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Generalized Poincaré conjecture

In the [mathematical] area of [topology] , the term Generalized Poincaré conjecture refers to a statement that a [manifold] which is a [homotopy sphere] 'is' a [sphere] . More precisely, one fixes a
[category] of manifolds: [topological] ( Top ), [differentiable] ( Diff ), or [piecewise linear] ( PL ). Then the statement is

:Every homotopy sphere (a closed n -manifold which is [homotopy equivalent] to the n -sphere) is isomorphic to the n -sphere in the chosen category, i.e. homeomorphic, diffeomorphic, or PL-isomorphic.

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] . 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] recipients [John Milnor] , [Steve Smale] , [Michael Freedman] and [Grigori Perelman] .

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

- Top : true in all dimensions.
- PL : true in dimensions other than 4; unknown in dimension 4, where it is equivalent to Diff.
- Diff : false generally, true in dimensions 1,2,3,5, and 6. First known counterexample is in dimension 7. The case of dimension 4 is unsettled ,.

A fundamental fact of [differential topology] is that the notion of isomorphism in Top, PL, and Diff is the same in dimension 3 and below; in dimension 4 PL and Diff agree, but Top differs. In dimension above 6 they all differ. In dimensions 5 and 6 every PL manifold admits an infinitely differentiable structure that is so-called Whitehead compatible .See

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