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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Authors: Donald M. Davis and Mark Mahowald
Journal: Trans. Amer. Math. Soc. 236 (1978), 361-383
MSC: Primary 57D40
DOI: https://doi.org/10.1090/S0002-9947-1978-0646070-X
MathSciNet review: 0646070
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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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Keywords: Immersions of projective spaces, obstruction theory, Adams spectral sequence
Article copyright: © Copyright 1978 American Mathematical Society