Generalized Poincare Conjecture

The generalized Poincare conjecture, in topology, states that a manifold that is a homotopy sphere is a sphere. Precisely, fixing a category of manifolds as topological, piecewise linear, or differentiable, every homotopy sphere in that category is isomorphic, in that category's own sense, to the standard sphere of the same dimension. The name derives from the original Poincare conjecture, made for three-dimensional manifolds, where being a homotopy sphere is equivalent to being simply connected and closed. The generalized conjecture is known to be true or false in a number of instances, through the work of topologists including the Fields medalists John Milnor, Steve Smale, Michael Freedman and Grigori Perelman. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/