Towards the poincare conjecture and the classification of 3. The poincare conjecture one hundred years later 2006. John willard milnor institute for mathematical sciences, stony brook university, new york for pioneering discoveries in topology, geometry and algebra. This work depends on the accumulative works of many geometric analysts. In this paper, we investigate a conjecture due to s. The full geometrization conjecture requires some more argument and i gather that perelman is supposed to at some point produce another preprint with more about this. We begin by constructing a morse function for w 02. The poincare conjecture is a topological problem established in 1904 by the french mathematician henri poincare. All of milnors works display marks of great research. For poincare series of binary polyhedral groups and coxeter polynomials there are obtained statements close to the euclid algorithm and orthogonal polynomials. Over time, the conjecture gained the reputation of being particularly tricky to tackle. The generalized poincare conjecture in higher dimensions.
Notes and commentary on perelmans ricci flow papers. The poincare conjecture, by john milnor in the millennium prize problems, clay mathematics institute and the american mathematical society, 2006. Ricci flow and the poincare conjecture siddhartha gadgil and harish seshadri the eld of topology was born out of the realisation that in some fundamental sense, a sphere and an ellipsoid resemble each other but di er from a torus the surface of a rubber tube or a doughnut. It was proved by vladimir voevodsky 1996, 2003a, 2003b. For example, in voevodskys original plan of attack on the milnor s conjecture 70, he proves that a certain map in the motivic stable category is nonzero see 70, proposition 3.
Lott posted a paper on the arxiv in may 2006 which filled in the details of perelmans proof of the geometrization conjecture. The annals of mathematics is currently published by annals of mathematics. A striking instance of this can be seen by imagining water. Sep 08, 2004 by now the situation seems to be that the experts are pretty convinced of the details of perelmans proof for the poincare conjecture. Generalized poincares conjecture in dimensions greater than. In mathematics, the milnor conjecture was a proposal by john milnor of a description of the milnor ktheory mod 2 of a general field f with characteristic different from 2, by means of the galois or equivalently etale cohomology of f with coefficients in z2z. Nov 11, 2002 recent progress on the poincare conjecture and the classification of 3manifolds, by john morgan, bull.
In fact there is a simple list of all possible smooth compact orientable surfaces. Milnor conjecture mc mc for n 2 is a theorem of merkurjev 1981the. Generalized poincare s conjecture in dimensions greater than four. Apr 23, 2014 the famed poincare conjecture the only millennium problem cracked thus far. These earlier successes in higher dimensions left the case of three dimensions in limbo.
The development of algebraic topology in the 1950s and 1960s was deeply influenced by the work of milnor. This problem was directly solved between 2002 and 2003 by grigori perelman, and as a consequence of his demonstration of the thurston geometrisation conjecture. In knot theory, the milnor conjecture says that the slice genus of the, torus knot is. A conjecture on poincarebetti series of modules of differential operators on a generic hyperplane arrangement snellman, jan, experimental mathematics, 2005. Notes on perelmans papers, by bruce kleiner and john lott, may 25, 2006. The generalized poincare 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. Work on the conjecture improved understanding of 3manifolds. If the address matches an existing account you will receive an email with instructions to reset your password. However, examples due to donaldson show that in general the simplyconnected hcobordism theorem fails for n 5.
The generalized poincare conjecture in higher dimensions by stephen smale1 communicated by edwin moise, may 20, 1960 the poincare conjecture says that every simply connected closed 3manifold is homeomorphic to the 3sphere s3. Homotopy theory, homology and cohomology, manifolds, and expository papers. A discussion of thurstons geometrization conjecture. The poincar e conjecture and the hcobordism theorem. From may to july 2006, several groups presented papers that filled in the details of perelmans proof of the poincare conjecture, as follows. For n 3 it follows from the 3dimensional poincar e conjecture proved by perelman. Poincare conjecture, in topology, conjecturenow proven to be a true theoremthat every simply connected, closed, threedimensional manifold is topologically equivalent to s3, which is a generalization of the ordinary sphere to a higher dimension in particular, the set of points in.
These are related to the delicate question of nonvanishing of lfunctions associated to peri. Perelman and the poincare conjecture not even wrong. In particular, we will not discuss the important and quite different results on this conjecture in four and higher dimensions by smale, freedman, etc. John willard milnor international mathematical union. John milnor commented that sometimes the errors in false proofs can be rather subtle and difficult to detect. Collected papers of john milnor, john willard milnor volume 19, issue 4 of collected works american. A class of examples closely related to the 3sphere are the 3dimensional spherical spaceforms, i. Terence tao perelmans proof of the poincare conjecture. The conference to celebrate the resolution of the poincare conjecture, which is one of the clay mathematics institutes seven millennium prize problems, was held at the institut henri poincare in paris. Collected papers of john milnor john willard milnor. Mc is a theorem due to voevodsky 2003 and orlovvishikvoevodsky 2007. Jun 29, 2011 john milnor stating the poincare conjecture at the beginning of his talk spheres given at the symposium in honor of the abel prize 2011 laureate. Our ma,in task, however, is to describe in some detail a proof of the theorem. Towards the poincare conjecture and the classification of 3manifolds john milnor 1226 notices of the ams volume 50, number 10 t he poincare conjecture was posed ninetynine years ago and may possibly have.
Experts in the field were often reluctant to announce proofs, and tended to view. John milnor states the poincare conjecture youtube. This webpage is meant to be a repository for material related to perelmans papers on ricci flow. It characterises threedimensional spheres in a very simple way. Notes and commentary on perelmans ricci flow papers introduction. In this paper, we give a complete proof of the poincar. This socalled smooth poincare conjecture, in dimension four, remains open and is thought to be very difficult. Our main attempt is to accomplish this by using structure results for limits spaces of manifolds with ricci curvature bounded. It is in a similar vein to the thom conjecture it was first proved by gauge theoretic methods by peter kronheimer and tomasz mrowka. Mc as stated above is a consequence of two conjectures of milnor relating milnor ring k f with mod 2 galois cohomology ring and the graded witt. Articles are listed by date of first public availability. Milnor s exotic spheres show that the smooth poincare conjecture is false in dimension seven, for example. In this collection of papers the reader finds those original papers and some previously unpublished works. Experts in the field were often reluctant to announce proofs, and tended to view any.
Towards the poincare conjecture and the classification of. There is one problem on this list in common with hilberts list, the riemann. On a conjecture of milnor about volume of simplexes. For example, in voevodskys original plan of attack on the milnors conjecture 70, he proves that a certain map in the motivic stable category is nonzero see 70, proposition 3.
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