Can any one give me an example of surjective homomorphism on braid groups on the sphere that is not injective? Such that B_{n}(S²)$B_{n}(S^2)$ is generated by σ₁,σ₂,...σ_{n-1}$\sigma_1,\sigma_2, \dots, \sigma_{n-1}$ which are subject to the following relations: {
$\sigma_{i} \sigma_{j} = \sigma_{j} \sigma_{i}$ if $|i-j| > 1$,
$\sigma_{i} \sigma_{i+1} \sigma_{i} = \sigma_{i+1} \sigma_{i} \sigma_{i+1}$ for all $i = 1,2, \dots, n-2$, and
$\sigma_1 \sigma_2 \dots \sigma_{n-1}^2 \dots \sigma_2 \sigma_1 = 1$.
σ_{i}σ_{j}=σ_{j}σ_{i} if |i-j|>1 σ_{i}σ_{i+1}σ_{i}=σ_{i+1}σ_{i}σ_{i+1} for all i=1,2,...,n-2 σ₁σ₂...σ_{n-1}²...σ₂σ₁=1 ┊ Let ϕ:B_{n}(S²)→B_{n}(S²)Let $\phi: B_{n}(S^2) \rightarrow B_{n}(S^2)$ be a surjective homomorphism. isIs there a situation that ϕwhere $\phi$ is not injective?