Should he help

So we have a month old I stay home with the baby currently. My fiancé works Monday-Friday 6am-2pm. He does not get up at night to help me with the baby. He just sleeps and doesn’t even give me a chance to sleep at night. Baby is up ever 3 to 4 hours to eat stays up about 30-40 min then back to sleep. My question is should he be getting up to help with baby shit I want sleep also. We put a full size bed in baby room so we could be close to baby while baby sleeps in crib but he refuse to sleep in there with me he sleeps in our bedroom so has never slept next to his child. ( baby is formula feed)