The Stable Homotopy Types of Stunted Lens Spaces mod $4$

Let $L^{n+k}_n$ be the mod $4$ stunted lens space $L^{n+k}/L^{n-1}$. Let $\nu(m)$ denote the exponent of $2$ in $m$, and $\phi (k)$ the number of integers $j$ satisfying $j\equiv 0,1, 2, 4 (\operatorname{mod}8)$, and $0< j\leq k$. In this paper we complete the classification of the stable homotopy types of mod $4$ stunted lens spaces. The main result (Theorem 1.3 (i)) is that, under some appropriate conditions, $L^{n+k}_n$ and $L^{m+k}_m$ are stably equivalent iff $\nu(n-m)\geq \phi(k)+\delta$, where $\delta=-1, 0$ or $1$.