The Arf-Kervaire Invariant via QX

Abstract

The objective of this chapter is to study π_i(∑^∞X) by identifying it with π_i(QX) where QX is the space introduced in Chapter 1 § 1.5.8. Since we are interested in the two-primary part of the stable homotopy groups of spheres we would like to concentrate on the case when X = S⁰. However, for technical reasons concerned with the mod 2 homology of QX we have to consider the case when X is connected. Fortunately, by virtue of the Kahn-Priddy theorem of Chapter 1 Theorem 1.5.10, it is no restriction to consider only connected X since we can study the case when X = ℝℙ^∞