If you had physical custody more than half the nights of the year, then you are entitled to claim the child as a dependent. If you and your wife both had physical custody more than half the nights of the year (because you separated in October, for example) then the parent who has the right to claim the child is the parent where the child lived the greater number of nights.
If you are the parent entitled to claim the child as a dependent, you will simply file a correct tax return. If you are blocked from e-filing, then print your return, sign it, and mail it in. The IRS will eventually investigate the duplicate dependent claim and require both taxpayers to furnish proof. (You do not submit proof when you file your tax return.)
If your wife is the parent who was entitled to claim the child as a dependent because she had custody more nights than you, but your wife allowed a friend to claim the child instead, then you can’t file a tax return to trigger an automatic review. You could file a fraud report with the IRS and that might start the necessary investigation.
https://www.irs.gov/individuals/how-do-you-report-suspected-tax-fraud-activity
The parent who had custody the larger number of nights can transfer the dependent claim to the other parent by signing a form 8332. Sometimes this is voluntary, if the parents can cooperate, and sometime this can be ordered by the court as part of a custody settlement. However, a dependent can never be transferred to an unrelated person using this form. The only way in which an unrelated person could claim your biological child as a dependent is if the child lived in that person’s home for all 365 nights of the year and you never had custody even one night (plus a couple other conditions that I’m skipping over for now).
