According to the experts, Poincaré made huge leaps in his proofs, often leaving lesser mathematicians scratching their heads. I'm wondering if some of those leaps later turned out to be false and if so how often.
It's not clear which experts you are relying on for this opinion about Poincare, nor do you cite any sources for these opinions. http://en.wikipedia.org/wiki/Henri_Poincaré Poincare was, by all accounts, not one to be overly bound by the logical underpinnings of mathematics. Unlike Frege or Bertrand Russell, he strongly disagreed that mathematics was a branch of logic, and he took a more intuitive approach in his mathematical researches.
Fine, June Barrow-Green which can be found here: http://www.bbc.co.uk/programmes/p0038x8l It's toward the last five minutes of the program. I didn't think anyone would bother disputing it so obviously no need to back up sources. I still am wondering if his big leaps were later proved false.
Not being familiar with the complete works of Poincare I cannot say that every one of his works is completely error free. In fact, it would be miraculous if Poincare managed to avoid making any blunders! I can say, however, that a good portion of the techniques he invented are still utilized (in some form or another) today.
At PF, the more sources you provide, the better discussion you will obtain. June Barrow-Green throws in a brief aside comparing Perelman with Poincare in the last few seconds of the program, specifically about how the two men's proofs allegedly contain gaps which are hard for lesser mortals to follow. If you are looking for tabloid-like headlines which say "French mathematician's proof goes poof!", I don't think you'll find any. Poincare's reputation AFAIK remains undiminished a century after his death, and I know of nothing in his work which has been disproven. He was in many respects, a man ahead of his time, and in some fields like relativity, his work predates that of Einstein. IMO, his untimely death robbed him of some of the recognition which might have made his name as widely known as Einstein's. Some mathematicians, like P.S. LaPlace were notorious for omitting a lot of the details in their work. When Nathaniel Bowditch, no slouch when it came to mathematics, undertook to translate LaPlace's 'Celeste Mecanique' into English, he found that the work of the Frenchman was extremely abbreviated, although it took five volumes to print it in the original. Bowditch not only translated LaPlace's work, but he also decided to check all of LaPlace's mathematical derivations himself. After some time was spent on his project, Bowditch remarked, "Whenever I meet in LaPlace with the words 'Thus it plainly appears', I am sure that hours, and perhaps days, of hard study will alone enable me to discover how it plainly appears."
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. . . Riemann, his paper on counting primes. Really tough to follow even for the expert, Edwards, to go through.