How can we compare two shapes with the help of topology?

In topology, the most difficult issue is determining whether two forms are equal. Except for Euclidean geometry, which would be concerned with measuring distances among points on a form, topology is concerned with the uniformity & continuation of certain geometrical characteristics.

It’s simple to sort a collection of forms. Triangles in their pile, circles here, squares there. But there’s a bunch more to it if we accept the job seriously. In reality, one of the biggest subdisciplines of mathematics, topology, is dedicated to precisely this sort of work, yet mathematicians aren’t even close to completing it despite decades of concentrated labor. Topologists investigate the characteristics of manifolds, which are generalized representations of shapes. Their animating aim is to sort them into categories. There seem to be a few important differences to be made in this endeavor. 

Here are the main distinctions among the manifolds:

Manifolds are forms of any dimension, ranging between zero-dimensional points to the one lines and two-dimensional levels (such as a ball’s surface) to 100-dimensional areas (and beyond) that are difficult to see but mathematically true. 3 Three- and 4-dimensional manifolds are studied by mathematicians for a variety of reasons, including the fact that they offer the setting for human lives.

Manifolds are divided into three kinds after the flatness requirement is satisfied. The “topological” manifold is the simplest. It has the basic feature of allowing you to trace your finger over it without having to raise your finger. This indicates that it is continuous since there are no sudden transitions from one place to the next. Because continuity is part of the concept of a manifold, all manifolds are topological by default.

The “smooth” manifold is the most complicated. It contains all of the characteristics of a topological manifold, such as flatness and continuity, but it also possesses something more. When you trace your finger over it, you’ll notice that the route is always smooth: you’ll never come to an abrupt corner like you would on a topological manifold. The consistent smoothness has far-reaching implications. It enables you to assign a distinct tangent plane to each location on a flat manifold, allowing you to conduct calculus on them.

Four manifolds from algebraic curves: ■ circles, ■ parabola, ■ hyperbola, ■ cubic. Cr: Wikipedia

Here between bare-bones topological manifolds as well as the highly intricately structured smooth ones, the third kind of manifold has a level of complexity. It’s known as a piecewise linear manifold, hence it will be very easiest to see it as a collection of polygon-like tiles. Piecewise linear manifolds, like topological ones, may contain corners, but the piecewise structure limits where they can appear: at only the vertices where tiles meet.

Manifolds of Topology

They have the straightforward feature of being continuous. Every manifold includes topological manifolds by definition.

In terms of complexity, piecewise linear manifolds are in between topological as well as smooth manifolds, but they’re also a little out to the side. Many of the most significant issues in topology revolve around the difference among topological & smooth manifolds, with piecewise linear manifolds being excluded. From dimension 5 onwards, they become more significant for the higher-dimensional manifolds. Homotopy equivalence is the most fundamental concept of similarity. This standard considers two manifolds to be the same if you can pull, compress, & expand one into the form of another without ripping it.

Equivalence of Homotopy

The three-dimensional sphere is homotopy equal to a specific point, which means we can bend it to the point indefinitely without tearing it.

There are two additional, more complex concepts of sameness than homotopy equivalence. Each one corresponds to a distinct kind of manifold: topological manifolds and smooth manifolds, respectively. Understandably, they would have their concept of equivalence: After all, you wouldn’t compare books using the same measure you’d use to evaluate crustaceans. A homeomorphism is an appropriate standard for topological manifolds. This is a transformation (or “morphism”) that connects every point through one topological manifold to a single point in another while maintaining a feeling of distance between them. If you do have 2 points that are near together over there, then they should be close together and when you examine their pictures over there.

Equivalence of Homeomorphy

A smooth ball and a spiky ball are homeomorphic.

The equivalency standard for smooth manifolds is more complicated. It’s referred to as a diffeomorphism. Close points with one manifold must couple to close points in the other, just as they must previously, but this time in a manner that preserves the uniform topology of both manifolds. In other words, if it’s impossible to couple points without creating a corner, the two manifolds aren’t diffeomorphic.

Illustration using a sphere to explain homotopy equivalence.
Cr: Quantamagazine

Wrapping up

Freedman’s approach brought two very distinct types of equivalence together. Freedman demonstrated that the very first loose type of equivalency necessarily implies the second, much stronger type of equivalence in his particular context. As a result, they are unable to execute one of the most fundamental categorization tasks of all: determining whether a uniform four-dimensional manifold is a sphere.

Sources :

https://www.quantamagazine.org/in-topology-when-are-two-shapes-the-same-20210928/

https://science.jrank.org/pages/6862/Topology-Topological-equivalency.html

https://www.dailyadvent.com/news/c1721f0504dd59890bccb3b4b79afff0-In-Topology-When-Are-Two-Shapes-the-Same

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