Can any one give me an example of surjective homomorphism on braid groups on the sphere that is not injective? Such that $B_{n}(S^2)$ is generated by $\sigma_1,\sigma_2, \dots, \sigma_{n-1}$ 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

• $\sigma_1 \sigma_2 \dots \sigma_{n-1}^2 \dots \sigma_2 \sigma_1 = 1$.

Let $\phi: B_{n}(S^2) \rightarrow B_{n}(S^2)$ be a surjective homomorphism. Is there a situation where $\phi$ is not injective?