Can someone buy a home with a fake Social Security number?

I guess your question most likely is that if you can get a mortgage and/or home equity line of credit (HELOC) to purchase a home, because there are no restrictions on purchasing a home in the United States by anyone as long as they can pay for it.  Generally speaking foreigners and illegal immigrants buy homes either for living or as investments by paying in cash.  No one would bother with your legal status.  Obviously, you will need to pay property taxes but to do that a SSN is not necessary (though, if you do so, it is a good idea to get an ITIN, particularly if you have other sources of incomes in the United States and definitely if you rent the house so that you can file your taxes on the rental income and claim the property taxes as a deduction).  Going back to the question of financing the home, typically most banks give loans primarily to legal residents who have a substantial credit history, job, other assets, etc.  The conclusion is that submitting a false SS# will most likely be identified by the bank (which are fairly good at detecting identity theft), but if not, it is still a crime to submit fraudulent information to a bank, because if something goes wrong the bank can sue you for fraud, and that can cause a lot of complications.