Can my child's father put his name on the birth certificate even if I don't want him to?

See the statutes below.  The father has to consent in writing to have his name placed on the certificate.  The father can fill out paternity forms in your state to determine parentage -- usually genetic testing may be involved.  Under Kansas statute 65-2416, the father must give his consent.  Under Kansas statute 38-1119, the determination of parentage can be made using many tests and or factors, one of which includes any evidence that is relevant to the issue of paternity of the child, including but not limited to voluntary acknowledgment of paternity made in accordance with Kansas statute 38-1138.

If there are serious reasons you don't wish to have the father be named on the birth certificate, you should raise it now.  These factors could enable the state registrar of vital statistics to disallow his name being placed on the birth certificate.