The second real Johnson-Wilson theory and nonimmersions of $RP^n$

摘要

Hu and Kriz construct the real Johnson-Wilson spectrum, $ER(n)$, which is $2^{n+2}(2^n-1)$-periodic, from the $2(2^n-1)$-periodic spectrum $E(n)$. $ER(1)$ is just $KO_{(2)}$ and $E(1)$ is just $KU_{(2)}$. We compute $ER(n)^*(RP^{infty})$ and set up a Bockstein spectral sequence to compute $ER(n)^*(-)$ from $E(n)^*(-)$. We combine these to compute $ER(2)^*(RP^{2n})$ and use this to get new nonimmersions for real projective spaces. Our lowest dimensional new example is an improvement of 2 for $RP^{48}$.