Hu and Kriz construct the real Johnson-Wilson spectrum, ER(n), which is 2n+2(2n-1)-periodic, from the 2(2n-1)-periodic spectrum E(n). ER(1) is just KO(2) and E(1) is just KU(2). We compute ER(n)*(RP∞) and set up a Bockstein spectral sequence to compute ER(n)*(-) from E(n)*(-). We combine these to compute ER(2)*(RP2n) and use this to get new nonimmersions for real projective spaces. Our lowest dimensional new example is an improvement of 2 for RP48.
Homology, Homotopy and Applications, Vol. 10 (2008), No. 3, pp.223-268.