Since you are being audited, let's go through this in detail.
First your fiance has to have a qualifying child as his dependent. Your fiance cannot be the dependent of anyone else.
For a qualifying child, he needs to meet the relationship test (birth certificate will do), the age test (met as long as the birth certificate shows 2009) and residency test. If the 3 of you live together and the hospital paperwork shows his address on it, you should be fine.
Once he has a qualifying child, then your fiance had to qualify for HOH. To meet that test, he has to show that he paid more than half the cost of keeping up his home. He does that by producing rent and utility receipts and copies of his payments. The IRS will also consider whether his income was high enough to have paid these himself.
If, on the other hand, you were living with your parents all year (and he with his parents) and they claimed you as a depenent, the IRS is going to ask why a breast feeding baby wasn't living with you.