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# Homology, Homotopy and Applications

## Volume 10 (2008)

### Number 3

### Proceedings of a Conference in Honor of Douglas C. Ravenel and W. Stephen Wilson

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

Pages: 223 – 268

DOI: https://dx.doi.org/10.4310/HHA.2008.v10.n3.a11

#### Authors

#### Abstract

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 $RP^{48}$.

#### Keywords

real projective space, nonimmersions, Johnson-Wilson theories

#### 2010 Mathematics Subject Classification

55N20, 55N91, 55T25, 57R42

Published 1 January 2008