Generalized Poincaré conjecture
Encyclopedia
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
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. 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
.
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.
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.
Michael Freedman
solved the case n = 4 (in TOP) 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 August 2006 and the Millenium Prize
from the Clay Mathematics Institute
in March 2010.
s: you can interpret these as non-standard smooth structures on the standard (topological) sphere.
Thus the homotopy sphere
s that Milnor produced are homeomorphic (Top-isomorphic, and indeed piecewise linear homeomorphic) to the standard sphere Sn, but are not diffeomorphic (Diff-isomorphic) to it, and thus are exotic sphere
s: 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. 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.
s, 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
area of topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, the term Generalized Poincaré conjecture refers to a statement that a manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
which is a homotopy sphere
Homotopy sphere
In algebraic topology, a branch of mathematics, a homotopy sphere is an n-manifold homotopy equivalent to the n-sphere. It thus has the same homotopy groups and the same homology groups, as the n-sphere...
'is' a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
. More precisely, one fixes a
category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
of manifolds: topological
Topological manifold
In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...
(Top), differentiable
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
(Diff), or piecewise linear
Piecewise linear manifold
In mathematics, a piecewise linear manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions.An isomorphism of PL manifolds is called a PL...
(PL). Then the statement is
- Every homotopy sphere (a closed n-manifold which is homotopy equivalentHomotopyIn topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
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
Poincaré conjecture
In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
, which was made for (topological or PL) manifolds of dimension 3, where being a homotopy sphere is equivalent to being simply connected. 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
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
recipients John Milnor
John Milnor
John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. He won the Fields Medal in 1962, the Wolf Prize in 1989, and the Abel Prize in 2011. Milnor is a distinguished professor at Stony Brook University...
, Steve Smale, Michael Freedman
Michael Freedman
Michael Hartley Freedman is a mathematician at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the Poincaré conjecture. Freedman and Robion Kirby showed that an exotic R4 manifold exists.Freedman was born...
and Grigori Perelman
Grigori Perelman
Grigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology.In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston's geometrization conjecture...
.
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 unsettled .
A fundamental fact of differential topology
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
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.
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
Stephen Smale
Steven Smale a.k.a. Steve Smale, Stephen Smale is an American mathematician from Flint, Michigan. He was awarded the Fields Medal in 1966, and spent more than three decades on the mathematics faculty of the University of California, Berkeley .-Education and career:He entered the University of...
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
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
for his work in 1966. Shortly after Smale's announcement of a proof, John Stallings
John R. Stallings
John Robert Stallings Jr. was a mathematician known for his seminal contributions to geometric group theory and 3-manifold topology. Stallings was a Professor Emeritus in the Department of Mathematics at the University of California at Berkeley where he had been a faculty member since 1967...
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
Max Newman
Maxwell Herman Alexander "Max" Newman, FRS was a British mathematician and codebreaker.-Pre–World War II:Max Newman was born Maxwell Neumann in Chelsea, London, England, on 7 February 1897...
extended PL engulfing to the topological situation and proved that for n ≥ 5 a topological homotopy n-sphere is homeomorphic to the n-sphere.
Michael Freedman
Michael Freedman
Michael Hartley Freedman is a mathematician at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the Poincaré conjecture. Freedman and Robion Kirby showed that an exotic R4 manifold exists.Freedman was born...
solved the case n = 4 (in TOP) in 1982 and received a Fields Medal in 1986.
Grigori Perelman
Grigori Perelman
Grigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology.In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston's geometrization conjecture...
solved the last case n = 3 (where TOP, PL, and DIFF all coincide) in 2003 and was awarded a Fields Medal in August 2006 and the Millenium Prize
Millennium Prize Problems
The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of September 2011, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize being awarded by the institute...
from the Clay Mathematics Institute
Clay Mathematics Institute
The Clay Mathematics Institute is a private, non-profit foundation, based in Cambridge, Massachusetts. The Institute is dedicated to increasing and disseminating mathematical knowledge. It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998...
in March 2010.
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 sphereExotic sphere
In differential topology, a mathematical discipline, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere...
s: you can interpret these as non-standard smooth structures on the standard (topological) sphere.
Thus the homotopy sphere
Homotopy sphere
In algebraic topology, a branch of mathematics, a homotopy sphere is an n-manifold homotopy equivalent to the n-sphere. It thus has the same homotopy groups and the same homology groups, as the n-sphere...
s that Milnor produced are homeomorphic (Top-isomorphic, and indeed piecewise linear homeomorphic) to the standard sphere Sn, but are not diffeomorphic (Diff-isomorphic) to it, and thus are exotic sphere
Exotic sphere
In differential topology, a mathematical discipline, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere...
s: they can be interpreted as non-standard differentiable structures on the standard sphere.
Michel Kervaire
Michel Kervaire
Michel André Kervaire was a French mathematician who made significant contributions to topology and algebra. He was the first to show the existence of topological n-manifolds with no differentiable structure , and computed the number of exotic spheres in dimensions greater than four...
and John Milnor showed that the oriented
Orientability
In mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...
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. 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.
PL
For piecewise linear manifoldPiecewise linear manifold
In mathematics, a piecewise linear manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions.An isomorphism of PL manifolds is called a PL...
s, 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.