Joseph Bauer, the 55-year-old Baltimore Ravens fan who suffered a brain injury in a fight with Oakland Raiders fans Sunday at M&T Bank Stadium, showed encouraging signs according to his sister on Tuesday.

Bauer, who remains in critical condition on a respirator at Maryland Shock Trauma Center, gave a thumbs-up sign when asked to do so by doctors.

“We have been encouraged by some signs we believe indicate he recognizes his wife,” Susan Bauer, his sister, told the Baltimore Sun. “The trauma doctor said she is ‘cautiously optimistic’ about his progress and he is trending in the right direction.”

Scott Smith, 30, of Mount Vernon, N.Y., and Andrew Nappi, 31, of Eastchester, N.Y., were each charged with first- and second-degree assault and were released Monday on bond. Smith posted $50,000 and Nappi $25,000. Smith, a rookie firefighter in Mount Vernon, has been suspended without pay pending an investigation.

A police officer working the game in uniform was summoned to an exit ramp near section 529, according to the police report obtained by The Post, and found the alleged victim unconscious. The officer reported that Nappi and Smith were stopped as they walked away. Nappi said that, after the men had had a verbal altercation, Bauer threw a bottle that struck Smith in the head. The report indicates that Nappi said he then struck Bauer in the face with a closed fist. Bauer’s wife, Sharon, told police that Bauer hit his head when Smith shoved him to the concrete. According to the police report, a witness confirmed the account.

Bauer said her brother, who works at Schuster Concrete, has two grown children and has been a Ravens fan since the team came to Baltimore.

“Here’s a man and a family that have been longtime season ticket holders, that know the people here that work in the building, that love the Ravens and have been to countless numbers of games at M&T Bank Stadium without incident,” Coach John Harbaugh said Monday. “Then all of a sudden, there’s an incident where a gentleman is assaulted by somebody. That’s just not what we’re about. We’re a family. It’s a family atmosphere.”