Talk to a lawyer, or someone at a legal-aid clinic. Probably the laws are different in different places. I don't think it is up to you to stay in touch, it would be up to him to approach you and try to make that happen, if he wants parental rights.
If he's the father he has a right to see his child (and the child has a right to a relationship with his father). You can deny his paternity and force him to go to court for a DNA test before giving him access to the baby,but it sounds as though he wouldn't bother.