Higher Real K-theory cohomology of Complex Projective spaces

I will present some results on $KO^*(CP^n)$, $EO_{p-1}^*(CP^n)$ for odd primes $p$, using homotopy fixed point spectral sequence. Using these results we will discuss the existence of exotic structures on complex projective spaces, existence of smooth $S^1$, $S^3$ actions on exotic spheres, existence of complex hyperbolic manifolds with certain curvature properties.