In the last post I’ve presented the term of **homotopy**, defined the **homotopy category** and **homotopic equivalent spaces**. In this post I would like to present some more properties and definitions related to homotopy and get some results. Just one note before I start: I will only deal with continuous maps from now on – Unless I say otherwise

## Contractible spaces

I’ll start with a definition: Let be two topological spaces, and suppose that is a (continuous) map between them. We say that is **null-homotopic** if it is homotpic to a constant map.

Intuitivly, it means that you can continuously transform into a constant function. Let’s see an example: Pick a map . Then the function:

H:X\times I\to \mathbb{R}^n,H(x,t)=(1-t)f(t)

is indeed a homotopy between to the constant function . Thus, every function into is null-homotopic.

With the term of a null-homotopic map we can define what a contractible space is: We say that is a **contractible space** if is null-homotopic. Using the example in the above, we can subsitute and conclude that is null homotopic.

I want to give some more intuition to what a contractible space is using the following statement:

X\text{ is contractible}\iff X\text{ is homotopic equivalent to a point}

This statment allows us to think of as a space that we can continuously **shrink** to a single point.

Let’s prove it: Suppose that is contractible and is a space with a single point, we know that, is homotopic to a constant function (where for every ).

Let’s define a map where (we actually don’t really have that much of a choice here…). We need to prove that is an homotopic equivalence. i.e. we need to find a map such that:

f\circ g\sim Id_{\{p\}},g\circ f\sim Id_{X}

I’ll pick the map . Let’s check if it valid:

f\circ g(p)=f(g(p))=f(a)=p\Longrightarrow f\circ g = Id_{\{p\}}\Longrightarrow f\circ g\sim Id_{\{p\}}

Great, now on the other hand:

g\circ f(x) = g(f(x))=g(p)=a\Longrightarrow g\circ f\sim K_a

And by our assumption, , and since is transitive, we get that . And we have proved that is a homotopic equivalence, as we wanted.

Now, suppose that is homotopic equivalent to a single point space . Therefore, the function is an homotopic equivalence (since it is the only map from to ). Therefore, there is a map such that:

f\circ g\sim Id_{\{p\}},g\circ f\sim Id_{X}

Let , so:

g\circ f(x)=g(f(x))=g(p)=a\Longrightarrow g\circ f = K_a

And we know that , thus so by definition, is contractible.

## Homotopy with respect to subspace

Great, we now have a nice intuition to contractible space, this property will be important to us later. I want to define now a new kind of homotopy: Given two topological spaces , a subspace and two maps , we will say that **is homotopic to** **with respect to** , and denote if there is a homotopy from to if for every . There isn’t really anything new here, this is just a regular homotopy with some restrictions, the value of elements of the set shall stay the same at any given point at time (). There is a nice animation in the wikipedia of such an homotopy:

Those are two paths on the plane () which are homotopic equivalent with respect to the set .

It’s not hard to see that is an equivalnce relation and a necessary condition for and to be homotopy equivalent with respect to is: . This kind of homotopy will be important later, so remember it!

## Retract

I am now going to talk about special case of a subset, I am going to use the **inclusion map**: where and .

As always, is a topological space and . We will say that is a **retract** of if there is a map such that .

Let’s try to understand what we are facing here. We can think about retract as a subspace that we can continuously ‘squish’ the whole space into, while not moving the the subspace itself. It kind of makes sense to think of as a dominant subspace that can ‘carry on his back’ the whole space.

I don’ t if you noticed, but I really like demonstrating thing in topology with animations, so I’ll give an example Consider the “8” space:

(Formally, It is a qutient space of two intervals where we identify the endpoints as the same point), The lower / upper circles are both retracts. I’ll show here two ways to map the space into the lower circle, by ‘shrinking’ the upper circle or by ‘folding’ the space.

A topolical space and it’s retract have some shared properties, for example, if is contractible, and is a retract, then is contractible as well. Let’s prove it:

By definiton, we know that for some . Since is a retract, there is a map such that . Suppose that is a homotopy from to . Let’s take now the map:

r\circ H\circ (i\times Id_{[0,1]}):A\times I\overset{i\times Id_{[0,1]}}{\longrightarrow}X\times I\overset{H}{\longrightarrow}X\overset{r}{\longrightarrow}A

It is continuous as a composition of continuous maps, and:

r\circ H\circ (i\times Id_{[0,1]})(a,0)=r\circ H((i\times Id_{[0,1]})(a,0))

=r\circ H(a,0)=r\circ Id_X(a)=r(a)=a

Thus, (notice that since ). Moreover:

r\circ H\circ (i\times Id_{[0,1]})(a,1)=r\circ H((i\times Id_{[0,1]})(a,1))

=r\circ H(a,1)=r\circ K_p(a)=r(p)\in A

Thus, and we proved that , therefore, is contractable, as we wanted.

Now I Want to give a counter-exmaple: Pick and . I state that is **not** a retract. Suppose it is, then there is a continuous map such that , we can exapnd this function’s image to the whole interval by defining: . And now we have a contradiction to the Intermediate value theorem (think why!).

I want to present now a stronger type of retract: Suppose that is a topological space, . We say that is a **deformation retract** of if there is a map such that:

- (nothing new here)

This is a stronger property, we are not only asking if such a map exists, we also want to be a **homotopy equivalence with respect to **, therefore, is homotopic equivalent to with respect to .

Notice that in order to check that a retract is a deformation retract we only need to find a homotopy with respect to between and the identity on .

As you may guess, it is now time to give an example. I’ll pick and:

A=S^{n-1}=\{(x_1,\dots,x_n)\in\mathbb{R}^n:\sum_{k=1}^nx_k^2=1\}

We need to define a homotopy:

H:(\mathbb{R}^n-\{0\})\times I\to (\mathbb{R}^n-\{0\})

such that . Wait a minute, what is ? I’ll use the fact that the points in are exactly the points with norm one, so let’s take the function that normalizes it’s input: . This function indeed satisfies the first property, and we can now define the homotopy:

H(x,t)=(1-t)\cdot x+t\cdot\frac{x}{||x||}

That is indeed a homotopy with respect to , thus is a deformation retract.

I don’t know about you, but I really like this animation, and it seems like good place to stop.

## Summary

I’ve shown some new definitions here, and made another step towards my goal. In the next post we are going to use some of the definitions here in order to construct **The** **fundemental group**(s), which will allow us to finally start involving some **algebra** to our theory.