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 $(\dim M \geq 4)$ 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 $M^n$ with the infinite fundamental group $\mathbb{Z}_2 \ast \mathbb{Z}_2$, for any $n\geq 4$, admitting no real projective structure.