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Poincare is one of the greatest and most creative mathematicians and physicists in the world. He nearly discovered the special theory of relativity before Einstein. He almost founded an extremely important branch of modern mathematics-algebraic topology. Poincare has extensive knowledge and great achievements, and his research covers several branches of mathematics, as well as astromechanics, modern physics and even psychology, so he is known as the last great scientific generalist in the world.

Poincare, much like Riemann, likes to carry out his own research based on basic principles, rather than based on the work of others or even his own previous work. Nowadays most mathematicians regard him as one of the greatest geniuses of all time. He also has a strong interest in the nature of mathematical thinking. In 1908, on the basis of reflecting on his thought process, he made a famous speech on the creativity of mathematics, entitled "the invention of mathematics".

Poincare himself wrote ∶, "We prove it by logic and create it by intuition." In particular, he disagrees with Hilbert's view that ∶ mathematical reasoning can be axiomatized and (in principle) "mechanized". It was a plan that Poincare thought impossible, and Godel later proved him right.

Poincare's first major contribution to mathematics is the creation of the concept and theory of automorphic functions, which are special functions from complex to complex numbers. In his subsequent career, Poincare made a further study of functions involving complex numbers, and he was generally regarded as the founder of the theory of analytic functions of multiple complex variables. He also studied number theory and geometry.

Poincare conjecture in Poincare's study of topology, there is a world problem ∶ Poincare conjecture. In 1895, Poincare published his book position Analysis. In this book, Poincare introduces almost all the concepts and main methods of topology.

Topology is a kind of hypergeometry. Mathematicians study the very general properties of surfaces and other mathematical objects through topology. In topology, mathematicians are mostly interested in three-dimensional or higher-dimensional mathematical objects. Poincare mistakenly believes that a fact that is obvious about two-dimensional objects will also be true for similar objects with three dimensions or higher dimensions (this is the source of Poincare's conjecture).

Two-dimensional topology is sometimes referred to associatively as "rubber membrane geometry".

Rubber membrane Geometry in 1931, Baker of England designed what is now the London Underground. This map is considered to be one of the best maps, and many people try to improve it without success. This map combines convenience with the beauty of appearance and is now a symbol of London and a model of subway maps all over the world.

The application of topology ∶ London Underground Map. This map shows the great power of topology. In fact, this map is inaccurate in every respect. But it accurately describes the information a passenger needs to get from the map-where to get on the bus, where to get off, where to change lanes, at the expense of all other details.

This example illustrates the nature of two-dimensional topology. If the subway map is printed on an extremely elastic rubber film, it can be stretched and compressed so that every detail is correct, resulting in a standard, geographically accurate map. In mathematical terms, the reason is that the layout of this network structure (defined as a set of points connected by different lines) is a topological property. To put it simply, the network is a topological object. You can distort and stretch any connection in a network without changing its overall layout. To change the network, you have to disconnect a connection or add a new one. This is true for circuit diagrams, circuits themselves, computer chips, telephone networks and the Internet.

This is why rubber film geometry is one of the most important branches of mathematics in the world today. In the case of a subway map, as long as it is topologically accurate, it does not matter whether the map is accurate or not. Similarly, for the design of circuits or computer chips, the important thing is the layout of the network. If the layout is topologically accurate, the location of the wires can be changed at will to meet other design requirements. The same is true in the design of computer chips, the key is that the circuits etched on silicon wafers must be topologically accurate.

Generally speaking, two-dimensional topology (rubber film geometry) is the study of such a property of a figure. ∶ draws the figure on a (hypothetical) rubber film with excellent elasticity, and then distorts and stretches the film, and this property remains the same. In fact, the development of topology is not driven by the needs of any field of applied mathematics. Instead, it comes from within pure mathematics, from the struggle to understand why calculus works.

Calculus and Topology from the moment Newton and Leibniz invented calculus in the mid-17th century, mathematicians widely used it. But no one really understands why calculus works. It was finally explained by a large number of mathematicians who worked hard for 300 years (they made a detailed analysis of the nature of real numbers and infinite processes, as well as mathematical reasoning itself).

But it also makes mathematics more and more abstract. A large number of new types of objects and patterns emerged in the 19th century, which are not part of everyday experience. Among the new objects and patterns studied by mathematicians in the last 200 years, there are non-Euclidean geometry (parallel lines can intersect), four-dimensional and higher-dimensional geometry, infinite dimensional geometry, algebra that uses symbols to represent graphic symmetry (called group theory), algebra that uses symbols to represent logical thinking (propositional logic), and algebra (vector algebra) that uses symbols to represent motion in two-dimensional or three-dimensional space.

In the abstract expansion, topology also appeared. At first, the idea was to invent a kind of "geometry" to study the properties of graphics that will not be destroyed by continuous deformation, so this geometry does not depend on the concepts of lines, circles, cubes, nor on measures such as length, area, volume, and angle. In topology, the object of study is called topological space.

The connection between topology and "how calculus works" is subtle. In essence, both depend on being able to grasp the infinitesimal. But what does topological transformation have to do with infinitesimal? The key is that ∶ intuitively says that the essence of topological transformation is that two points that are "infinitely close" before the transformation are still "infinitely close" after the transformation. In particular, no matter how stretch, compress or twist a rubber film, it will not destroy this closeness. The two points that are initially close to each other remain close after the completion of the operation.

Note that the concept of proximity here is relative to all other points in the topological space. We can stretch the film so that the two points that were close together in the first place no longer seem to us to be close together. But in this case, the change in closeness is a geometric change that we impose from the outside. From the point of view of the rubber film, the two points are still close together.

The only way to destroy closeness is to cut or tear the film-an operation that is prohibited in topology.

To develop topology, mathematicians must find a way to grasp the key idea of relative proximity. To this end, they set out to find a way to clarify the hypothetical concept of two points of "infinite proximity". Intuitively, topological transformation has this property.

If the two points are infinitely close to each other at first, they will remain the same after this transformation.

The problem with this approach is that the concept of infinite proximity is not a well-defined (well-defined) concept. However, by considering topological transformations in this way, mathematicians have found a way to give a precise definition of topological transformations (don't expect me to define them here). At this time, we can use the concept of topological transformation to accurately analyze the intuitive concept of "infinite proximity". In this way, they developed calculus in a strict sense.

This is the main reason why Poincare and other mathematicians founded topology. People who encounter topology for the first time will have a problem ∶ about topological space. Topological space has not only no straight line, no concept of fixed shape, but also no distance of any type. All you can say is when two o'clock are close to each other.

There are a lot of things you didn't think of.

Topology is one of the most colorful, charming and important branches of contemporary mathematics, which has many applications in mathematics, physics and other fields. Here is only one important application: ∶ topology is the mathematical basis of superstring theory, which is the latest theory about the nature of the universe.

Let's take a look at what topologists study. For simplicity, I will confine myself to the two-dimensional case. And think about what properties in high school geometry can be transferred to topology. Because stretching and twisting the rubber film will turn straight lines into curves and change distances and angles, these familiar geometric concepts are meaningless in topology.

So what's left? There are also lines and closed circles (rings). If you draw a circle on a rubber film with excellent elasticity, the circle is still a circle no matter how much you stretch, compress and twist the rubber film. What else is there?

To answer this question, let me show you the first topology achievement. It is attributed to the great Swiss mathematician Euler. In 1735, Euler solved a long-standing problem-the Konigsberg Bridge. Many citizens of Konigsburg are used to coming here for a walk with their families. They often have to cross several bridges. So there is a frequently discussed question: does ∶ have a route that happens to be crossed only once per bridge?

Euler realized that the exact location of the island and the bridge did not matter, but how the bridges were connected, that is, the network formed by the bridge. This problem is a topology problem, not a geometry problem.

So Euler argues as follows. Take any network and assume that there is a route that happens to be passed only once on each side. Any node, as long as it is not the beginning or end of the route, must have an even number of edges intersecting here, because these edges can be paired in one way and out the other. But in this network of bridges, all four vertices have an odd number of edges intersecting there. Therefore, there can be no such route. The conclusion is that the exact route of each bridge through Konigsberg does not exist.

The node of the grid represents land, which is the answer to the Konigsberg Bridge problem, which produces the world's first topological theorem. Euler proved that for any network drawn on a plane, if V is the total number of vertices (nodes), E is the total number of edges (or lines), and F is the total number of "faces" (a closed area surrounded by three or more edges), then the following simple formula holds ∶.

For example, with regard to the network of the Konigsberg Bridge, there are Vantage 4, Magic 7, and Fidd4, so

Although Euler solved the first topology problem and proved the first topology theorem, it was not until the late 19th century that topology really started. Because at this moment, Poincare took the stage!

Through the surface in topology, we study the properties of graphics and objects that remain unchanged under a continuous deformation. By continuous, we mean that the deformation does not involve any cutting, tearing or pasting.

For example, in topology, rugby is the same as football, and they are the same as tennis, because any of these three balls can be transformed into the other two by continuous deformation. There is only one "ball" in topology. Usually we identify the differences between all kinds of balls, are related to the size and shape, but these are not topological properties.

Poincare was a leading figure in the early study of topology, which looked for ways to show when two shapes were topologically different.

For example, although any two spheres have the same topology, and any two torus (circle, ellipse, or whatever) have the same topology, any sphere is topologically different from any torus. Intuitively, this seems obvious. After all, you don't seem to have a way to transform a sphere continuously to get a torus. The problem is that the unimportant word-- it seems. How do you know there's no way to do that? For example,

In the sub-loop intelligence problem shown above, can you find a way to continuously transform the figure (a) into the figure (b)? The easy way to think of is to cut off one of the two interlocking rings (but this is not allowed), as shown in (c). But you don't have to cut the ring to do it. This should convince you that it is indeed an important task to find a variety of completely reliable ways to prove that two objects have the same or different topologies.

By the way, it is impossible to confirm that the topologies of the two objects are different just by failing to find a continuous deformation that turns one object into another. What is needed here is to find some topological property that one of the two objects has but the other does not-that is, the property that remains unchanged after continuous deformation.

We have come across a nature like this. For any network, the value of V-EigenF is a topological property. This amount is the same for any network.

For a grid on a two-dimensional plane, VmureEffect F = 1; for a network on a sphere (to cover the whole sphere, not a part of the sphere), Vmure Found2; and for a network on a torus (which is also required to cover the whole torus), Vmure Elastic F = 0. Therefore, we can assert with absolute certainty that two-dimensional planes, spheres and torus are topologically different. For a network drawn on a double torus (shaped like the number 8), V-E+F=-2, so we also know that the double torus is topologically different from the plane, sphere, and torus.

For any network drawn on a particular surface, the value of the expression V-E+F is an example of what mathematicians call surface topology invariants. If we make a topological transformation (that is, continuous deformation) of the surface, this value will remain the same. In order to commemorate the contribution of Euler, the value of V-E+F is called the Euler characteristic number of the surface. Topologists have found many topological invariants that can be used to determine whether two specific surfaces are topologically equivalent, and the Euler characteristic number is one of them.

Another topological invariant is the chromatic number of a surface. It originates from a classic problem about map coloring. In 1852, a young English mathematician named Guthrie raised a seemingly insignificant question, ∶.

At least how many colors do you need to be able to color each area on any map?

The only rule is that any two areas that share a common boundary must be colored differently. (if two regions only touch each other at one point, then this point cannot be regarded as a common boundary. ) it is easy to draw maps that require four different colors, but is there a map that requires five colors? The answer is no, but it took mathematicians more than 100 years to prove it, which involves not only ingenious mathematical reasoning, but also the important application of computers. In fact, the four-color theorem is the first mathematical conjecture that can only be proved by using a computer.

The four-color theorem is obviously a topological result, because the continuous deformation of the paper with a map does not change the pattern of the common boundary. Two areas that share a common boundary before deformation remain the same after deformation, and vice versa.

The four-color theorem, and the original question it answers, are all for maps drawn on the plane. But you can ask the same question about a map drawn on any surface. The chromatic number of a surface is at least the number of colors needed to color any map painted on the surface. According to the four-color theorem, the chromatic number of a plane is 4. The chromatic number of the sphere is also 4. The chromatic number of the torus is 7.

Another topological invariant originates from the concept of "sidedness"-whether a surface has one side or two sides. Any surface has two sides, isn't it? The answer is no. It is easy to construct a surface with only one side. Take a long, narrow piece of paper tape, twist it for half a week, and then glue the two ends together to form a twisted paper circle

This twisted circle of paper is a curved surface with only one side, which is called the Mobius belt. In addition to having only one side, the Mobius belt has only one side.

Mobius takes this example to tell us that laterality is closely related to marginality. Usually, mathematicians focus on surfaces without edges-they call them closed surfaces. Moreover, the more interesting topological properties are related to the internal structure of the surface (how the surface is twisted and flipped). In fact, for every surface with one or more edges, there is generally a closed surface with almost the same properties. For example, a sphere is similar to a finite plane (such as a flat desktop). When we prove a topological result about a sphere, we usually have a result about the plane immediately, and vice versa.

Intuitively, this is because we can take a fully stretchable piece of flat paper and fold its edges to form a closed bag-topologically this is a sphere.

The closed surface corresponding to the Mobius belt is called a Klein bottle. The Klein bottle has no edges and has neither the inside nor the outside. In theory, you can take two Mobius belts and glue them together along their single edge to form a Klein bottle. I say "theoretically" because you can't do this in an ordinary three-dimensional space. Klein bottles (as an object of mathematics) exist only in four-dimensional space. In our three-dimensional world, you'd better allow this surface to pass through itself.

In four-dimensional space, there is no need for the bottle to pass through itself. For ordinary people, an object that exists only in four-dimensional space is certainly not real, but mathematicians don't think so. After all, everyone "knows" that negative numbers have no square root, but that doesn't stop mathematicians from creating plural numbers and using them in practical applications. Many of the great powers of mathematics come from the fact that ∶

We can use it to study objects beyond the imagination of creatures in a three-dimensional world.

For example, we can study the properties of the network painted on the Klein bottle. We find that the Euler number of the Klein bottle is 0, which is the same as the torus. Does this mean that the Klein bottle is topologically equivalent to the torus? No! The Euler index number can not distinguish between the Klein bottle and the torus, but the chromatic number of the Klein bottle is 6 and the torus is 7.

The topological property of Klein bottle "corresponding to its surface unilateralism" is a strange concept called non-directionality. It means that on the surface of the Klein bottle you cannot distinguish between left-handed and right-handed or clockwise and counterclockwise rotation. If you draw a left hand on the surface of a Klein bottle, and then slide the figure far enough along the surface (far enough means that if the Klein bottle is in three-dimensional space, then the hand has to pass completely through the bottleneck of self-intersection), so when it returns to the starting point, you will find that it has incredibly become a right hand.

This experiment is easier to do on the Mobius belt. Draw a small left hand on the surface, then copy the figure near it, and repeat the process until you get back to where you started. Then you will find that the left hand becomes the right hand. Or, draw a small circle on the surface of the Klein bottle or on the Mobius belt and use an arrow to indicate clockwise rotation. If you slide along the surface or copy the figure until you return to the starting point, you will find that the arrow points counterclockwise.

Surfaces that cannot turn the left hand into the right hand or counterclockwise by sliding along the surface are called orientable. For example, a sphere (or plane) is orientable, as are torus and double torus. A surface that can make these changes, such as a Klein bottle or Mobius belt, is said to be unorientable. Orientability (or non-directionality) is a topological invariant.

One of the first achievements of surface classification topology is to prove that any two closed surfaces can be distinguished as long as there are two topological invariants, Euler property number and orientability. That is to say, if two closed surfaces have the same Euler characteristic number, and both are orientable or unorientable, then they are in fact equivalent. This result is called the surface classification theorem because it says that as long as you use these two features, you can classify all surfaces (in the topological sense).

To put it simply, the proof of the surface classification theorem is made by taking the sphere as the basic surface and measuring the difference between any given surface and the sphere (that is, what has to be done to the sphere in order to transform the sphere into that surface). This is consistent with our intuition that the ∶ sphere is the simplest and most basic closed surface.

In this case, the operation on the sphere in order to transform the sphere into some other surface goes beyond the conventional topological operation of continuous deformation. Indeed, if you change the sphere by twisting, bending, stretching, and compressing, the resulting object is still a sphere in the topological sense. To figure out how the surface is constructed from the sphere to classify the surface, cutting and stitching must be allowed in addition to the usual torsion, stretching, etc. Topologists call this process "cutting and mending". A typical cutting and mending technique involves cutting one or more pieces from a sphere, twisting, flipping, stretching, or compressing them, and then sewing them back onto the sphere.

The classification theorem tells us that any orientable surface is equivalent to a spherical topology with a certain number of "ring handles" stitched on the surface. You can get a ring handle by cutting two holes in the sphere and connecting them with a tube. as shown on the left side of the image below, any unorientable surface is equivalent to a sphere with a certain number of "cross sets" stitched together. You can cut a hole in the sphere and sew a Mobius band on the edge of the hole to get a cross set, as shown on the right side of the image below. As in the case of Klein bottles, this cannot be done without letting Mobius pass through itself in ordinary three-dimensional space.

The surface classification theorem says that any smooth closed surface is equivalent to a spherical topology with a certain number of ring handles or cross sets.

In the early 20th century, Poincare and other mathematicians began to classify the high-dimensional analogues of surfaces, which they called "manifolds". The method they try is similar to the one that already works for two-dimensional surfaces. They take a three-dimensional analogue of a sphere (called "three-dimensional sphere") as the basis, and measure the difference between any three-dimensional manifold and the three-dimensional sphere (3-manifold for short) in order to try to classify all three-dimensional manifolds.

Note that a regular surface, such as a sphere or torus, is a two-dimensional object. Although the part surrounding the surface is three-dimensional, the surface itself is two-dimensional.

Any surface other than a plane can only be constructed in a three-dimensional or higher-dimensional space. Therefore, any closed surface requires three-dimensional or higher-dimensional space. For example, to construct a sphere or torus is to take a three-dimensional space, and to construct a Klein bottle is to take a four-dimensional space. However, a sphere, a torus, or a Klein bottle is a two-dimensional object-a surface without thickness, which can in principle be made of a flat sheet with excellent elasticity.

But just as a sphere can be seen as a two-dimensional analogue of a circle (it is an one-dimensional object-a curve, in two-dimensional space), we can also imagine a three-dimensional analogue of a sphere (in four-dimensional space). Of course, in fact, we can't imagine. But we can write an equation to determine such an object and study it mathematically. In fact, physicists usually study such imaginary objects and use these results to help understand our universe. A 3-manifold, a three-dimensional analogue of a surface (which exists in a four-dimensional or higher-dimensional space), is sometimes called a hypersurface, while a three-dimensional analogue of a sphere is called a hyperspherical surface.

You can write equations that determine three-dimensional, four-dimensional, five-dimensional, six-dimensional or any dimensional manifolds. The mathematical theory of matter currently studied by physicists regards our universe as having 11 dimensions. According to these theories, what we are directly aware of is the three dimensions of these dimensions. Other dimensions represent themselves as different physical properties, such as electromagnetic radiation and the force that binds atoms together.

Poincare tries to classify three-dimensional and higher-dimensional manifolds by taking the "sphere" of each dimension as a basic figure and then applying the cut-and-complement technique. In this attempt, the first step is naturally to find a simple topological property that tells you when a given (hyper) surface is equivalent to a (hyper) sphere topology.

Remember, what we're doing here is topology. Even in the simple case of a conventional two-dimensional surface, a surface may be extremely complex, but the result can still be transformed into a sphere by continuous deformation.

In the case of two-dimensional surfaces, there is such a property. Suppose you take a pencil and draw a simple closed circle on a sphere. Now imagine the circle shrinking smaller and smaller, keeping sliding on the sphere as it shrinks. Is there a limit to how small this circle can be? Obviously not. You can shrink the circle so that it can't be distinguished from the dot. Mathematically, you can actually shrink it into a point.

If you start by drawing a circle on a torus, the same thing doesn't necessarily happen. The property that a circle drawn arbitrarily on a surface can shrink to a point is a surface topology property that only a sphere has. That is to say, if there is a surface on which every circle ("every" is important here) can contract to a point without leaving the surface, then the surface is equivalent to the spherical topology.

Is this also true for a three-dimensional hyperspherical surface? This is the question raised by Poincare at the beginning of the 20th century, which is the first step on his way to a classification theorem for three-dimensional hypersurfaces. He created a systematic method (called homotopy) to study what happens to circles when they move and deform on a manifold.

In fact, this is not the case. At first, Poincare assumed that the ring contraction property of a three-dimensional manifold was indeed a feature of a three-dimensional sphere. After a while, however, he realized that his assumptions might not be valid. In 1904, he published his questions.

Consider a three-dimensional compact manifold V without boundary. Even if V is different from a three-dimensional sphere, can the basic group of V be trivial?

Dimension reduction is said to be

Is it possible that a three-dimensional manifold with circular contraction is not equivalent to a three-dimensional sphere?

Poincare conjecture was born!

In 1960, the American mathematician Stephen Smale proved that the Poincare conjecture is correct for all manifolds with more than five dimensions. In this way, if a five-dimensional or higher-dimensional manifold has the property that any closed cycle drawn on it can be contracted to a point, then the manifold is topologically equivalent to a hyperspherical surface of the same dimension.

Unfortunately, the method used by Smeer can not be applied to three-dimensional or four-dimensional manifolds, so the original Poincare conjecture has not been solved. Then, in 1981, another American, Friedman, discovered a way to prove the Poincare conjecture about four-dimensional manifolds.

The problem has not been solved yet. The Poincare conjecture has been proved to be true for every dimension, except for three dimensions. Smeer and Friedman both won the Fields Prize for their achievements. There is no doubt that the only remaining person who first proves Poincare's conjecture will receive the same honor.

In 2003, the Russian mathematician Grigory Perelman proved the three-dimensional case of the Poincare conjecture, and in 2006, the mathematical community finally confirmed that Perelman's proof solved the Poincare conjecture.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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