On the \(ER(2)\)-cohomology of some odd-dimensional projective spaces. (English) Zbl 1408.55002
Summary: N. Kitchloo and W. S. Wilson [Homology Homotopy Appl. 10, No. 3, 223–268 (2008; Zbl 1160.55002)] have used the homotopy fixed points spectrum \(ER(2)\) of the classical complex-oriented Johnson-Wilson spectrum \(E(2)\) to deduce certain non-immersion results for real projective spaces. \(ER(n)\) is a \(2^{n+2}(2^n-1)\)-periodic spectrum. The key result to use is the existence of a stable cofibration \(\Sigma^{\lambda(n)}ER(n)\to ER(n)\to E(n)\) connecting the real Johnson-Wilson spectrum with the classical one. The value of \(\lambda(n)\) is \(2^{2n+1}-2^{n+2}+1\). We 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 spaces \(\mathbb{R}P^{16K+9}\). This enables us to solve certain non-immersion problems of projective spaces using obstructions in \(ER(2)\)-cohomology.
MSC:
| 55N20 | Generalized (extraordinary) homology and cohomology theories in algebraic topology |
| 55N22 | Bordism and cobordism theories and formal group laws in algebraic topology |
| 55N91 | Equivariant homology and cohomology in algebraic topology |
| 57R42 | Immersions in differential topology |
Citations:
Zbl 1160.55002References:
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