# Poincaré conjecture

The Poincaré conjecture says that a geometric object, as long as it has no hole, can be deformed into a sphere (i.e. shrunk, compressed, inflated, etc.). And that applies not only to a two-dimensional surface in three-dimensional space, but also to a three-dimensional surface in four-dimensional space.

The Poincaré conjecture is one of the best known, long unproven mathematical theorems , and was considered to be one of the most important unsolved problems in topology , a branch of mathematics. Henri Poincaré put it up in 1904. In 2000, the Clay Mathematics Institute ranked the Poincaré conjecture among the seven most important unsolved mathematical problems, the Millennium Problems , and awarded a million US dollars for its solution. Grigori Perelman proved the conjecture in 2002. In 2006 he was to receive the Fields Medal for his evidence, which he refused. On March 18, 2010, he was also awarded the Clay Institute's Millennium Prize, which he also declined. In a three-dimensional space, a surface is then homeomorphic to a (not bounded two-dimensional) spherical surface if every closed loop on this surface can be drawn together to a point .
The Poincaré conjecture claims that this also applies in the case of a four-dimensional space, if the surface is described by a 3-dimensional manifold (e.g. by a 3-sphere , a non-visual “surface of a 4-dimensional spherical equivalent ").

## Wording and description

Every simply connected , compact , unbounded , 3-dimensional manifold is homeomorphic to the 3-sphere .

In addition, there is a generalization of the conjecture on n -dimensional manifolds in the following form:

Every closed n-manifold with the homotopy type of an n-sphere is homeomorphic to the n-sphere.

In the event this generalized conjecture is consistent Poincaré conjecture with the original. ${\ displaystyle n = 3}$ In simplified terms, the Poincaré conjecture can be described as follows: The surface of a sphere is 2-dimensional, limited and borderless, and every closed curve can be drawn together to a point that is also on the sphere. It is (from a topological point of view) the only two-dimensional structure with these properties. The Poincaré conjecture is about the 3-dimensional analogue: This is about a 3-dimensional "surface" of a 4-dimensional body.

## Explanations

Manifold
A 3-dimensional manifold is something that looks like a 3-dimensional Euclidean space in a neighborhood of each point on the manifold .
Closed
In this context, closed means that the manifold is limited (that is, it does not extend into infinity) and that it has no boundary . A three-dimensional sphere is roughly a 3-manifold, but it has an edge (the surface). Therefore it is not closed. On the other hand, its surface is a closed 2-dimensional manifold. The Poincaré conjecture only makes one claim for closed manifolds.
Simply connected
Simply connected means that any closed curve can be contracted to one point. A rubber band on a spherical surface can always be moved on the surface in such a way that it becomes a point. On a torus (e.g. a bicycle tube), for example, the contraction does not always work: If the rubber band runs around the thinner side of the bicycle tube, it can never be drawn together to one point (you would have to cut the tube, which is not allowed in the topology) . Hence a torus is not simply connected.
3 sphere
In general, an n-sphere (designation:) is the edge of an (n + 1) -dimensional sphere . A 1-sphere is the circular line of a circular area . A 2-sphere is the surface of a 3-dimensional sphere. A 3-sphere is the surface of a 4-dimensional sphere. This object can of course no longer simply be imagined because it actually "lives" in a 4-dimensional space. Mathematically, the 3-sphere can easily be described by a formula, namely as the set of all points in 4-dimensional real space that are at a distance of 1 from the zero point: ${\ displaystyle S ^ {n}}$ ${\ displaystyle S ^ {3} = \ {(x_ {1}, x_ {2}, x_ {3}, x_ {4}) | x_ {1} ^ {2} + x_ {2} ^ {2} + x_ {3} ^ {2} + x_ {4} ^ {2} = 1 \}}$ A 2-sphere consists of two (hollow) hemispheres that are joined together at the edges. Topologically, these hollow hemispheres are actually circular surfaces (if you press them flat from above, two disks are created). With this you can get a 2-sphere by gluing two circular surfaces together at the edges. In the same way one can construct a relatively clear picture of a 3-sphere. You take two spheres (corresponds to the circular areas in 2-dimensional) and "glue" them together at the corresponding points on the surface. A path on the 3-sphere thus begins in one of the two spheres. When you get to the edge, you jump to the corresponding point on the second ball and vice versa. In this way one can describe paths on the 3-sphere in 3-dimensional space. You can also see in this way that there is nowhere an edge. The 3-sphere is thus closed.

## The conjecture in higher dimensions

We call a manifold M as m-connected if every picture of a k -sphere to M for k <= m can be contracted to a point. For m = 1 this gives exactly the concept of 'simply connected' described above. A formulation of the n -dimensional Poincaré conjecture now says the following:

A compact unbounded n -dimensional manifold is (n-1) -connected if and only if it is homeomorphic to the n -sphere.

An argument with Poincaré duality shows that one can also replace (n-1) with (n-1) / 2 here . For n = 3 , the formulation of the Poincaré conjecture given above results exactly.

There are a number of other equivalent formulations that are often found in the literature. One replaces the condition (n-1) -contiguous by requiring that the manifold is already homotopy-equivalent to the n -sphere. These two conditions are equivalent according to Hurewicz's theorem. Homotopy equivalence is a coarser equivalence relation than homeomorphism, but it is often easier to check. The Poincaré conjecture says that these two relations fortunately coincide in the case of the sphere.

Another equivalent condition is that the manifold is simply connected and has the same homology as an n -sphere. While this description is more technical, it has the advantage that it is often relatively easy to compute the homology of a manifold.

While it has long been known in dimension 3 that every manifold that is homeomorphic to the sphere is also diffeomorphic to the sphere, this is not the case in higher dimensions. From dimension 7 there are so-called exotic spheres that are homeomorphic, but not diffeomorphic to the standard sphere . Thus, in the Poincaré conjecture of n> 6, 'homeomorphic' cannot be replaced by 'diffeomorphic'.

## history

Originally Poincaré had put forward a somewhat different assumption: He believed that every 3-dimensional closed manifold that has the same homology as a 3-sphere must already be topologically a sphere. While Poincaré initially believed he had a proof that made do with this weaker assumption, the requirement that the manifold is simply connected turned out to be indispensable. With the Poincaré homology sphere, Poincaré himself found a counter-example to his original conjecture: It has the same homology as a 3-sphere, but is not simply connected and therefore cannot even be homotopy-equivalent to a 3-sphere. Therefore, he changed his assumption to the statement known today.

It is interesting that the n-dimensional Poincaré conjecture has very different proofs in different dimensions, while the formulation is general.

For the statement is considered classic; in this case even all (closed) 2-dimensional manifolds are known and classified . ${\ displaystyle n = 2}$ In the case , Stephen Smale's conjecture was proven in 1960 (for smooth and PL-manifolds) using techniques from Morse theory . It follows from his H-cobordism theorem . For this proof, among other things, he received the Fields Medal in 1966 . Max Newman later extended his argument to topological manifolds. ${\ displaystyle n> 4}$ Michael Freedman solved the case in 1982. He also received the Fields Medal for it in 1986 . ${\ displaystyle n = 4}$ The case has proven (unsurprisingly) to be the most difficult. Many mathematicians have presented evidence but it turned out to be false. Still, some of this flawed evidence has broadened our understanding of the low-dimensional topology. ${\ displaystyle n = 3}$ ## proof

At the end of 2002, reports surfaced that Grigori Perelman from the Steklov Institute in Saint Petersburg had proven the suspicion. He uses the analytical method of the Ricci flow developed by Richard S. Hamilton to prove the more general conjecture of the geometrization of 3-manifolds by William Thurston , from which the Poincaré conjecture follows as a special case. Perelman published his chain of evidence, which spanned several publications and a total of around 70 pages, in the online archive arXiv . The work has since been reviewed by mathematicians around the world, and in recognition of the accuracy of his proof, Grigori Perelman was awarded the Fields Medal at the 2006 International Congress of Mathematicians in Madrid , which, as he previously announced, he did not accept.

Since Perelman himself shows no interest in a more detailed presentation of his proof, different groups of mathematicians have taken on this: Bruce Kleiner and John Lott published their elaboration of many details soon after Perelman's work became known and added them to 192 pages several times. John Morgan and Tian Gang published a complete 474-page report on the arXiv in July 2006 . Also Huai-Dong Cao and Zhu Xiping in 2006 published a proof of the Poincaré conjecture and the geometrization, Holding forth exactly worked out the proof of Perelman on 300 pages.

## Meaning of the guess

The proof of the Poincaré conjecture is an important contribution to the classification of all 3-manifolds. This is because Perelman actually proves the more general geometrization conjecture over closed 3-manifolds, which contains the Poincaré conjecture as a special case.