Special Holonomy on Special Spaces

We characterise simply-connected biquotients which potentially admit metrics of holonomy G_2. We prove that there are at most three real homotopy types of rationally elliptic such manifolds—all of them being formal. In the course of this examination we classify rationally elliptic homotopy types and characterise 7-dimensional simply-connected biquotients from a rational point of view. Moreover, we also investigate further manifolds of special holonomy, like manifolds of holonomy Spin(7) or Sp(n)\Sp(1) in special situations provided by rational ellipticity or geometric formality.