S(Ω)). Our main result of this section is that every possible Betti number is realized by the examples constructed by the admissible weight matrices of Proposition 2.21.

Theorem 4.2: Let n = k + 2 and Ω ∈ A_{k×k+2}(ℤ). Then

H_{2}(S(Ω); ℚ) ≅ ℚ^{k} ≅ H_{5}(S(Ω); ℚ).

Remark 4.3: Theorem 4.2 actually holds in the more general case of 3-Sasakian orbifolds where the the second admissibility condition fails as explained in remark 2.19.

Proof: We can use the natural projection ρ_{S} : S(Ω)⟶Q(Ω) to construct a Leray spectral sequence converging to H_{∗}(S(Ω); ℚ). Let B_{i} be the natural CW decomposition of Q(Ω) given in section 3 so that, in the notation of Lemma 3.11,

B_{0} = Q_{0}(Ω), B_{1} = Q_{0}(Ω) ∪ Q_{1}(Ω), and B_{2} = Q(Ω).

Then, we can filter S(Ω) by X_{i} = ρ_{S}^{−1}(B_{i}) to obtain the increasing filtration

X_{0} = S_{0}(Ω), X_{1} = S_{0}(Ω) ∪ S_{1}(Ω), and X_{2} = S(Ω).

The Leray spectral sequence associated to this filtration has E^{1} term given by

E_{s,t}^{1} ≅ H_{s+t}(X_{t}, X_{t−1}; ℚ)

with differential d_{1} : H_{s+t}(X_{t}, X_{t−1}; ℚ) ⟶ H_{s+t−1}(X_{t−1}, X_{t−2}; ℚ) where we use the convention that X_{−1} = ∅.

To compute these E^{1} terms notice that all the pairs (X_{t}, X_{t−1}) are relative manifolds so that one can apply the Alexander-Poincaré duality theorem. Hence, by Lemma 3.9,

\begin{aligned}H_{s}(X_{0}; ℚ) & ≅ H_{s}((Sp(1)^{k+2}; ℚ); \\ H_{s}(X_{1}, X_{0}; ℚ) & ≅ H^{5−s}((S^{1} × Sp(1))^{k+2}; ℚ); \\ H_{s}(X_{2}, X_{1}; ℚ) & ≅ H^{7−s}(S^{1} × S^{1} × Sp(1); ℚ). \end{aligned}

Hence,

E_{s,t}^{1} ≅ \{ \begin{matrix}ℚ^{k+2} & if t = 0 and s = 0, 3; \\ ℚ^{k+2} & if t = 1 and s = 0, 1, 3, 4; \\ ℚ & if t = 2 and s = 0, 2, 3, 5; \\ ℚ^{2} & if t = 2 and s = 1, 4; \\ 0 & otherwise. \end{matrix}

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The d_{1} differentials, which move horizontally one column to the left in diagram 4.4, are completely determined by equation 4.1. Thus,

E_{s,t}^{2} ≅ \{ \begin{matrix}ℚ & if t = 0 and s = 0, 3; \\ ℚ^{k} & if t = 1 and s = 1, 4; \\ ℚ & if t = 2 and s = 3, 5; \\ 0 & otherwise. \end{matrix}

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Finally, there is only one possibly non-trivial d^{2} differential

H_{4}(X_{2}, X_{1}; ℚ) ⟶^{d_{4,2}^{2}} H_{3}(X_{0}; ℚ)
↓≅  ↓≅
ℚ  ⟶  ℚ.

Equation 4.1 implies d_{4,2}^{2} must be an isomorphism, there are no other possible differentials, so E^{3} ≅ E^{∞}, and the result follows.

It should be possible to refine the computations of this section to compute the integral cohomology ring H^{∗}(S(Ω); ℤ), at least in the n = k + 2 case. It appears that, just like in the k = 1 case, that there is torsion in H^{4}(S(Ω); ℤ) which depends non-trivially on the weight matrix Ω. This would imply that these S(Ω) examples run through infinitely many distinct homotopy types for each fixed k.

After completing this work we received a preprint from R. Bielawski [Bi] where he computes the Betti numbers for 3-Sasakian orbifolds of any possible dimension arising as toral quotients of spheres. As expected the second Betti number of such an orbifold is equal to the dimension of the torus. However, the existence of smooth manifolds with n > k + 2 is much more delicate. For example, the subset A_{k×k+3}(1, ℤ), defined in section two, is empty for k > 4. Bielawski’s result implies for example that, in dimension 11, the Betti number relation b_{2} = b_{4} valid for regular 3-Sasakian manifolds [GS] is not true in general.

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Department of Mathematics and Statistics October 1996 University of New Mexico Albuquerque, NM 87131 email: cboyer@math.unm.edu, galicki@math.unm.edu mann@math.unm.edu

Department of Mathematics and Statistics University of Edinburgh Edinburgh, Scotland, UK EH9 3JZ email: elmer@maths.ed.ac.uk