## A note on smooth toral reductions of spheres.(English)Zbl 0913.53020

The main result of this paper is the following Theorem. 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 neither 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 $$\dim S=7$$.
This result extends previous results of the authors and E. G. Rees [Invent. Math. 131, 321-344 (1998; Zbl 0901.53033); Balkan J. Geom. Appl. 1, 1-7 (1996; Zbl 0889.53030)].

### MSC:

 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 57S25 Groups acting on specific manifolds

### Keywords:

3-Sasakian structure; reduction; orbifold; Betti numbers

### Citations:

Zbl 0901.53033; Zbl 0889.53030
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