What's the best current proof of the Poincaré conjecture? Is there a current "gold standard" proof of the Poincaré Conjecture? I am interested in reading through the full proof, and some day perhaps reading Perelman's proof/notes. For now, though, I was wondering what the most "polished" version is, which would represent any advances during the 17 years since his publications and the 14 years since the formalization by various groups. I can see that Terry Tao has published some notes recently, but I don't believe he has published a full proof.
 A: Still, the only (essentially) self-contained treatment of Perelman's proof of the Poincare Conjecture is
Morgan, John; Tian, Gang, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs 3. Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute (ISBN 978-0-8218-4328-4/hbk). xlii, 521 p. (2007). ZBL1179.57045.
The only major part of the proof which they only sketch but do not provide details of is the short-time existence theorem for RF. One thing they do not address is the equivalence  between the topological and smooth category for 3-dimensional manifolds, but, as long as one is only interested in classifying smooth manifolds up to diffeomorphism, this does not matter.
Since then, there were numerous advances in understanding Ricci Flow in dimension 3 as well as in higher dimensions, but these did not lead to a substantial simplification of the proof. Needless to say, I do not recommend reading either the above book or other accounts of the proof of the 3-dimensional Poincare Conjecture (and, more generally, the Geometrization Conjecture) to an undergraduate student. My suggestion instead would be to read a textbook on Differential (Riemannian) Geometry.
