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

Abstract

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