Smooth Manifolds with Infinite Fundamental Group Admitting No Real Projective Structure

It is an important question whether it is possible to put a geometry on a given manifold or not. It is well known that any simply connected closed manifold admitting a real projective structure must be a sphere. Therefore, any simply connected manifold M which is not a sphere (dim M ≥ 4) does not admit a real projective structure. Cooper and Goldman gave an example of a 3-dimensional manifold not admitting a real projective structure and this is the first known example. In this article, by generalizing their work, we construct a manifold Mn with the infinite fundamental group Z2 ∗ Z2, for any n ≥ 4, admitting no real projective structure