[nybsjytm]

> Folks knew the problem was near a solution once the monotonicity proof of the W functional came out.

This isn't true, it was a major accomplishment but by far the easiest part of Perelman's papers and not actually even part of the proof of the Poincaré conjecture.

[hodgehog11]

Interesting, this was the big 'a-ha' moment in the grad course I took on the subject. Surgery was clearly important too, but this seemed to be more apparent from Hamilton's work. The W functional was what provided the control needed. Also, calling it 'easy' feels like it dismisses the insights necessary to have developed the functional in the first place.

Happy to be corrected on this; I wasn't an active mathematician at the time, so everything I know comes from other accounts.

[nybsjytm]

I called it the easiest part of his papers, not easy. Either way, it's actually not relevant to the proof. For example, I believe that Morgan and Tian's 500 page exposition of the proof doesn't mention it even once.

Moreover I'd strongly dispute that there was any particular point where it was clear that the problem was "near a solution." The W functional is an example where experts could very quickly verify that Perelman had made at least one major new discovery about the Ricci flow. But the proof of the Poincaré conjecture was on another order of complexity and required different (more complicated) new results about Ricci flow.