 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.Here is a summary of the status of the generalized Poincaré conjecture in various settings.