On the double suspension homomorphism at odd primes

Abstract:

We work with the ${E_1}$-term for spheres and the stable Moore space, given by the $\Lambda$-algebra at odd primes. Writing $W(n) = \Lambda (2n + 1)/\Lambda (2n - 1)$ and $M(0) = {H_ \ast }({S^0}{ \cup _p}{e^1})$, we construct compatible maps ${f_n} \cdot W(n) \to M(0)\tilde \otimes \Lambda$ and prove the Metastability Theorem: in homology ${f_n}$ induces an isomorphism for $\sigma < 2({p^2} - 1)(s - 2) + pqn - 2p - 2$ where $\sigma = \text {stem degree}$, $s = \text {homological degree }$ resulting from the bigrading of $\Lambda$ and $q = 2p - 2$. There is an operator ${\upsilon _1}$ corresponding to the Adams stable self-map of the Moore space and ${\upsilon _1}$ extends to $W(n)$. A corollary of the Metastability Theorem and the Localization Theorem of the second author is that the map ${f_n}$ induces an isomorphism on homology after inverting ${\upsilon _1}$.