IN HIS HOTEL room the morning after, Muhammad Ali nursed a swollen jaw as the room service waiter arrived with breakfast and good wishes for the man he called champ.

“I’m not the champ,” Ali corrected him. “Joe Frazier is the champ.”

Indeed he was, and if anyone needed a reminder the morning papers provided it with a picture as shocking to Ali’s adoring fans as his defeat the night before. There was Ali glassy-eyed and struggling to get up from the canvas in the 15th round as Frazier walked to a neutral corner more certain than ever of victory.

“There were a couple of knockdowns really,” Frazier said, chuckling at the thought. “They called the first one a slip. But it was the left hook that made him slip.”

It was 40 years ago Tuesday that Ali and Frazier met at Madison Square Garden in a fight so big it was simply referred to as The Fight. It was Frazier’s heavyweight title that was on the line, but a lot of boxing fans still considered Ali the champion because he was stripped of the title and sent into boxing exile for refusing to be drafted.

Frazier was undefeated and in his prime, a relentless aggressor with a vicious left hook. Ali, in just his third fight since the 3½-year layoff, was a polarizing figure who was hated by some just as much as he was loved by others.

“A lot of ‘em want me whipped because of the draft,” Ali said before the fight.

“A lot of ‘em want me whipped because of religion. A lot of ‘em want me whipped because I’m black … and for other reasons that I might not even know about.”

They would go on to meet two more times, including the memorable Thrilla in Manilla. But nothing could match the stakes that March night at the Garden, where Frank Sinatra shot pictures at ringside, celebrities jockeyed for prime seats, and almost as many men as women wore full-length fur coats.

Prime seats were $150 — an astonishing sum at the time — and they could have sold them for twice that price. Tickets were so scarce that Frank Costello, reputed boss of the Luciano crime family and a fight regular, could only get two seats instead of his usual four.

There seemed no way the fight could live up to the hype. But Ali and Frazier made sure it did.

Frazier was on the attack the whole night, stalking Ali and hitting him with left hooks anytime he got close. Ali clowned at times, but his jab kept finding the mark and he landed some right hands to the head that would have dropped other fighters.

Frazier, though, was not going to be denied. He hurt Ali badly in the 11th round — the same round Ali slipped on a wet spot and went to the canvas — and landed a crushing left hook 25 seconds into the 15th round that put Ali down for real. Ali got up and finished the fight, but his jaw was swollen like a grapefruit and, when the scorecards were announced, Frazier had won a unanimous decision.

“Ali lost but it was still one of his greatest moments,” said retired Associated Press boxing writer Ed Schuyler Jr, who was ringside. “He had fought only twice after coming back from more than three years off and he went 15 rounds with a guy at the top of his game.”

That Ali wasn’t quite the fighter he was before his layoff may not have made a difference. Ali still had the skills to beat almost every other fighter, but he couldn’t overcome Frazier’s sheer determination.

“If Joe Frazier would have fought King Kong he would have knocked him out that night,” said Gene Kilroy, who was a friend of both men and later became Ali’s business manager.

“Nothing was going to stop Joe Frazier.”

Ali is 69 now, living in Arizona and suffering from the debilitating effects of Parkinson’s syndrome, a movement disorder causing tremors of the hands, arms and legs and stiffness. Frazier, who slurs his words at times, is 67 and lives in Philadelphia, where, on a recent day, he answered the phone after doing 25 minutes on an exercise bicycle.

Frazier said he planned to go to parties in New York and Philadelphia to celebrate the night that still defines his life today.

“I can’t go nowhere where it’s not mentioned,” Frazier said. “That was the greatest thing that ever happened in my life.”

Not that all his moments with The Greatest were all that great. Frazier was bitter over things Ali said about him while promoting their fights, and he carried a grudge for many years.

But he says he’s moved on.

“I forgive him,” Frazier said. “He’s in a bad way.”

Asked what they would talk about if they met, Frazier just laughed. They would, of course, talk about the same thing everyone wants to talk to Joe Frazier about.

The greatest night of his life.

And a fight so great it remains The Fight.