Stable Homotopy Around the Arf-Kervaire Invariant pp 155-197 | Cite as

# Upper Triangular Technology and the Arf-Kervaire Invariant

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## Abstract

The objective of this chapter is to give applications of the upper triangular results of Chapter 3 and Chapter 5. § 1 gives some background concerning Adams operations, recapitulates the results and proves the main one (Theorem 8.1.2) assuming the results of § 4. § 2 contains technical computations including some multiplicative relations in the collapsed Adams spectral sequences which will be used later. The most important corollary of this is Proposition 8.2.10 which implies that the maps *l*^{ k,k-1 } corresponding to the super-diagonal entries in the matrix for 1∧ ψ^{3} induce almost injective homomorphisms on the connective K-theory groups which we shall study (see Proposition 8.3.7). §3 analyses the Adams spectral sequences associated with the summands in the Mahowald decomposition (see Chapter 3) of (*bu* ∧ *bo*)_{*}(*X*) for the examples which will be used in § 4. § 4 applies the upper triangular technology to the case when *X* is a projective space and the mapping cone of a homotopy class \(
\Theta _{8m - 2} \in \pi _{8m - 2} (\sum ^\infty \mathbb{R}\mathbb{P}^{_{8m - 2} } )
\). The central results in the latter case are Theorem 8.4.6, Theorem 8.4.7 and Corollary 8.4.9.

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