Let’s play a game, a game called Topology. In this game, we manipulate some objects using certain rules:
- You can shrink and stretch objects.
- You can’t paste different parts of the objects.
- You can’t cut different parts of the objects.
So, using these rules we will take an object and convert it in other (this is why topology is sometimes called ‘glue geometry’). Our first example, shown in picture 1, is the Donut-Mug convertion. None part was cutted/pasted, just stretching and shrinking. We’ll say then that donut and mug are topologically equivalent (or using technical language, homeomorphic).
Now, your goal is determine if the elastic-man shown in picture 2 can separate his hands using the topology rules, and if that is possible, describe how. In other words, we need to determine if the elastic-man of the picture 2 is topologically equivalent to the free elastic-man of picture 3. Think about it, the solution is below.
Finally, we’ll use topology for a mathematical problem: what’s the difference between a torus (a donut) and a sphere? (We are thinking these objects as surfaces, with an empty inside). The answer is obvious: one has a hole, the other doesn’t. But that’s not a very mathematical answer. Let’s see how topology helps here:
The sphere has a property that torus not: every closed loop on it can be shrunk into a point, using the rules described above*. An example is shown in picture 4. Meanwhile, on the torus (picture 5), the loops a and b can’t be shrunk into a point without cutting the torus. Just loops as c can do it. Objects with this property are called simply-connected.
The Poincaré’s conjecture, recently solved by Perelman (picture 6), asserts that the only simply-connected objects are sphere-shaped. For his work, he won a million dollars, prize he rejected.
Now, the answer to the glue-man problem: Answer here.
*The formal rules are slightly different.