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 2(n+2) (2(n) - 1)-periodic spectrum. The key result to use is the existence of a stable cofibration Sigma(ER)-E-lambda(n)(n) -> ER(n) -> E(n) connecting the real Johnson-Wilson spectrum with the classical one. The value of lambda(n) is 2(2n+1) 2(n+2) + 1. We extend Kitchloo-Wilson's 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. (C) 2013 Elsevier B.V. All rights reserved.