Factoring 2k on stunted projective spectra and the root invariant

Let x∈π∗S0. In this paper we estimate the root invariant of 2wx in terms of the root invariant of x. For a stunted projective space RP2n2k−1, we use Todaʹs calculation of the smallest integer ε(n,k), such that 2ε(n,k) times the identity map on RP2n2k−1 is null homotopic. To calculate the root invariant, defined by Mahowald, we find factorizations of 2ε(n,k)−1 times the identity map on RP2n2k−1 for small n−k and use these factorizations to estimate R(2ε−1x) in terms of R(x). In some cases, it is shown there are common elements in R(2ε−1x) and certain Toda brackets. For instance, we prove the following theorem. m. For x :Sr−1→S−1, f∈R(x),(i) x)|−|x|≡1 (mod 2), then 〈f,2,α4k〉∩R(24kx)≠∅, or R(24kx) is in a higher dimension than 〈f,2,α4k〉. x)|−|x|≡0 (mod 2), then α4k∘f∈R(24kx), or R(24kx) is in a higher dimension than α4k∘f. 4k is the element of order 2 in the image of J in dimension 4k−1, and 〈f,2,α4k〉 is the Toda bracket.