Factoring 2 k on stunted projective spectra and the root invariant

Abstract

Let x∈π ∗S 0 . In this paper we estimate the root invariant of 2 w x in terms of the root invariant of x. For a stunted projective space RP 2 n 2 k−1 , we use Toda's calculation of the smallest integer ε( n, k), such that 2 ε( n, k) times the identity map on RP 2 n 2 k−1 is null homotopic. To calculate the root invariant, defined by Mahowald, we find factorizations of 2 ε( n, k)−1 times the identity map on RP 2 n 2 k−1 for small n− k and use these factorizations to estimate R(2 ε−1 x) in terms of R( x). In some cases, it is shown there are common elements in R(2 ε−1 x) and certain Toda brackets. For instance, we prove the following theorem. Theorem. For x :S r−1→S −1 , f∈ R( x), (i) If |R(x)|−|x|≡1 ( mod 2) , then 〈 f,2, α 4 k 〉∩ R(2 4 k x)≠∅, or R(2 4 k x) is in a higher dimension than 〈 f,2, α 4 k 〉. (ii) If |R(x)|−|x|≡0 ( mod 2) , then α 4 k ∘ f∈ R(2 4 k x), or R(2 4 k x) is in a higher dimension than α 4 k ∘ f. Here α 4 k is the element of order 2 in the image of J in dimension 4 k−1, and 〈 f,2, α 4 k 〉 is the Toda bracket.