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. 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. [ Grigori Perelman declined the award.]

tatus

Here is a summary of the status of 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 (as of 2006).

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 [http://www.msp.warwick.ac.uk/gt/gtmcontents6.html Fragments of Geometric Topology from the Sixties] by Sandro Buoncristiano, in Geometry & Topology Monographs, Vol. 6 (2003)]

History

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

Stephen Smale solved the cases $n\; geq\; 7$ in 1960 and subsequently extended his proof to $n\; geq\; 5$ (in Top and PL); he received a Fields Medal for his work in 1966.

Michael Freedman solved the case $n\; =\; 4$ (in Top, but not PL) in 1982 and received a Fields Medal in 1986.

Grigori Perelman solved the last case $n\; =\; 3$ (where Top, PL, and Diff all coincide) in 2003 and was awarded a Fields Medal in 2006, which he declined.

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, the exotic spheres: you can interpret these as non-standard smooth structures on the standard (topological) sphere.

Thus the homotopy spheres that 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 John 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. [Michel A. Kervaire; John W. Milnor. " [http://links.jstor.org/sici?sici=0003-486X%28196305%292%3A77%3A3%3C504%3AGOHSI%3E2.0.CO%3B2-R Groups of Homotopy Spheres: I] " in "The Annals of Mathematics", 2nd Ser., Vol. 77, No. 3. (May, 1963), pp. 504-537. This paper calculates the structure of the group of smooth structures on an n-sphere for n > 4.] 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. [ Herman Gluck, "The embedding of two-spheres in the four-sphere,", Trans. Amer. Math. Soc. 104 (1962) 308-333.]

PL

For piecewise linear manifolds, the Poincaré conjecture is true except possibly in 4 dimensions, 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. [ See [http://www.msp.warwick.ac.uk/gt/gtmcontents6.html Fragments of Geometric Topology from the Sixties] by Sandro Buoncristiano, in Geometry & Topology Monographs, Vol. 6 (2003)]

References