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

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

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.

The case n = 1 and 2 has long been known, by classification of manifolds in those dimensions.

For a PL or smooth homotopy n-sphere, in 1960 Stephen Smale proved for n ≥ 7 that it was homeomorphic to the n-sphere and subsequently extended his proof to n ≥ 5; he received a Fields Medal for his work in 1966. Shortly after Smale's announcement of a proof, John Stallings gave a different proof for dimensions at least 7 that a PL homotopy n-sphere was homeomorphic to the n-sphere using the notion of "engulfing".E. C. Zeeman modified Stalling's construction to work in dimensions 5 and 6. In 1962, Smale proved a PL homotopy n-sphere was PL-isomorphic to the standard PL n-sphere for n at least 5. In 1966, M.H.A. Newman extended PL engulfing to the topological situation and proved that for n ≥ 5 a topological homotopy n-sphere is homeomorphic to the n-sphere.

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