On the ER(2)-cohomology of some odd-dimensional projective spaces

Kitchloo and Wilson have used the homotopy fixed points spectrum ER(2) of the classical complex-oriented Johnson-Wilson spectrum E(2) to deduce certain non-immersion results for real projective spaces. ER(n) is a 2n+2(2n−1)-periodic spectrum. The key result to use is the existence of a stable cofibration Σλ(n)ER(n)→ER(n)→E(n) connecting the real Johnson-Wilson spectrum with the classical one. The value of λ(n) is 22n+1−2n+2+1. We extend Kitchloo-Wilsonʼs results on non-immersions of real projective spaces by computing the second real Johnson-Wilson cohomology ER(2) of the odd-dimensional real projective spaces RP16K+9. This enables us to solve certain non-immersion problems of projective spaces using obstructions in ER(2)-cohomology.