Mathematics – Algebraic Topology
Scientific paper
2012-04-18
Mathematics
Algebraic Topology
27 pages
Scientific paper
We introduce the homotopy fixed points of the $\mathbb{Z}/2$ equivariant cohomology theory $E\mathbb{R}(n)$ as a computational tool. These equivariant cohomology theories were first introduced by Hu and Kriz. Kitchloo and Wilson have used the homotopy fixed points spectrum ER(2) to deduce certain non-immmersion results for real projective spaces. ER(2) is a $2^{n+2}(2^n-1)$-periodic spectrum. The key result to use is the existence of a stable cofibration $\Sigma^{\lambda(n)}ER(n) \rightarrow ER(n) \rightarrow E(n)$ connecting the real Johnson-Wilson spectrum with the classical one. In this paper we give a proof of the existence of the cofibration sequence by showing that the Tate spectrum associated to $E\mathbb{R}(n)$ is trivial. We also 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 space $RP^{16K+9}$. This enables us to solve certain non-immersion problems of projective spaces using obstructions in ER(2)-cohomology.
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