Not according to the law.

And who's responsible for baby then?

Based on equality? Nothing about what op described is equal. She's paying the brunt of the load for their child, doing all the house work, and raising the kid solo.

Honestly sounds borderline financially abusive to me. I would collect as much evidence as you can to prove that you have been providing most of the income for your child, and then start talking to some good lawyers. At the very least you'll get split custody and you'll have more time on your hands to focus on work while kiddo is with dad.. better case scenario he'll have to pay you child support which is better off than your current scenario.

I mean If he's having sex with you then he's obviously sexually attracted to you??