In the run-up to the Paralympics, which start next week in Paris, Christie Raleigh Crossley’s coach asked her a serious question.

“Do you really have a hole in your brain?”

They were talking about an anonymous online accusation that Raleigh Crossley had exaggerated the disability that qualified her for the Paralympics.

Raleigh Crossley was taken aback. This was her coach, the woman she had trained with for much of the last year.

Yes, she answered, she really did have a hole in her brain. She could show pictures to prove it. “I had a blood tumor in my brain that had been killing the brain as it took over that area,” she said. “And then they removed it. And so now I’ve just got a nice hole.”

A competitive swimmer since she was 3 years old, Raleigh Crossley always dreamed of going to the Olympics. Now, at 37, a single mother raising three children, she is heading to Paris as a first-time Paralympian, hoping to bring home four medals, maybe more.

On a recent afternoon at her home in Toms River, N.J., Raleigh Crossley walked using a forearm crutch, dragging her left leg. As she does every day, she had hit the pool at 5 a.m. so she could be home before her children, ages 3 to 13, woke up.