Blogpost #6: Topology, what the heck is that?

It's amazing to me: I am close to graduation with a degree in pure mathematics, and this is the first time (at least that I can recall) that I have heard information about the subject Topology (I am sure that there are many other subjects in math that are typically not discussed/covered in great detail at the undergraduate level). Like many other disciplines in math, Topology (seems to) originate from a mathematician who was trying to solve a particular problem, but maybe did not have the correct tool or approach needed to solve the problem, and while working on the problem, a new subject is created! Here we will take a look at the beginnings of Topology, the two major sub disciplines of Topology, and what types of creations/technologies have been created and might still be used today because of Topology.

Perhaps the first work that is usually considered to be the beginnings of Topology was Euler's work on the "Bridges of Konigsberg" problem. (Here is a link to the specifics of the problem  https://nrich.maths.org/2484 ). Euler wrote about his solution to this problem in his paper that was titled "  The solution of a problem relating to the geometry of position". The geometry of position?!? This was a new way of thinking about geometry: distances and length measurements are irrelevant. Not only did Euler prove that the desired trip in the Konigsberg bridge problem was impossible, he realized what would be necessary in order for this type of trip to be made, in a general case, regardless of how many "bridges" and "land masses", or in modern day math, how many vertices and edges a graph or map has; Euler determined that the only way the Konigsberg path could be done, is if there was exactly two land masses that had an odd number of bridges connecting each land mass, respectively, to other land masses. In modern graph theory, this is stated as the following:

 "A graph has a path traversing each edge exactly once if exactly two vertices have odd degree." 

Euler's creation of this solution is the beginning of Topology.

There are two major sub disciplines of Topology: general, or point-set, topology, and the other is called algebraic topology. 

General topology deals a lot with the congruence of two or more objects to each other. In particular in topology, two or more objects are equivalent if they can be continuously deformed into one another through such motions in space as bending, twisting, stretching, and shrinking while tearing and gluing together parts is not allowed. This link to an animation shows again what was discussed in my MTH 495 class, showing the equivalence between a doughnut and a coffee cup (now tell me how easy it would be to see this congruence with just basic geometry!)

https://www.britannica.com/topic/topology/images-videos/Because-both-a-doughnut-and-a-coffee-cup-have-one/90829

The coffee cup and doughnut are examples of objects that are simply connected, meaning we can continually shrink and morph any portion of the cup or doughnut into the other without running into any problems. There are other objects, like the torus, that are not simply connected. lets look at the following picture as to why: 

Here we can see, that loop c can be shrunk down to a point without any problems, while loops a and b cannot because they are affected by the torus's hole.

Euler had derived a formula for the polyhedra, that is known as the Euler Characteristic, which states that

v - e + f = 2

where v is the number of vertices, e the number of edges, and f the number of faces. Thisformula was used to prove many theorems dealing with polyhedra's. However, a mathematician in 1813 named Antoine-Jean Lhuilier noticed that Euler's formula did not work for solids that had holes in them. Lhulier's adaptation to the Euler Characteristic was 

v - e + f = 2 - 2g

Where g is the number of holes in the solid (source from http://www-history.mcs.st-andrews.ac.uk/HistTopics/Topology_in_mathematics.html ). This was the first known result of a topological invariant (meaning the object stays the same no matter if we continuously deform the geometric object) 

The real importance of topology from a mathematical perspective seems to be the consideration of objects in higher-dimensional spaces, or even, in particular abstract objects that are sets of elements. Topology focuses on space and spacial functions, determining of objects are isotopic to a larger space (topological equivalence). This is why Topology is considered one of the abstract fields of mathematics (along with abstract algebra, analysis, and others). 

Algebraic Topology uses tools from abstract algebra to study topological spaces. The goal 

is to find Algebraic invariants that classify topological spaces up to homeomorphism (objects can be deformed into each other by a continuous, invertible mapping), though most classify up to homotopy equivalance (a class below homeomorphism. Does not satisfy invertible mapping). I think of these as classes from abstract algebra; a homeomorphism invariant is a field, and a homotopy equivalance is a integral domain (or less, I am not completely sure) Where if we recall from abstract algebra that integral domains do not satisfy the Multiplicative Inverses Property, while Fields do, which seems to some coincide with the idea of homeomorphism and homotopy invariants. Very interesting stuff!! of course I not even really skimming the surface of Topology, this was just a little bit of insight  I have learned about the subject.

Modern day Topology is used in many fields today, from Cosmology, to DNA research, to Crystallography  (used by chemists) to even Networking (along with Graph Theory, ya know, things like Facebook, twitter, etc) as well as medical imaging software and technology, just to name a few. 

What an interesting topic Topology is!