Your matter is complicated and requires experienced counsel to review the agreements and documents to determine if there is a viable challenge to the agreement.
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Before worrying about suing company A for falsifying your resume, I would first find out if the non-compete agreement is enforceable, if I were you. If the non-compete is unenforceable, then you'd be able to have your new job, which is the most important thing I assume. Many attorneys will review your non-compete agreement, more than likely for a small fee.
As for suing for falsifying your resume, you would have to show that you suffered some form of damages. It sounds like you not being able to take the position with the new company stems more from the non-compete than anything else.
Bottom line: You should have the non-compete reviewed before doing anything else.
See prior response. You should confer with an attorney for legal advice for your personal issues. You appear to have multiple legal issues. If you were hypothetically to seek recourse as to misrepresentations and falsehoods, a concern is what damage (harm) you've suffered. Tricia Dwyer Esq.
MINNEAPOLIS/ST. PAUL - ST CLOUD. This law firm may accept avvo posters as clients but this post is not legal advice and does not create an attorney-client relationship. This post is to be considered general information which may or may not apply to your personal situation. Please do seek private attorney counsel as to your personal legal issues and needs.