Erika Decker knew three things about John Looker when he reached out to her on Facebook in 2014: He was a cult figure in Columbus, Ohio, where they both lived . He had become a face of the Pelotonia bike-a-thon, which has raised millions of dollars to fight cancer. And he had the same brain cancer as her young daughter, Lily.

Mr. Looker had read about Lily in The Columbus Dispatch and wanted to show his support. “I know I’m a stranger to you and your family, but please know that I am with you all in spirit today,” he wrote.

A few days later, another note: “I would love to come visit with you and your family.”

Ms. Decker, 46, was curious. So one night she scoured his Facebook page. It was loaded with health updates, inspirational quotations and community adoration. But something didn’t seem right.

She wrote back and asked for the name of his oncologist.

“I generally don’t share it because several years ago I did, and she got bombarded by phone calls from the Pelotonia community asking her to do more for me,” he wrote. Ms. Decker repeated her question. Mr. Looker “unfriended” her on Facebook.