# Die Poincaré-Vermutung | Mathewelten | ARTE

Jul 01, 2022
the world of math is exotic and utterly strange they speak a strange language talk about home office men multiple or trans paying rent but you also stumble upon epic realms awesome ideas and even quite useful sometimes so let's go on a little excursion everyone you guys I need a tour guide that is me and a smart brain and don't worry we won't really do math just a bit of sightseeing in the middle of the world of math not far from the familiar land of geometry lies a particularly strange. area where the fundamental properties of spaces are studied this curious inverse topology here in 1904 the french mathematician uniform carré made a conjecture that bears his name the parkallee conjecture generations of mathematicians have tried in vain to prove that at one time there was even a million dollar reward for solving this math problem, so what is this famous guess about? you can understand what it is about it is possible you have to familiarize yourself with the strange world of theologians and their vocabulary but for that I am welcome in the universe of topology the shapes we find are called manifolds they are one dimensional two dimensional and even 42 dimensional manifolds in topology, two objects count as identical if one can be deformed into the other without tearing or sticking a circle is also a triangle and a triangle is a square, however as soon as you tear or pierce an object it becomes a different object three dimensional means that a sphere is also a cube a pyramid a ufo a potato and a horseshoe for a topologies all these shapes sorry diverse can become one another by deformation therefore they are identical or in home language topologies of sphere and cube are humor for diversity too a vase or in valleys are nothing more like a deformed sphere that's why some topologies also try placing your flowers in a completely flat vase but unfortunately flowers have nothing to do with ecology now that we have to master the home office we have to deal with the spherical surface of the sun- We call sphere differently than one might think that the sphere only has two dimensions We take, for example, a ball on which we place a topology We shrink it to the size of an ant The curvature of the ball is no longer perceptible to it The world seems finite and flat it can be moved from left to right and back like on a surface is s i mean a two dimensional manifold the proof is that our sphere can be represented exactly the map of the world is such a flat sphere even though this map is flat it is still a sphere in the topological sense because it has no edge to the left and the right edge of the map is purely virtual you try to get over it you end up on the other side the top and bottom edge also s on virtual it is actually about points the north and south pole unlike a square or a circular disk you can go straight on a ferry without ever reaching an edge if you keep walking you just go back to the starting point to describe this state the vocabulary of the topology is exceptionally understandable they say our diversity is untreated and our sphere does not extend to infinity so they label it compact at this point the animal might think a cube is a plate which is a vase which in turn is a sphere not everything is a sphere in the end but it is not like that and that is why now I present to you the tore the tore has the shape of a donut or a bicycle lock, but it can also be a final swimming ring, a cup or a shaped balloon of sphere, the tus is a two-dimensional, crude and compact variety, there is a basic difference by which the two can be distinguished, I note that the turus has a hole in the middle but how can one prove that the tu rus has a hole and it will not be visible but we do not accept it as an argument because we would be in the buttocks logan quickly downwards fortunately they have an infallible tool to solve the problem, the topological one we are going to throw this loop in our sphere and pull it without no matter where we put the lasso the result will always be this it is the same we tighten the lasso and it ends in a point the sphere cannot be grasped with the topological lasso and that is proof that the sphere is simply united the same applies to the cube see the vase and plate on the bull upside down.
If you put the lasso in the right place, it is impossible to pull it to the end. We have a firm grip on the tors with the lasso. Therefore, there is a fundamental difference between the ball and the tors. The ball is only related. The bulls are not next. In addition to the sphere and towers, there is a whole series of other tales of two-dimensional multiplicity with two, three or even more holes, or the Möbius band that does not have a clear lid and bottom, or the little bottle where the interior is at the same time exterior for a topology, the question that naturally arises is which sphere is the only two-dimensional manifold that is not treated as compact and simply connected at the same time t this question reminds us a lot of porr kari's assumption or yes, but not quite because all these objects are multiple two-dimensional and in the second dimension volker reeh proved that only the sphere is simultaneously groundless compact and simply connected, but the assumption that he could.