Unstable splitting of $V(1) ^ V(1)$ and its applications

Abstract

$Let P^n(p)$ be an $n$- dimensional mod $p$ Moore space and $V^n$ be the mapping cone of an Adams map $A : P^{n-1}(p) \rightarrow P^{n-2p+1}(p)$. This paper gives an unstable splitting of $V^m \bigwedge V ^n$ for a prime $p \geq 5$. The proof is based on explicit calculations of $[V^{n+2p-1}, V^n]$. As an application, we define a Samelson product on $[V^*, \Omega X]$ and prove that it satisfies anticommutativity and the Jacobi identity.