It is easy to sort a collection of shapes. You can find triangles in your own pile, squares elsewhere, and circles here.
If you take this task seriously, you will find that there is much more to it. Mathematicians are far from being done. In fact, this is the biggest subdiscipline in mathematics topology.
Topologists research the properties of manifolds, which are generalized versions of shapes. Their primary goal is to classify them. There are some key distinctions that can be made in this effort. What are manifolds and what concept of similarity do we use when comparing them?
These are the main differences.
Manifolds are shapes that can be of any dimension. They can range from zero-dimensional points to one dimensional lines to two-dimensional surfaces like the surface of a ball to 100-dimensional spaces. Although they are difficult to visualize, they are mathematically real. They are studied by mathematicians, among others, because they provide the setting for our lives.
They look just like the place we live on Earth, or in space. Maggie Miller, a Stanford University postdoctoral fellow, suggested that the universe may be an intriguing multiverse.
All manifolds share a common feature: a general flatness. You would find the entire space flat around you if you were to stand on the surface of a manifold. This is what we actually experience on Earth. You might be forgiven for thinking that the planet's surface is a two-dimensional manifold. It is obvious that the planet is flat.
A manifold's global features, such as its shape like a sphere, or whether it contains a hole similar to a doughnut, can't be seen from a ground eye view. This definition excludes shapes like two cones touching tip-to-tip, such as an hourglass. You would be able to see something unusual at the tip where they meet if you lived in this area.
Manifolds can be divided into three types once the flatness condition has been met. The topological manifold is the simplest. The simple property of the topological manifold is that you can trace your finger across it without lifting your finger. It is continuous because it does not make abrupt jumps between points. Since continuity was an integral part of the definition for a manifold, all manifolds automatically qualify as topological manifolds.
The smooth manifold is the most complicated type. The smooth manifold has all the characteristics of a topological-manifold flatness and continuity, but it also has another feature. It is smooth and you can trace your finger along it. You will never encounter an abrupt corner like on a topological manifold.
This uniform smoothness can have big implications. This allows you to assign a unique tangent plan to each point on a smoothmanifold. It also means that you can do calculus on smooth manifolds.
This third type of manifold is somewhere in the middle between the simple topological manifolds with no structure and the more complex smooth ones. It is called a piecewise-linear manifold and can be pictured as a collection of polygonlike tiles. Although piecewise linear manifolds may have corners just like topological ones do, the piecewise structure limits where these corners can occur. Only at the vertices are tiles joined.
Miller stated that the corners of a topological manifold are not as isolated as they are in a piecewise-linear manifold.
Although they are a little more complicated than smooth manifolds, piecewise linear manifolds can be considered to be between topological and smooth in terms of their complexity. Many of the most important topological questions are based on the distinction between smooth and topological manifolds. They do not include piecewise linear ones. These distinctions are more relevant in the study higher-dimensional manifolds from dimensions 5 to 10.
Once you understand what manifolds mean, you can ask when one is identical to another.
Homotopy equivalence is the most basic concept of similarity. This standard considers two manifolds to be the same if they can be compressed, pulled and expanded into the desired shape without ever being ripped.
This loose standard means that many shapes, even if they look different, are treated the same. A three-dimensional ball (like a ball) is homotopy equivalent of a single point because it can be deformed continuously to a point without ever ripping the ball. A doughnut, however, is not homotopy equivalent of a point due to the hole in its middle, which cannot be removed no matter how tight you shrink it.
There are two more advanced notions of similarity that go beyond homotopy equivalence. Each corresponds with a different type manifold: one is for topological manifolds, the other for smooth manifolds. Each would have its own notion of equivalence. After all, you wouldn't use the same metric for comparing novels as you do to compare invertebrates.
Sarah Blackwell, a University of Georgia graduate student, stated that you can easily guess the definition of an equivalence when you know the definition of a new type of space. It should naturally match the space's definition.
The relevant standard for topological manifolds is called a "homeomorphism". This transformation (a morphism), pairs each point of a topological manifold with one unique point in another, maintaining a sense distance between them.
Blackwell stated that if you have two points that are very close together, you should see their images together.
The standard of equivalence for smooth manifolds is more complicated. This is called a diffeomorphism. Similar to before, the close points of one manifold must pair with the close points of another manifold. However, this pairing must be done in a way that preserves both manifolds' smooth structures. The other way around is that if it's impossible to pair points without inserting a corner, then they are not mutually diffeomorphic.
These distinctions are the foundation of the grand topology classification project. This classification has been made possible by mathematicians in all dimensions, except for dimension 4.
Michael Freedman's 1981 proof that the four-dimensional Poincar conjecture was homotopy equivalent to a four-dimensional globe by Michael Freedman is the most important classification result. It established that any topological manifold with a four-dimensional topology that is homotopy equivalent is also homeomorphic the four dimensional sphere. Quanta recently explained that this proof was so complex and poorly communicated that it was slowly disappearing from mathematics, until a new book brought it back.
Two very different types of equivalence were linked by Freedman's work. Homotopy equivalence is where a ball is equivalent in value to a point. This is very different from homeomorphic, which requires that points are paired in a precise way. In his particular case, Freedman showed that the first loose form equivalence always implies the second, stronger form.
Freedmans proof did not include the smooth four-dimensional Poincar conjecture. This states that any smooth manifold four-dimensional that is homotopy equal to the four dimensional sphere, is also diffeomorphic with the four dimensional sphere. This statement is stronger than that of Freedman, since a diffeomorphism can be considered a stronger form than a homeomorphism. It is also one that mathematicians don't know how to settle.
They are now in a strange situation of not being able to recognize when a smooth, four-dimensional manifold is actually a sphere.
