In this post we are going to face the very last universal problem, it won’t be so hard though it won’t be as easy as the previous problem. However, the solution for the problem is great! we will use it a lot, since it can be really comfortable in some cases. Ok, let’s do it:Continue reading “Generators and Relations”

# Category Archives: Algebraic Topology

## The Free Group

So, in this post we will face the fifth universal problem. It’s really not difficult, though it will be extremely useful, and not just for our purposes! We will define the free group (over a given set). Since this one is really important, After solving the universal problem, I want to show some properties ofContinue reading “The Free Group”

## Free product with amalgamation

We are now facing the fourth universal problem – the most challenging one yet! Don’t worry though, I’ll try to explain it as simple and organized as possible. Moreover, despite it being challenging, it is also the most rewarding problem, as we’ll see in the future. Let’s do it: The problem Now there are threeContinue reading “Free product with amalgamation”

## Free product

We’ve already solved two universal problems! In this post, we will solve the third, which is not more complicated than the last problem. However, the next problem is pretty complicated and it will use what we’ll do here, so it’s important to understand this problem as good as possible. Ok, let’s not spend any moreContinue reading “Free product”

## Abelianization

After solving our first universal problem, It’s time for the second one. And as you may guess from the title, we are going to meet a new term which is called “Abelianization“. We’ll see exactly what does that mean: The universal problem Now the rules of the game are: We are given only one groupContinue reading “Abelianization”

## Universal Problems

After calculating the fundamental group of the circle, and deducing lot’s of results, where one of them was really imortant, it’s time to gain some new skills. Untill now, we had two main tools for calculating the fundamental group of a topological space: Check if the space is contractible or simply connected. If it is,Continue reading “Universal Problems”

## The fundamental theorem of algebra – a topological proof

Now that we know what the fundamental group of the circle is . We can use this knowledge to prove the fundamental theorem of algebra! The theorem states that if is a polynomial over with a degree greater than zero, then there exists such that . This is actually kind of wierd. So far, IContinue reading “The fundamental theorem of algebra – a topological proof”

## What can we learn from the fundamental group of the circle?

In the last post we finally revealed the true identity of the fundamental group! It is – aka the group of integers / the infinite cyclic group. This group is first non-trivial group that we’ve met so far. While The circle, is first non-contractible space we’ve met. In this post we are going to seeContinue reading “What can we learn from the fundamental group of the circle?”

## The Fundamental Group Of a Circle

After discussing about simply connected spaces, the next definition that is going to be useful for us is the one of covering spcae. Intuitivley speaking, just by the name we can kind of understand what a covering space is – it is a space that covers another space. Though the intuition is simple, the definitionContinue reading “The Fundamental Group Of a Circle”

## Simply connected spaces

In the last post we’ve proved a great theorem: If two spaces are homotopy equivalent, then they have the same fundamental group (up to isomorphism). So far, all the fundamental groups we were able to calculate turned out to be trivial. I’ve also said that my goal now is to calculate the fundamental group ofContinue reading “Simply connected spaces”