The heartbreaking story began last Sunday when two visitors from Oklahoma strolling along the East River in Manhattan spotted a tiny body floating in the water near the Brooklyn Bridge. A criminal complaint that was made public on Friday offered new details of how the police tracked Mr. Currie’s movements and cited statements he is said to have made in response to worried text message from Mason’s mother.

The police said that Mr. Currie, 37, a cleaner for the New York City Transit Authority, went to the Bronx home of Mason’s mother to pick up the boy on Saturday afternoon. Surveillance video showed that Mr. Currie and the baby entered his apartment in Co-Op City soon afterward, the police said, adding that Mason was alive at that time.

But the police said they believed that Mason was dead when Mr. Currie left his apartment at 1:30 p.m. the next day. A detective wrote in the criminal complaint that Mr. Currie was wearing a backpack across his front like a chest harness baby carrier, adding that a blanket covered “a large bump approximately the size of an infant.”

“There does not appear to be any movement beneath the blanket,” the detective, Ohmeed Davodian, added.

Soon after the video showed Mr. Currie leaving his home with the backpack, Mr. Currie’s MetroCard was used to board a bus in the Bronx and then to enter a subway station at 23rd Street in Manhattan, the complaint said.