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# Homology, Homotopy and Applications

## Volume 8 (2006)

### Number 1

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

Pages: 169 – 186

DOI: http://dx.doi.org/10.4310/HHA.2006.v8.n1.a5

#### Author

#### 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 $V^m \wedge V^n$ 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.

#### Keywords

$V(1)$, Samelson product

#### 2010 Mathematics Subject Classification

55P15, 55Q15