The immersion conjecture for 𝑅𝑃^{8𝑙+7} is false

Davis, Donald M.; Mahowald, Mark

doi:10.1090/S0002-9947-1978-0646070-X

The immersion conjecture for $RP^{8l+7}$ is false
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by Donald M. Davis and Mark Mahowald PDF

Trans. Amer. Math. Soc. 236 (1978), 361-383 Request permission

Abstract:

Let $\alpha (n)$ denote the number of l’s in the binary expansion of n. It is proved that if $n \equiv 7$ (8), $\alpha (n) = 6$, and $n \ne 63$, then ${\mathbf {R}}{P^n}$ can be immersed in ${{\mathbf {R}}^{2n - 14}}$. This provides the first counterexample to the well-known conjecture that the best immersion is in ${{\mathbf {R}}^{2n - 2\alpha (n) + 1}}$ (when $\alpha (n) \equiv 1$ or $2 \bmod 4$). The method of proof is obstruction theory.

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Semi-principal bundles and stable non-orientable obstruction theory

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