Hurewicz Images, BP-theory and the Arf-Kervaire Invariant

Abstract

The objective of this chapter is to prove the conjecture of [30] in its original form, as stated in Chapter 1, Theorem 1.8.10. This result is reiterated in this chapter as Theorem 7.2.2. The conjecture states that an element of $\pi _{2^{n + 1} - 2} (\sum ^\infty \mathbb{R}\mathbb{P}^\infty )$ corresponds under the Kahn-Priddy map to the class of a framed manifold of Arf-Kervaire invariant one if and only if it has a non-zero Hurewicz image in ju-theory. I shall prove this result in three ways (§§7.2.3-7.2.5)-one of which uses an excursion into BP-theory.