Fundamental Group of the Torus-Figure 8

[sammycaps]

So I'm revamping the question I had posted here, after a bit of work.

I'm concerned with the homomorphism induced by the inclusion of the Figure 8 into the Torus, and why it is surjective. There seem to be a lot of semi-explanations, but I just wanted to see if the one I thought of makes sense.

So, we know that the fundamental group of the Figure 8 is isomorphic to the free product on 2 generators (i.e. of two copies of the integers), and the fundamental group on the torus is isomorphic to the cartesian product of two copies of the integers.

So, I don't know if there is a homomorphism j* such that this diagram commutes, for f and g isomorphisms from above, but if there is then this diagram commutes,

$\pi$₁(Figure 8) $\stackrel{i*}{\longrightarrow}$ $\pi$₁(Torus)
$\:\:\:\:\:\:\:$$f\downarrow$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$g\downarrow$
$\:\:\:\:\:\:\:\:$$Z$*$Z$$\:\:\:\:$$\stackrel{j*}{\longrightarrow}$ $\:$$Z×Z$

And then we can do something from there.

Is that going somewhere, or not at all?

 

[mathwonk]

would n't you represent a torus as a square with identifications, then push off any loop in the square onto the boundary?

 

[sammycaps]

would n't you represent a torus as a square with identifications, then push off any loop in the square onto the boundary?

So, any loop is homotopic to a loop on the boundary? And then any loop on the boundary is a loop of the figure 8? So then would we say the homomorphism induced by inclusion is $i*([a])=[i\circ a]=[a]$, so then this induced homomorphism is surjective?

 

[lavinia]

So, any loop is homotopic to a loop on the boundary? And then any loop on the boundary is a loop of the figure 8? So then would we say the homomorphism induced by inclusion is $i*([a])=[i\circ a]=[a]$, so then this induced homomorphism is surjective?

you can think of the torus as a figure 8 with a disk attached. The boundary of the disk is attached to the loop aba$^{-1}$b$^{-1}$ on the figure 8. This is what Mathwonk is saying.

Van kampen's Theorem then gives you the result you are looking for.

BTW: Think about the same ideas for tori with more than one handle.

 

[blue_raver22]

I can say that your approach is certainly on the right track. The key concept here is that of a homomorphism, which is a function that preserves the algebraic structure of a group. In this case, we are looking at the fundamental groups of two topological spaces - the Figure 8 and the Torus - and trying to understand how they are related.

Your diagram shows the homomorphism i* induced by the inclusion map from the Figure 8 into the Torus. This means that if we take a loop in the Figure 8 and map it into the Torus, the resulting loop in the Torus will have the same algebraic structure as the original loop. In other words, the fundamental group of the Figure 8 is a subgroup of the fundamental group of the Torus.

Your next step is to consider the homomorphism j* from the free product of two copies of the integers to the Cartesian product of two copies of the integers. This homomorphism essentially combines the two generators of the free product into a single generator in the Cartesian product. This is important because it allows us to see that the fundamental group of the Figure 8 - which is isomorphic to the free product of two copies of the integers - is a subgroup of the fundamental group of the Torus - which is isomorphic to the Cartesian product of two copies of the integers.

So, in essence, your idea of using homomorphisms to understand the relationship between the fundamental groups of the Figure 8 and the Torus is a valid approach. It allows us to see that the inclusion map is surjective, meaning that every element in the fundamental group of the Torus has a corresponding element in the fundamental group of the Figure 8. This helps us understand why the fundamental group of the Torus is a "bigger" group than the fundamental group of the Figure 8, as it contains all the elements of the Figure 8 and more.

In conclusion, your approach is a good start and can definitely lead to a better understanding of the homomorphism induced by the inclusion of the Figure 8 into the Torus. Keep exploring and asking questions, as that is the essence of scientific inquiry.