More precisely, one fixes a category of manifolds: topological (Top), piecewise linear (PL), or differentiable (Diff).
Then the statement is The name derives from the Poincaré conjecture, which was made for (topological or PL) manifolds of dimension 3, where being a homotopy sphere is equivalent to being simply connected and closed.
The generalized Poincaré conjecture is known to be true or false in a number of instances, due to the work of many distinguished topologists, including the Fields medal awardees John Milnor, Steve Smale, Michael Freedman, and Grigori Perelman.
In dimensions 5 and 6 every PL manifold admits an infinitely differentiable structure that is so-called Whitehead compatible.
Freedman gave a series of lectures at the time, convincing experts that the proof was correct.
A project to produce a written version of the proof with background and all details filled in began in 2013, with Freedman's support.
The project's output, edited by Stefan Behrens, Boldizsar Kalmar, Min Hoon Kim, Mark Powell, and Arunima Ray, with contributions from 20 mathematicians, was published in August 2021 in the form of a 496-page book, The Disc Embedding Theorem.
[11][12][13] He was offered a Fields Medal in August 2006 and the Millennium Prize from the Clay Mathematics Institute in March 2010, but declined both.
The generalized Poincaré conjecture is true topologically, but false smoothly in most dimensions.
[15] For piecewise linear manifolds, the Poincaré conjecture is true except possibly in dimension 4, where the answer is unknown, and equivalent to the smooth case.