In the last post I have presented a great theorem that allows us to **translate **homotopy equivalence to isomorphism. Let’s give a quick reminder:

If is a homotopy equivalence then for every :

f_{*}:\pi_1(X,a)\to\pi_1(Y,f(a))

is a group isomorphism.

In this post, I am going to prove this theorem, and present some conclusions. Ok, so I shall begin now.

## Proving The Theorem

Before I’ll start proving this theorem I want to give two quick reminders from elementary set theory.

#### Quick preparations

- If such that is one to one, then is one-to-one.
- The proof is pretty straightforward: Suppose that where . Then . However, is one-to-one, therefore . Thus, is one to one as well.

- With the same functions , we now assume that is
**onto**, then is onto as well.- Again, this is also pretty easy. Let be some element, since is onto, there is some such that . Thus, is an element such that . Thus is onto.

The last thing I want to remind is a theorem from the previous post:

If such that and , Then there is a path from to such that . We also proved that is an **isomorphism** (here).

Ok, we are now ready to the proof

## The proof

We know that is an homotopic equivalence (The definition is here). By definition, there is some such that:

- .
- .

We are going to use both of this facts. Let’s pick some and start with the first:

#### Using Fact 1

Since then for some . Notice that:

- (A proof can be found here)
- .

Moreover:

It’s easier to look on a diagram of the situation:

That’s a bit better right? notice that I’ve added to a little to it, and a little to . Those are just symbols that will make things clearer.

Recall that is an isomorphism thus it is **onto**. Therefore:

F_\gamma=g_{*}^{f(a)}\circ f_{*}^a\text{ is onto}\Rightarrow g_{*}^{f(a)}\text{ is onto}

Great, that’s enough for part 1.

#### Using Fact 2

I am going to extend the diagram using the second fact:

f\circ g \sim Id_Y

Therefore, there exists some such that:

f_*\circ g_*=(f\circ g)_*=F_\delta\circ Id_Y = F_\delta

However, we are going to use the same we used in the last part, i.e. .

Therefore, now maps the group to the group (get ready for the confusing base-point) .

The diagram is now looks like:

Pause and ponder on the diagram for a few seconds. Explain yourself what is going on here, even out loud if necessary. Once you understand what’s going on here, you will realize that everything here makes perfect sense!

Ok, So the second part of the diagram yields:

f_*^{g(f(a))}\circ g_*^{f(a)}=F_\delta

Recall that is an **isomorphism** and in particular, is one-to-one. Therefore:

F_\delta=f_*^{g(f(a))}\circ g_*^{f(a)}\text{ is one-to-one }\Rightarrow g_*^{f(a)}\text{ is one-to-one}

#### Putting the parts together

From both parts we conclude that:

- is
**onto**. - is
**one-to-one**.

In other words, is an **isomorphism**! In addition, we also know that:

g_{*}^{f(a)}\circ f_{*}^{a}=F_\gamma

Since has an inverse map (which is also an isomorphism) we get:

f_{*}^{a}=( g_{*}^{f(a)})^{-1}\circ F_\gamma

Therefore, is a composition of isomorphisms, thus it is an isomorphism itself, and that completes the proof!

## Conclustions

Yes! we’ve finally found some real connection between groups and spaces that actually teaches us about the space itself!

You want to prove that two spaces are **not** homotopic (and in particular – homeomorphic)? then you can calculate the fundamental groups of them, check whether they are isomorphic or not – if not, then that’s it!

Ok, let’s try to find some concrete exapmles:

What is the fundemental group of a single point space ? Since there is only one point in the space, there is only one loop in it – the constant loop . Therefore, it’s fundemental group is just the set . i.e. it is the **trivial** group.

Now, recall that a **Contractible space** is a space that is homotopic to a single point space. Therefore, the fundamental group of every contractible space is trivial.

What contractible spaces have we met so far? for every . Therefore, thier fundamental group is trivial.

We’ve also seen that is a deformation retract of the space . Therefore, those spaces are homotopic equivalent to each other, thus, they have the same fundamental group!

But what is this group exactly? is trivial? isn’t it? we don’t know yet (we will though). We still don’t even know if those space are contractible or not? If we will manage to prove that their fundamental group is **not** trivial, then we automatically prove that there **exist** non-contractible spaces!

To end this post, I just want to tell you the plan for the future. My main goal now is to calculate the fundamental group of a **circle** – . We will have to work hard in order to do that, so prepare yourself!