Please post about this on legal. If you've paid towards the mortgage you may have a share of the equity in the house (housework doesn't count, needs to be money) but this is the wrong thread for that. Law in the area is complicated and nobody here is qualified to give an answer - you need the law thread, here There are several family lawyers posting there who will give you good advice. Relationships is not the place for it, IMO. We aren't lawyers and the law varies a lot by circumstance.
I'm not a lawyer, but I do know that if you're married, you automatically have rights over the other person's property when/if you part, whether or not it's in joint names and whether or not you've paid towards it. We don't have community property when married in England and Wales but we do on separation. The law takes things such as length of marriage, and whether there are kids etc, into account. (They almost never take behaviour into account. You have to try to kill someone or something before they do - someone can sleep with your sister and it's irrelevant to the money side). Usually my understanding is that the primary carer of the child is often the one who gets to stay in a marital home. But neither party can oust the other until a legal agreement has been reached, either, afaik.
If you aren't married, then the normal law applies, which is a bit of an issue, to put it mildly, when people have kids outside marriage. The person who cares for those kids can be screwed if she stays at home etc. However, the normal law looks at what the intentions of the parties was when sharing the house, and if you paid towards the mortgage than that strengthens the argument that you may have some rights to the equity. The main thing they consider is whether the two of you had an understanding that you would have some rights over the property, I think. But as I say you need to ask a family lawyer in legal for advice on all this as it's a specialised area, especially if you aren't married.