The Simple Loop Conjecture is as follows.

> Every non-injective map from a surface group to a 3-manifold group kills a simple closed curve.

As there are all sorts of 3-manifolds with abelianisation of rank two, I think the answer to your question is unknown.

**UPDATE:** Sorry, I wrote the above too hastily.  I should have said 'I think that the kernel is not known to contain such a loop.'  On the other hand, there may well be examples of such maps with no simple loops in the kernel.  You could try looking at Louder's recent preprint on the [Simple loop conjecture for limit groups][1], for instance.


  [1]: http://arxiv.org/abs/1106.1350