On the -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 2 n + 2 ( 2 n − 1 ) -periodic spectrum. The key result to use is the existence of a stable cofibration Σ λ ( n ) ER ( n ) → ER ( n ) → E ( n ) connecting the real Johnson–Wilson spectrum with the classical one. The value of λ ( n ) is 2 2 n + 1 − 2 n + 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 RP 16 K + 9 . This enables us to solve certain non-immersion problems of projective spaces using obstructions in ER ( 2 ) -cohomology.