Yes, Eschenburg constructed an infinite family of simply connected 7-dimensional examples and proved that many of them are not homotopy equivalent to homogeneous spaces. His examples are biquotients however. In fact the only known examples of closed positively curved manifolds are biquotients or spaces of cohomogeneity one and above dimension 13 only homogeneous examples (and space forms) are known.

Yes, Eschenburg constructed an infinite family of simply connected 7-dimensional examples and proved that many of them are not homotopy equivalent to homogeneous spaces. His examples are biquotients however. In fact the only known examples of closed positively curved manifolds are biquotients or spaces of cohomogeneity one and above dimension 13 only homogeneous examples (and space forms) are known.

Yes, Eschenburg constructed an infinite family of 7-dimensional examples and proved that many of them are not homotopy equivalent to homogeneous spaces. His examples are biquotients however. In fact the only known examples of closed positively curved manifolds are biquotients or spaces of cohomogeneity one and above dimension 13 only homogeneous examples are known.

Yes, Eschenburg constructed an infinite family of simply connected 7-dimensional examples and proved that many of them are not homotopy equivalent to homogeneous spaces. His examples are biquotients however. In fact the only known examples of closed positively curved manifolds are biquotients or spaces of cohomogeneity one and above dimension 13 only homogeneous examples (and space forms) are known.