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,

\pi₁(Figure 8) \stackrel{i*}{\longrightarrow} \pi₁(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.