A NOTE ON SMOOTH TORAL REDUCTIONS OF SPHERES

In this paper we show that there exist mod 2 obstructions to the smoot hness of 3-Sasakian reductions of spheres. Specifically, if S is a smo oth 3-Sasakian manifold obtained by reduction of the 3-Sasakian sphere S4n-1 by a torus, and if the second Betti number b(2)(S) greater than or equal to 2 then dim S = 7, 11, 15, whereas, if b(2)(S) greater tha n or equal to 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.