In the mathematical area of topology, the generalized Poincaré conjecture is a statement that a manifold that 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
Every homotopy sphere (a closed n-manifold which is homotopy equivalent to the n-sphere) in the chosen category (i.e. topological manifolds, PL manifolds, or smooth manifolds) is isomorphic in the chosen category (i.e. homeomorphic, PL-isomorphic, or diffeomorphic) to the standard n-sphere. 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.
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, with the first known counterexample in dimension 7 (Milnor sphere). True in some dimensions including 1, 2, 3, 5, 6, 12, 56 and 61. This list includes all odd dimensions for which the conjecture is true. For even dimensions, it is true only for those on the list, possibly dimension 4, and possibly some additional dimensions ≥ 64 {\displaystyle \geq 64} (though it is conjectured that there are none such). The case of dimension 4 is equivalent to PL. Thus the veracity of the Poincaré conjectures is different in each category Top, PL, and Diff. In general, the notion of isomorphism differs among the categories, but it is the same in dimension 3 and below. In dimension 4, PL and Diff agree, but Top differs. In dimensions above 6 they all differ. In dimensions 5 and 6 every PL manifold admits an infinitely differentiable structure that is so-called Whitehead compatible.
History The cases n = 1 and 2 have long been known by the classification of manifolds in those dimensions. For a PL or smooth homotopy n-sphere, in 1960 Stephen Smale proved for n ≥ 7 {\displaystyle n\geq 7} that it was homeomorphic to the n-sphere and subsequently extended his proof to n ≥ 5 {\displaystyle n\geq 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 that a PL homotopy n-sphere is 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 {\displaystyle n\geq 5} a topological homotopy n-sphere is homeomorphic to the n-sphere. Michael Freedman solved the topological case n = 4 {\displaystyle n=4} in 1982 and received a Fields Medal in 1986. The initial proof consisted of a 50-page outline, with many details missing. 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. Grigori Perelman solved the case n = 3 {\displaystyle n=3} (where the topological, PL, and differentiable cases all coincide) in 2003 in a sequence of three papers. He was offered a Fields Medal in August 2006 and the Millennium Prize from the Clay Mathematics Institute in March 2010, but declined both. In the smooth category for n > 4 {\displaystyle n>4} , studying the Poincaré conjecture comes down to determining the elements of the Kervaire-Milnor short exact sequence of groups
0 → b P n + 1 → Θ n → π n S / ( I m a g e ( J n ) ) → 0 {\displaystyle 0\to bP_{n+1}\to \Theta _{n}\to \pi _{n}^{S}/(\mathrm {Image} (J_{n}))\to 0\,}
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