1) If he is a tenant on a written lease, then while you need to make sure the full rent is paid to the landlord (to avoid eviction), you can sue your boyfriend for his share of the rent, which he is obligated to under the lease.2) You can also seek child support from him (not spousal support or alimony, since you are not married) as the father of your baby.You should consult with an attorney about the claims you have. Good luck.