This book gives a short, self-contained complete proof of the (topological) 4-Dimensional Poincaré Conjecture, after Michael Freedman. This is one of the crowning achievements of 20th century topology, but the details of the proof are appreciated by distressingly few mathematicians. Part of the reason seems to be the result of two widely-held impressions: that the proof is very abstract, technical and obscure; and that the results and techniques that go into the proof are orthogonal to the mathematical mainstream. Both impressions are entirely false.
The book is (or aims to be) clear and simple, self-contained, but with enough details to allow the motivated reader to extract complete and rigorous proofs. It's short - short enough (I hope) that the curious reader can - and will want to - sit down and read it through for pleasure.
The book, its focus and its overall structure borrows heavily from Freedman's 2013 UCSB lectures (archived online at MPI Bonn) and from the excellent book of Freedman-Quinn, which introduces many substantial simplifications over Freedman's original JDG article, including notably the technique of gropes. Additional topics include the existence of exotic smooth structures on four-space, and the fact that every homology 3-sphere bounds a contractible 4-manifold.
The version of the book available here for download is still incomplete and unpolished at a few points, but is in the process of substantial rewriting. The final version will be published by Cambridge University Press, but some version of the book will remain freely downloadable after publication.
Download .pdf (preliminary version 12/10/2018)