Not if your fare includes a checked bag on some routes: US to Europe: Free. Within the US: $45–50 per bike, each way.
How much does American charge for a bike between the US and Europe?
Free (counts as a checked bag). Included in your free checked bag allowance. Basic Economy fares don't include a free checked bag.
How much does American charge for a bike within the US?
$45–50 per bike, each way. Counts as one of your checked bags. No extra bike fee. Economy fares within the US don't include a free checked bag. First bag: $45 if you pay online, $50 at the airport. Basic Economy: $50–55.