He wants a breakup you don't want to. I believe you've tried to explain to him and have promised it will never happen again - you'll be faithful in the relationship - but he doesn't want to. To complicate matters, you stay in the same house and he doesn't want to talk to you.
It is usually very difficult for a person to forgive their partner when they have been unfaithful. You'll have to find means of asking for another chance to display your faithfulness to the relationship and love for him. You can write a letter. However, you should be careful to avoid any form of confrontation or fights. Also, you shouldn't beg or plead a lot since you'll turn him off.
You might want to indicate in the letter or through text or when you call him (whichever you choose though in this situation written texts are better than calling) that you love him. You have to prove this. It's a fact he is hurt which you will have to consider. Then, let him know that you are waiting for him patiently because you don't want to lose him.
You want contact him or beg or plead because you'll annoy him which will aggravate the situation. You will have to exercise no contact. It is during those silent days you'll know whether he has already made up his mind to move on by not contacting you. It means he is firm with his decision. It's difficult to convince someone who has already made up his mind to follow through his decision.