In this paper we show that there exist mod 2 obstructions to the smoothness of 3-Sasakian reductions of spheres. Specifically, ifS is a smooth 3-Sasakian manifold obtained by reduction of the 3-Sasakian sphereS 4n−1 by a torus, and if the second Betti numberb 2(S)≥2 then dimS=7, 11, 15, whereas, ifb 2 (S)≥5 then dimS=7. We also show that the above bounds are sharp, in that we construct explicit examples of 3-Sasakian manifolds in the cases not excluded by these bounds.
Mathematics Subject Classification (1991)53C25 57S25
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