In the mathematical area of topology, the generalized Poincaré conjecture is 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
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, true in some dimensions including 1,2,3,5, and 6. First known counterexample is in dimension 7. The case of dimension 4 is equivalent to PL and is unsettled (as of 2019).
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.[1]
History
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 \( {\displaystyle n\geq 7} \) that it was homeomorphic to the n-sphere and subsequently extended his proof to \( {\displaystyle n\geq 5} \) ;[2] 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".[3] E. C. Zeeman modified Stalling's construction to work in dimensions 5 and 6.[4] In 1962, Smale proved a PL homotopy n-sphere was PL-isomorphic to the standard PL n-sphere for n at least 5.[5] In 1966, M. H. A. Newman extended PL engulfing to the topological situation and proved that for \( n\geq 5 \) a topological homotopy n-sphere is homeomorphic to the n-sphere.[6]
Michael Freedman solved the case n=4 (in Top) in 1982 and received a Fields Medal in 1986.[7]
Grigori Perelman solved the case n = 3 (where Top, PL, and Diff all coincide) in 2003 in a sequence of three papers.[8][9][10] He was offered a Fields Medal in August 2006 and the Millennium Prize from the Clay Mathematics Institute in March 2010, but declined both.
Exotic spheres
The generalized Poincaré conjecture is true topologically, but false smoothly in some dimensions. This results in constructions of manifolds that are homeomorphic, but not diffeomorphic, to the standard sphere, which are known as the exotic spheres: you can interpret these as non-standard smooth structures on the standard (topological) sphere.
Thus the homotopy spheres that John Milnor produced are homeomorphic (Top-isomorphic, and indeed piecewise linear homeomorphic) to the standard sphere \( S^{n}
\), but are not diffeomorphic (Diff-isomorphic) to it, and thus are exotic spheres: they can be interpreted as non-standard differentiable structures on the standard sphere.
Michel Kervaire and Milnor showed that the oriented 7-sphere has 28 different smooth structures (or 15 ignoring orientations), and in higher dimensions there are usually many different smooth structures on a sphere.[11] It is suspected that certain differentiable structures on the 4-sphere, called Gluck twists, are not isomorphic to the standard one, but at the moment there are no known invariants capable of distinguishing different smooth structures on a 4-sphere.[12]
PL
For piecewise linear manifolds, the Poincaré conjecture is true except possibly in dimension 4, where the answer is unknown, and equivalent to the smooth case. In other words, every compact PL manifold of dimension not equal to 4 that is homotopy equivalent to a sphere is PL isomorphic to a sphere.[1]
References
See Buoncristiano, Sandro (2003). "Fragments of Geometric Topology from the Sixties" (PDF). Geometry & Topology Monographs. 6.
Smale, Stephen (1961). "Generalized Poincaré's conjecture in dimensions greater than four". Ann. of Math. (2). 74 (2): 391–406. doi:10.2307/1970239. MR 0137124.
Stallings, John (1960). "Polyhedral homotopy spheres". Bulletin of the American Mathematical Society. 66: 485–488. doi:10.1090/S0002-9904-1960-10511-3.
Zeeman, Erik Christopher (1962). "The Poincaré conjecture for n greater than or equal to 5". Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961). Englewood Cliffs, NJ: Prentice–Hall: 198–204. MR 0140113.
Smale, Stephen (1962). "On the structure of manifolds". Amer. J. Math. 84 (3): 387–399. doi:10.2307/2372978. MR 0153022.
Newman, M. H. A. (1966). "The Engulfing Theorem for Topological Manifolds". Annals of Mathematics. (2). 84 (3): 555–571. doi:10.2307/1970460. MR 0203708.
Freedman, Michael (1982). "The topology of four-dimensional manifolds". Journal of Differential Geometry. 17 (3): 357–453. doi:10.4310/jdg/1214437136. MR 0679066.
Perelman, Grigori (11 November 2002). "The entropy formula for the Ricci flow and its geometric applications". arXiv:math.DG/0211159.
Perelman, Grigori (10 March 2003). "Ricci flow with surgery on three-manifolds". arXiv:math.DG/0303109.
Perelman, Grigori (17 July 2003). "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds". arXiv:math.DG/0307245.
Kervaire, Michel A.; Milnor, John W. (1963). "Groups of homotopy spheres: I". Annals of Mathematics. 2nd Ser. 77 (3): 504–537. doi:10.2307/1970128. JSTOR 1970128. MR 0148075. This paper calculates the structure of the group of smooth structures on an n-sphere for n > 4 {\displaystyle n>4} n>4.
Gluck, Herman (1962). "The Embedding of Two-Spheres in the Four-Sphere". Trans. Amer. Math. Soc. 104 (2): 308–333. doi:10.2307/1993581.
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