Compact 3-Sasakian 7-manifolds with arbitrary second Betti number

Boyer, Charles P.; Galicki, Krzysztof; Mann, Benjamin M.; Rees, Elmer

doi:10.1007/s002220050207

Cited by 50 publications

(102 citation statements)

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“…Up to coverings they are all obtainable by taking 3-Sasakian quotients of S 4n−1 by a torus T k , k = n − 2. See [8,5] for more details. Let (S, g) be a 3-Sasakian manifold.…”

Section: -Sasakian Reductionmentioning

confidence: 99%

“…We can define F 1,2 ⊂ CP 2 × (CP 2 ) * by And the complex contact structure is given by θ = q i dp i − p i dq i . Given [a, b] ∈ CP 1 the one parameter group (e iaτ , e ibτ ) induces the holomorphic vector field W τ ∈ Γ(T 1,0 F 1,2 ) and the quadratic divisor X τ = (θ(W τ )) given by [19] and further considered in [8]. These are circle quotients of HP 2 .…”

Section: = Cpmentioning

confidence: 99%

“…And there is a unique toric quasi-regular Sasaki-Einstein 5-manifold associated to every simply connected toric 3-Sasakian 7-manifold. Using 3-Sasakian reduction as in [8,7] an infinite series of examples is constructed of each odd second Betti number. They are all diffeomorphic to #kM∞, where M∞ ∼ = S 2 × S 3 , for k odd.…”

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confidence: 99%

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Some Examples of Toric Sasaki—Einstein Manifolds

Coevering

2009

Riemannian Topology and Geometric Structures on Manifolds

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Abstract. A series of examples of toric Sasaki-Einstein 5-manifolds is constructed which first appeared in the author's Ph.D. thesis [40]. These are submanifolds of the toric 3-Sasakian 7-manifolds of C. Boyer and K. Galicki. And there is a unique toric quasi-regular Sasaki-Einstein 5-manifold associated to every simply connected toric 3-Sasakian 7-manifold. Using 3-Sasakian reduction as in [8,7] an infinite series of examples is constructed of each odd second Betti number. They are all diffeomorphic to #kM∞, where M∞ ∼ = S 2 × S 3 , for k odd. We then make use of the same framework to construct positive Ricci curvature toric Sasakian metrics on the manifolds X∞#kM∞ appearing in the classification of simply connected smooth 5-manifolds due to Smale and Barden. These manifolds are not spin, thus do not admit Sasaki-Einstein metrics. They are already known to admit toric Sasakian metrics (cf. [9]) which are not of positive Ricci curvature. We then make use of the join construction of C. Boyer and K. Galicki first appearing in [6], see also [9], to construct infinitely many toric Sasaki-Einstein manifolds with arbitrarily high second Betti number of every dimension 2m + 1 ≥ 5. This is in stark contrast to the analogous case of Fano manifolds in even dimensions. IntroductionA new series of quasi-regular Sasaki-Einstein 5-manifolds is constructed. These examples first appeared in the author's Ph.D. thesis [40]. They are toric, and arise as submanifolds of toric 3-Sasakian 7-manifolds. Applying 3-Sasakian reduction to torus actions on spheres C. Boyer, K. Galicki, et al [8] produced infinitely many toric 3-Sasakian 7-manifolds. More precisely, one has a 3-Sasakian 7-manifold S Ω for each integral weight matrix Ω satisfying some conditions to ensure smoothness. This produces infinitely many examples of each b 2 (S Ω ) ≥ 1. A result of D. Calderbank and M. Singer [12] shows that, up to finite coverings, this produces all examples of toric 3-Sasakian 7-manifolds. Associated to each S Ω is its twistor space Z, a complex contact Fano 3-fold with orbifold singularities. The action of T 2 complexifies to T 2 C = C * × C * acting on Z. Furthermore, if L is the line bundle of the complex contact structure, the action defines a pencilwhere t C is the Lie algebra of T 2 C . We determine the structure of the divisors in E. The generic X ∈ E is a toric variety with orbifold structure whose orbifold anti-canonical bundle K −1 X is positive. The total space M of the associated S 1 orbifold bundle to K X has a natural Sasaki-Einstein structure. Associated to any [3]) that M is diffeomorphic to #k(S 2 × S 3 ) where b 2 (M ) = k. We have the following: Theorem 1.1. Associated to every simply connected toric 3-Sasakian 7-manifold S is a toric quasi-regular Sasaki-. This gives an invertible correspondence. That is, given either X or M in diagram 1.1 one can recover the other spaces with their respective geometries.This gives an infinite series of quasi-regular Sasaki-Einstein structures on #m(S 2 × S 3 ) for every odd m ≥ 3.In section...

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“…Up to coverings they are all obtainable by taking 3-Sasakian quotients of S 4n−1 by a torus T k , k = n − 2. See [8,5] for more details. Let (S, g) be a 3-Sasakian manifold.…”

Section: -Sasakian Reductionmentioning

confidence: 99%

Section: = Cpmentioning

confidence: 99%

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Some Examples of Toric Sasaki—Einstein Manifolds

Coevering

2009

Riemannian Topology and Geometric Structures on Manifolds

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“…(That is, if we enlarge C the property fails.) Obviously C = [k + 2], and C = ∅, we may choose t ∈ T [k+2] such that no element of t lies in C (2) or ([k + 2] − C) (2) . Since P | ( {p,q}∈t |∆ pq |), there exists {p, q} ∈ t such that P | ∆ pq .…”

Section: The Group G ωmentioning

confidence: 99%

“…Fix t ∈ (A (2) ) (n−1) − T A . Since t is not a tree, the corresponding graph on A is disconnected, and we may write A = B ∪ C with B, C nonempty and disjoint, and t = u ∪ v for u ⊂ B (2) and v ⊂ C (2) . Without loss of generality, B = {a 1 , .…”

Section: Lemma 25 the Group H 4 (Bt K ) Is Spanned By The Classesmentioning

confidence: 99%

The topology of certain 3-Sasakian 7-manifolds

Hepworth

2007

Math. Ann.

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Abstract. We calculate the integer cohomology ring and stable tangent bundle of a family of compact, 3-Sasakian 7-manifolds constructed by Boyer, Galicki, Mann, and Rees. Previously only the rational cohomology ring was known. The most important part of the cohomology ring is a torsion group that we describe explicitly and whose order we compute. There is a surprising connection with the combinatorics of trees.

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