What rights does the father of my child have?

I'm pregnant, and the father of my child only wants to be apart of our child's life just to spite me. His family is incredibly sue happy, also. I don't want him messing with my child, just because he's not mentally there. He's extremely bipolar and doesn't think straight - hence why I left him. I was thinking of not even putting his name on the birth certificate, but don't know if that would bite me in the butt later... Help please!