Can a man who has signed a birth certificate relinquish his parental rights to the child including child support. He makes no effort to be in the child's life and I see no reason to have any ties whatsoever with him. He is an alcoholic and wanted me to have an abortion when we found out she was on the way?

Relinquishing parental rights in no way gives them freedom from financially supporting the child, it only takes away their rights in making decisions in the child's interests.

If he is proven to be the dad (and signing the birth certificate usually establishes that proof), then the only way he can forfeit his rights and responsibilities is if someone adopts the child.

You can not make him see, call, write or even pulbicly acknowledge the child, and he can sign away any rights to have anything to do with the child.

But he can not sign away his obligation to pay child support, this is something the state views as a right of the child, and also if public assistance is ever received the state may wish to go after him for repayment of that assistance.

You should be filing for child support and use every possible way, including garnishment of his wages, to get that support for the child