A note on smooth toral reductions of spheres

Abstract

In this paper we show that there exist mod 2 obstructions to the smoothness of 3-Sasakian reductions of spheres. Specifically, if S is a smooth 3-Sasakian manifold obtained by reduction of the 3-Sasakian sphere S 4 n -1 by a torus, and if the second Betti number b2(S) ≥ 2 then dim S = 7, 11, 15, whereas, if b2(S) ≥ 5 then dim S = 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.