I, Koch: a decidedly unauthorized biography of the mayor of New York City.

In the end, Mayor tells you, entertainingly, what an egomaniac sees when he looks in the mirror. It glimpses of the inside of City Hall are vivid but unreliable. You can learn a lot more, while having much less fun, reading the hositle I, Koch, by three city reporters who have covered Koch's career for some years. He sent word to his aides and city officials that he wouldn't cooperate with the authors. Nevertheless, they have compiled a rich account of his life and fortunes, one that rings truer than his book. They are critical, but they draw a portrait of convincing subtlety.

Koch was born in 1924 of Polish-Jewish immigrant parents. He grew up in Newark, mostly, and studied law at New York University. Old acquantances recall him as a shy but loquacious loner with a biting tongue. He never took much interest in the opposite sex; but then, he never took much interest in his own sex either. He was just ferociously ambitious. He always wanted to be mayor of New York, even as a young man. He didn't need a Lady Macbeth to propel him.

During the Fifties he lived in Greenwich Village and tried to get into politics. At first he was part of the small and impotent reform Democrat movement. Then he seems to have flirted briefly with the old Tammany crowd, in the twilight of Carmine De-Sapio, only to discover that you couldn't rise quickly there. So it was back to the reform set. He made himself an ultraliberal on issues like sodomy, abortion, and divorce; he went to the Deep South with the Freedom Riders and opposed the Vietnam War.

All the while, Koch had his feet on the ground. He was no utopian. He even took pains to cultivate animal lovers--fanatically loyal constituents, he observed, and there were a lot of them. He crossed party lines to endorse John Lindsay in 1965, then was elected to Lindsay's old congressional seat in 1968. He seemed to belong to Congress's far Left; but he cozied up to old pols like Wayne Hays and made friends who counted.

Sensing a change in the political atmosphere, he began emphasizing the crime issue and denouncing racial busing. He opposed a city housing project because it would displace middle-class homeowners. His liberal colleagues began calling him a turncoat, a chameleon.

When the Beame administration floundered in scandal and financial crisis in 1977, Koch made his big move. He was a long shot with no real base, running against Beame, Cuomo, Bqlla Abzug, and a couple of others in the Democratic primary. But he ran one of the great races in New York political history, assisted by David Garth, the media magician (who gave him no better than a 40 to 1 chance to winning). Koch had no administrative experience, but Garth packaged him as the "issues" candidate, and they gambled money they didn't have on TV spots. (Koch to camera: "Mayor Beame is asking for four more years to finish the job. Finish the job. Hasn't he done enough?")

Then came the big breaks. On July 13, 1977, lightning struck a power station and every light in New York City went out. Looters went wild. Koch attacked Beame for failing to call in the National Guard. With a lunatic killer, "Son of Sam," roaming the city, he also called for the death penalty. The Australian publisher Ruper Murdoch, who had just bought the New York Post, gave Koch a spectacular front-page endorsement; a few days later the Daily News followed suit. Koch even got the sort of machine backing he had denounced in his reformist days.

Roaring up from behind, Koch won a plurality in the primary and faced Cuomo in a runoff. Now things got dirty. There were homosexual rumors about Koch, Mafia rumors about Cuomo. Garth persuaded Bess Myerson, a former Miss America, to accompany Koch in public. They held hands and made Nancy-eyes at each other for the cameras. It was all an act, but Miss Myerson put her heart into it. To a fault. She was so enraged by the homo rumors that she gave a newspaper interview to assail them, thereby getting them into print on the eve of the election and no doubt taking years off Garth's life. Now everyone was off balance, including Cuomo. When the dust settled, Koch was mayor.

How good a mayor has he been? The authors of I, Koch think the public overrates him. Crime is way up since 1977; the city is dirtier than evern; the subways and trains are more dangerous and less dependable. The truth is that the city is too chaotic for any simple measurement of that sort. It's terrible, of course, but nobody misses John Lindsay or Abe Beame. Ken Auletta of the Daily News put his finger on something in a prq-primary column last month: The voters respect Koch for having said no to all the special interests the other candidates were fawning over. His negatives have become positve. As long as Ed Koch is in Gracie Mansion, New Yorkers have assurance that the city isn't completely out of control.