A Note on Smooth Toral Reductions of Spheres

Charles P. Boyer Krzysztof Galicki Benjamin M. Mann

In previous work [BGMR1, BGMR2] the authors and E. Rees, using 3-Sasakian toral reductions of the standard (4n − 1)-dimensional sphere, constructed compact 3-Sasakian 7-manifolds with arbitrary second Betti number. These were the first examples of smooth compact Einstein manifolds of positive scalar curvature with arbitrary second Betti number. This toral reduction procedure actually constructs more general families of 3-Sasakian (4n − 1)-dimensional orbifolds S(Ω). However, to obtain (4n − 1)-manifolds with b_{2} > 1 one should answer two separate questions: (1) What are the orbifold Betti numbers of S(Ω)? and (2) When is S(Ω) smooth?

In [BGMR1] sufficient conditions for guaranteeing smoothness for general toral reductions were given, and in Proposition 1.4 below we show that these conditions are essentially also necessary. The Betti number computations in [BGMR1] were carried out only for dimension seven. In this case the we gave a 2k-parameter family of smooth 3-Sasakian 7-manifolds with second Betti number k. Contemporaneously, R. Bielawski [Bi1] computed Betti numbers of all these toral examples; more precisely, he showed that

b_{2i} = ( \begin{matrix}k + i − 1 \\ i \end{matrix} )

for i ≤ n − k − 1 for any 3-Sasakian orbifold obtained by 3-Sasakian reduction of S^{4n−1} by a k-dimensional torus. Since the odd Betti numbers b_{2i+1} of any 3-Sasakian manifold are known to vanish [GS] for i ≤ n − k − 1 (for orbifolds this follows from results of [BG]), Bielawski’s results and Poincaré duality would determine the rational homology of any 3-Sasakian manifold obtained from toral reduction. However, there remains the question as to when these higher dimensional orbifolds can be smooth manifolds. In this note we give mod 2 obstructions to smoothness which show, somewhat surprisingly, that if the reduced 3-Sasakian orbifold S is a smooth manifold of dimension greater than 7 then b_{2} ≤ 4, and if the dimension of the reduced 3-Sasakian manifold is greater than 15 then b_{2} ≤ 1. That is, we prove

Theorem A: Let S be a 3-Sasakian manifold obtained by 3-Sasakian reduction of S^{4n−1} by a torus T^{k}, and assume that S is not a sphere nor a smooth quotient of a sphere by a finite group. If k > 1 then dim S = 7, 11, 15. In addition, if k > 4 then dim S = 7.

After proving Theorem A we show that these bounds are sharp in that there exist infinite families of manifold examples in all cases not eliminated by Theorem A. This also shows that any additional mod p obstructions to smoothness for p an odd prime give no further constraints on the Betti numbers.

Acknowledgements We would like to thank R. Bielawski and J. Hurtubise for helpful discussions. We would also like to thank the referee for pointing out the need to discuss the degenerate cases in Proposition 1.4 and for other helpful suggestions. Finally, the second named author would like to thank the MPI in Bonn where final revisions of the paper were made.

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

§1. Proof of Theorem A

Recall from [BGMR1] that any k-torus action on S^{4n−1} that preserves the 3-Sasakian structure on the sphere is given by a quaternionic representation of the k-torus T^{k} on a quaternionic vector space ℍ^{n}. This action can be described by left multiplication on a quaternionic column vector u = (u_{1}, ⋯, u_{n})^{t} by a diagonal matrix h_{Ω}(τ_{1}, .., τ_{k}) ∈ T^{n} ⊂ Sp(n)

1.1  h_{Ω}(τ_{1}, .., τ_{k}) = ( \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. This representation gives rise to the integral weight matrix

1.2  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} ) .

The 3-Sasakian quotient space S(Ω) is obtained from a moment map μ_{Ω} : S^{4n−1}⟶𝔱_{k} ⊗ℝ^{3} by defining S(Ω) = μ^{−1}(0)/T^{k}. For the explicit form of μ_{Ω} we refer to [BGMR1]. It is a quadratic form on the quaternionic vector space ℍ^{n} and this fact makes its analysis accessible.

Two weight matrices Ω, Ω′ ∈ M_{k×n}(ℤ) are equivalent if there are A ∈ GL(k, ℤ) and w ∈ W(Sp(n)) such that Ω′ = AΩw. Equivalent weight matrices give rise to isomorphic 3-Sasakian quotients μ_{Ω′}^{−1}(0)/T^{k}(Ω′) ≃ μ_{Ω}^{−1}(0)/T^{k}(Ω). The question of whether or not the converse is true is more subtle and will be addressed elsewhere. The conditions on Ω that determine the behavior of the quotient are given by the proposition below. We say that Ω ∈ M_{k×n}(ℤ) is non-degenerate if all its k by k minor determinants are non-vanishing; i.e., if for all sequences 1 ≤ α_{1} < ⋯ < α_{k} ≤ n,

1.3  Δ_{α_{1}...α_{k}} = | \begin{matrix}a_{α_{1}}^{1} & … & a_{α_{k}}^{1} \\ ⋮ & & ⋮ \\ a_{α_{1}}^{k} & … & a_{α_{k}}^{k} \end{matrix} | ≠ 0.

Otherwise, Ω is called degenerate. Since we deal with an effective action of T^{k}, we can and shall assume that Ω has rank k.

Proposition 1.4: Let S(Ω) = μ_{Ω}^{−1}(0)/T^{k}(Ω) be a 3-Sasakian quotient space obtained by 3-Sasakian reduction [BGM1] from the standard sphere S^{4n−1} by a torus T^{k}. Then

(i) if Ω is non-degenerate, S(Ω) is an orbifold.

(ii) If Ω is degenerate, then either S(Ω) is singular or it is a manifold obtained by reduction of a lower dimensional sphere S^{4n−4r−1} by a torus T^{k−r}(Ω′) or a finite quotient of such, where 1 ≤ r ≤ k and Ω′ is non-degenerate. (When r = k the quotient is the sphere S^{4n−4k−1}).

(iii) Assuming that Ω is non-degenerate S(Ω) is a smooth manifold if and only if

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, where g denotes the k^{th} determinantal divisor of Ω.

Before giving the proof of this proposition, we recall some terminology from [BGM1]. An Ω ∈ M_{k×n}(ℤ) that satisfies the two conditions of Proposition 1.3 was called admissible in [BGMR1]. Recall also that if g > 1 then the T^{k} action is not effective, and there is a normal subgroup Γ of order g such that the factor group T^{k}/Γ ≃ T^{k} acts effectively. Thus, we may assume without loss of generality that g = 1; i.e., Ω is in reduced form. The subset of reduced admissible matrices in M_{k×n}(ℤ) is denoted by A_{k×n}(ℤ). It is invariant under the action of GL(k, ℤ) × W(Sp(n)). Furthermore, the unordered set {|Δ_{s}|} of absolute values of all k by k minor determinants is an invariant of A_{k×n}(ℤ).

Proof of Proposition 1.4: (i) and the if part of (iii) are proved in [BGMR1]. To prove the only if part of (iii), we show that if the gcd conditions are not satisfied, the quotient is not smooth. Assuming that Ω in reduced form, suppose there are k + 1 columns such that the gcd of the proposition is = r > 1. Then, putting the n−k −1 quaternionic coordinates that do not correspond to the k + 1 columns equal to zero, one shows as in Lemma 3.3 of [BGM2] that the leaf holonomy group of the foliation on μ_{Ω}^{−1}(0) generated by 𝔱_{k} is ℤ_{r}. But this is the local uniformizing group of an orbifold which cannot be reduced to the identity. This proves (iii).

To prove (ii) we make use of the hyperkähler cone C(S) over any complete 3-Sasakian orbifold S [BGM1]. C(S) is a hyperkähler orbifold with its cone metric which admits a non-isometric local homothety. The following lemma is a slight orbifold modification of Lemma 2 Chapter VI, section 3 of [KN].

Lemma 1.5: Let S be a complete 3-Sasakian orbifold and C(S) its hyperkähler cone, then if the hyperkähler metric g_{C} on C(S) extends smoothly to the cone point, it is flat. Moreover, if C(S) (including the cone point) is a smooth manifold, then S is a sphere.

Let Ω ∈ M_{k×n}(ℤ) be degenerate of rank k. By using an element of GL(k, ℤ) if necessary we can make a row with the maximal number of zero entries the last row.

We reduce our considerations to the case where all rows have at least two non-vanishing entries. If the last row has only one non-vanishing entry it must be a 1, and using the Weyl group W(Sp(n)) if necessary we can make this the last column. One now easily sees by solving the moment map equations for the last row, that this reduces S(Ω) to a reduction of S^{4n−5} by the k − 1 torus T^{k−1}(Ω′) with Ω′ equal to Ω with its last row and column removed. We continue in this way until we remove all rows with only one non-vanishing entry (up to a GL(k, ℤ) transformation) and the corresponding columns. This gives a Ω_{1} ∈ M_{k−r,n−r}(ℤ) with 0 ≤ r ≤ k. Note that if r = k we have reduced S^{4n−1} to S^{4n−4k−1} and we are done. So we can assume that Ω is not equivalent to any matrix with only one non-vanishing entry in any row.

Next consider the cone C(S(Ω)) where we can assume that every representative in the GL(k, ℤ) equivalence class [Ω] has at least two non-vanishing entries in every row. So Ω can be put in Hermite normal form:

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

where a_{l_{j}}^{j} ≠ 0, n − l_{k} ≥ 2, and if i < j then l_{i} < l_{j}. We can assume that the row with the most zeroes is the last row and by using permutations of the Weyl group W(Sp(n)) if necessary, that all zero entries are the first l_{k} − 1 entries with n − 2 ≥ l_{k} ≥ k + 1; i.e., all a_{j}^{k} ≠ 0 for l_{k} ≤ j ≤ n. Moreover, since Ω is degenerate l_{k} ≥ k + 1.

We consider the hyperkähler quotient reduction by the circle defined by the last row. We obtain ℍ^{l_{k}−1} × C(S(a_{l_{k}}^{k}, ⋯, a_{n}^{k})), where l_{k} − 1 ≥ k. Now the remaining components of the moment map induce a moment map ν : ℍ^{l_{k}−1} ×C(S(a_{l_{k}}^{k}, ⋯, a_{n}^{k}))⟶𝔱_{k−1} ⊗ℝ^{3}. Let Ω_{1} denote Ω with its last row removed, and let Ω_{2} ∈ M_{k−1×l_{k}−1}(ℤ) denote the matrix formed from the first l_{k} − 1 columns of Ω_{1}. Since l_{i} < l_{j} it follows from 1.6 that Ω_{2} has rank k − 1. So Ω_{2} induces a linear map L_{Ω_{2}} : 𝔱_{l_{k}−1}⟶𝔱_{k−1} whose kernel K has dimension l_{k} − k ≥ 1. Thus, if Σ is the subvariety of ν^{−1}(0) ⊂ ℍ^{l_{k}−1} × C(S(a_{l_{k}}^{k}, ⋯, a_{n}^{k})) obtained by choosing the cone point of the second factor, we have dim Σ ≥ 4. This subvariety is invariant under the T^{k−1}(Ω_{1}) action, so if we choose a point s ∈ Σ, the entire orbit T^{k−1}(Ω_{1}) · s lies in Σ. So if s is a singular point of the variety ν^{−1}(0) in the sense that its Zariski tangent space has dimension larger than its dimension, then this inequality passes to the space of orbits. So the quotient space μ^{−1}(0)/T^{k}(Ω) = ν^{−1}(0)/T^{k−1}(Ω_{1}) is singular.

On the other hand if s ∈ Σ is a Zariski smooth point of ν^{−1}(0), there is a sequence q_{i} ∈ ν^{−1}(0) converging to s ∈ Σ such that for i ≠ j q_{i} and q_{j} lie on distinct T^{k−1}(Ω_{1}) orbits. Furthermore, the T^{k−1}(Ω_{1}) action is locally free near s, so locally the projection π : ν^{−1}(0)⟶μ^{−1}(0)/T^{k}(Ω) is an orbifold Riemannian submersion with respect to a unique hyperkähler metric g on the quotient μ^{−1}(0)/T^{k}(Ω). From the construction the local uniformizing groups are isometries of the induced metric on ν^{−1}(0). It follows that the metric g on μ^{−1}(0)/T^{k}(Ω) is isometric to the horizontal component g_{H} of the metric on ν^{−1}(0). If there are more than two non-zero entries in the last row of Ω, that is if l_{k} < n − 1, it follows from [BGM1] that C(S(a_{l_{k}}^{k}, ⋯, a_{n}^{k})) is not flat. So Lemma 1.5 implies that g_{H} does not extend smoothly to s. Thus, g is not smooth at π(s). On the other hand if l_{k} = n − 1, then S = S^{3}/ℤ_{a_{n−1}^{k}+a_{n}^{k}}, so ℍ^{l_{k}−1} × C(S(a_{n−1}^{k}, a_{n}^{k})) is covered by the flat hyperkähler vector space ℍ^{n−2} × ℍ. Furthermore, since the deck transformations are hyperkähler isometries, the 3-Sasakian quotient space is just a finite quotient of a reduction by lower dimensional sphere. This completes the proof of Proposition 1.4.

By Proposition 1.4 we need only consider non-degenerate Ω. So let Ω ∈ A_{k×n}(ℤ). Since Ω is reduced there is a k by k minor determinant that is odd. By permuting columns if necessary this minor can be taken to be the first k columns. Now consider the mod 2 reduction M_{k×n}(ℤ)⟶M_{k×n}(ℤ_{2}). We have the following commutative diagram

1.7  \begin{matrix}GL(k, ℤ) × M_{k×n}(ℤ) & ⟶ & M_{k×n}(ℤ) \\ ↓ & & ↓ \\ GL(k, ℤ_{2}) × M_{k×n}(ℤ_{2}) & ⟶ & M_{k×n}(ℤ_{2}). \end{matrix}

Let Ω̃ ∈ A_{k×n}(ℤ_{2}) denote the mod 2 reduction of Ω ∈ A_{k×n}(ℤ). Since the first k by k minor determinant of Ω is odd, the mod 2 reduction of this minor in Ω̃ is invertible. Thus, we can use the GL(k, ℤ_{2}) action to put Ω̃ in the form

1.8  Ω̃ = ( \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} )

with a_{j}^{i} ∈ ℤ_{2}.

Lemma 1.9: The set A_{k×n}(ℤ) is empty for n > k + 2 and k > 4.

Proof: We analyze the second admissibility condition. Since Ω is reduced the second condition of Proposition 1.4 implies

1.10  (Δ̃_{α_{2}···α_{k+1}}, ..., Δ̃_{α_{1}···α̂_{s}···α_{k+1}}, ..., Δ̃_{α_{1}···α_{k}}) ≠ (0, ⋯, 0)

for all sequences 1 ≤ α_{1} < ⋯ < α_{s} < ⋯ < α_{k+1} ≤ n, where Δ̃ denotes the mod 2 reduction of Δ. This condition says that every subset of k + 1 columns of Ω̃ spans (ℤ_{2})^{k}, or equivalently, that every k by k + 1 submatrix of Ω̃ has rank k. By considering k − 1 of the first k columns and 2 of last n − k columns, this condition implies

1.11  (a_{l}^{j}, a_{m}^{j}) ≠ (0, 0)

for all j = 1, ⋯, k, and k + 1 ≤ l < m ≤ n. Similarly, by considering k − 2 of the first k columns and 3 of last n − k columns 1.10 implies

1.12  ( \begin{matrix}a_{l}^{i} & a_{m}^{i} & a_{r}^{i} \\ a_{l}^{j} & a_{m}^{j} & a_{r}^{j} \end{matrix} ) ≠ ( \begin{matrix}1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix} ) ,  ( \begin{matrix}a_{l}^{i} & a_{m}^{i} & a_{r}^{i} \\ a_{l}^{j} & a_{m}^{j} & a_{r}^{j} \end{matrix} ) ≠ ( \begin{matrix}0 & 1 & 1 \\ 0 & 1 & 1 \end{matrix} ) ,

where the last inequality is understood to be up to column permutation.

Hence, it follows that, up to column and row permutations, that any four triples of the last n − k columns of an admissible Ω̃ must have the form

1.13  ( \begin{matrix}1 & 1 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} ) .

So we see that we cannot add another row without violating the above conditions. It follows that k ≤ 4.

We can also use our analysis to prove:

Lemma 1.14: Let Ω ∈ A_{k×n}(ℤ) and let k > 1, then n − k ≤ 4.

Proof: By Lemma 1.9 we need only check this for k = 2, 3, 4. We consider the case k = 2 as the cases k = 3, 4 are similar. As before, let Ω̃ denote the mod 2 reduction of Ω. Then by 1.12, up to GL(2, ℤ) transformations and column permutations, any Ω ∈ A_{2×6}(ℤ) will have a reduced matrix of the form

Ω̃ = ( \begin{matrix}1 & 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 1 & 1 \end{matrix} ) .

One cannot add another column to this matrix since 1.11 implies that this must be ( \begin{matrix}1 \\ 1 \end{matrix} )

which is forbidden by 1.12.

Lemmas 1.9 and 1.15 now imply Theorem A.

§2. Some Examples in Dimension 11 and 15

In [BGM1] we constructed 3-Sasakian manifolds obtained by toral reduction for k = 1 for every n and in [BGMR1] we constructed 3-Sasakian 7-manifolds with arbitrary k. Thus, to show that Theorem A is sharp we need only construct 11 and 15 dimensional manifold examples when k = 2, 3, 4. These last constructions give explicit examples of non-regular 3-Sasakian manifolds for which the Betti number relations given in [GS] for regular 3-Sasakian manifolds fail as Bielawski [Bi1] showed that the relations can hold for a 3-Sasakian toral quotient if and only if k = 1. Some of the calculations performed in this section were tedious, especially for dimension 15, and were done with the aid of MAPLE symbolic manipulation program.

We begin by assuming that k = 4. We take Ω to be in the special form

2.1  Ω = (𝕀_{4} A),

where 𝕀_{4} denotes the 4 by 4 identity matrix and A ∈ M_{4×3}(ℤ) or A ∈ M_{4×4}(ℤ). It is straightforward to show Lemma 2.2: Let Ω be as above.

(i) The first admissibility condition implies that all the entries (a_{j}^{i}) of the matrix A are non-vanishing and that all minor determinants are non-vanishing.

(iia) If A ∈ M_{4×3}(ℤ) then the second admissibility condition is equivalent to

gcd(a_{l}^{j}, a_{m}^{j}) = 1

and

gcd(| \begin{matrix}a_{l}^{i} & a_{m}^{i} \\ a_{l}^{j} & a_{m}^{j} \end{matrix} |, | \begin{matrix}a_{l}^{i} & a_{r}^{i} \\ a_{l}^{j} & a_{r}^{j} \end{matrix} |, | \begin{matrix}a_{m}^{i} & a_{r}^{i} \\ a_{m}^{j} & a_{r}^{j} \end{matrix} |) = 1

for all 1 ≤ i < j ≤ 4, and 1 ≤ l < m < r ≤ 3. (iib) If A ∈ M_{4×4}(ℤ) then the second admissibility condition is equivalent to the conditions

in (iia) for all 1 ≤ i < j ≤ 4, and 1 ≤ l < m < r ≤ 4, together with

gcd(M_{i1}, ⋯, M_{i4}, det A) = 1

for i = 1 ⋯ 4, where M_{ij} denotes ij minor determinant of A. These equations are still somewhat complicated in their full generality, so we shall

present two three parameter families of solutions, namely

2.3  A_{1} = ( \begin{matrix}1 & 1 & 1 \\ 2 & 1 & 1 + 2^{l} \\ 1 & 16 & 1 + 2^{m} \\ −1 & 3 & 2c \end{matrix} ) ,  A_{2} = ( \begin{matrix}1 & 1 & 1 & 2 \\ 2 & 1 & 1 + 2^{l′} & −1 \\ 1 & 16 & 1 + 2^{2n} & 3 \\ −1 & 3 & 2c′ & −1 \end{matrix} ) ,

where l, l′, m, n ∈ ℤ^{+}, and c, c′ ∈ ℤ. Proposition 2.4: Let Δ = 2(31c + 6 + 19 · 2^{l−1} − 7 · 2^{m−1}).

(i) Ω_{1} = ( 𝕀_{4} A_{1} ) is admissible if and only if c ≠ 0 and is not divisible by 3, and Δ ≠ 0 and is not divisible by 7, 19 nor 31.

(ii) Ω_{2} = ( 𝕀_{4} A_{2} ) is admissible if and only if c and all minor determinants of A_{2} are nonvanishing, and c′ ≢ 0 (mod 3), l′ ≢ 0 (mod 4), c′ ≢ 5 (mod 7), and 11, 19, 37, 71 do not divide det A = 19 · 2^{2n} − 63 − 148c′ − 11 · 2^{l′}, and the following conditions hold:

gcd(3, 4c′ + 2^{l′} + 1, 2c′ − 2^{l′} − 1) = 1
gcd(7, 2^{2n+1} − 2^{l′} + 1, 3 · 2^{l′} + 2^{2n} + 4) = 1
gcd(19, 2^{2n} − 2^{l′+4} − 15, 3 · 2^{l′} + 2^{2n} + 4) = 1
gcd(25, 32c′ − 3 · 2^{2n} − 3, 6c′ + 2^{2n} + 1) = 1.

The conditions in this proposition guarantee that the quotient spaces denoted by S(c, l, m) and S(c′, l′, n) are smooth manifolds of dimension 11 and 15, respectively, which by [Bi1] have b_{2} = 4. One can then obtain smooth manifolds with b_{2} = 3 or 2 by simply removing one or two rows from the matrices A_{1} or A_{2} (together with the corresponding columns in 𝕀_{4} in Ω_{1} or Ω_{2}). It is routine to verify that the three parameter infinite family given by

2.5

c ≡ 14 (mod 21),
l ≢ 1 (mod 5),
m ≢ α(c) (mod 18),

where 2^{α(c)} = 22(31c + 6) (mod 18) satisfies the conditions in (i) of Proposition 2.4. This gives examples in dimension 11. (Notice that as 2 is a primitive root of 19 the equation defining α(c) has a unique solution (mod 18) for each value of c.) Similarly, it is straightforward to verify that the infinite family given by

2.6  \begin{aligned}c′ & = 2, \\ l′ & = 1, \\ n & ≡ 21 (mod 90), \end{aligned}

satisfies the conditions (ii) of Proposition 2.4. We have arrived at:

Theorem 2.7: There exist smooth 3-Sasakian manifolds in dimensions 11 and 15 with second Betti number b_{2} equal to 2, 3 and 4.

An important corollary of this theorem is that the linear Betti number relations obtained by Galicki and Salamon [GS] for regular 3-Sasakian manifolds fail in the nonregular case. Bielawski [Bi1] showed that these relations can hold for a 3-Sasakian toral quotient if and only if k = 1. Since the Betti number relations in [GS] hold only for dimensions greater than or equal to 11, Theorem 2.7 exhibits the first examples of nonregular 3-Sasakian manifolds where these relations fail. The examples occur in dimensions 11 and 15 where the Betti number relations [GS] are particularly simple, namely b_{2}(S) = b_{4}(S) for dim S = 11, and b_{2}(S) = b_{6}(S) for dim S = 15.

An important problem not considered in this note is the equivalence problem, that is, given two 3-Sasakian manifolds (S(Ω), g, ξ^{a}) and (S(Ω′), g′, ξ′^{a}) when is there a diffeomorphism f : S(Ω)⟶S(Ω′) such that f^{∗}g′ = g and f_{∗}ξ^{a} = ξ′^{a}. The best approach to this problem is through combinatorial geometry. This is currently under study. In this regard, Bielawski and Dancer [BD] have given a combinatorial approach to hyperkähler reductions by a torus, and Bielawski [Bi2] has recently proven a Delzant-type theorem [Gui] for both hyperkähler and 3-Sasakian toral reductions. Explicitly, Bielawski proves that any compact 3-Sasakian manifold S of dimension 4n − 1 having a locally smooth effective action of an n-torus T^{n} preserving the 3-Sasakian structure is T^{n} equivariantly homeomorphic to a 3-Sasakian quotient of the sphere S^{4d−1} by a subgroup of T^{d}. If this result could be strengthened to a diffeomorphism, then it would follow from this and Theorem A that there would be no compact 3-Sasakian manifolds of dimension 4n − 1 with n > 4 and b_{2} > 1 whose automorphism group of 3-Sasakian isometries contain an n-torus T^{n}.

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Department of Mathematics and Statistics December 1996 University of New Mexico revised June 1997 Albuquerque, NM 87131 email: cboyer@math.unm.edu, galicki@math.unm.edu mann@math.unm.edu

Max-Planck-Institut für Mathematik Gottfried Claren Strasse 26 53225 Bonn, Germany email: galicki@mpim-bonn.mpg.de