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

Authors

Nitu Kitchloo (Department of Mathematics, University of California at San Diego)

W. Stephen Wilson (Department of Mathematics, Johns Hopkins University, Baltimore, Maryland, U.S.A.)

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

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