- #1
sammycaps
- 91
- 0
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,
[itex]\pi[/itex]1(Figure 8) [itex]\stackrel{i*}{\longrightarrow}[/itex] [itex]\pi[/itex]1(Torus)
[itex]\:\:\:\:\:\:\:[/itex][itex]f\downarrow[/itex][itex]\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:[/itex][itex]g\downarrow[/itex]
[itex]\:\:\:\:\:\:\:\:[/itex][itex]Z[/itex]*[itex]Z[/itex][itex]\:\:\:\:[/itex][itex]\stackrel{j*}{\longrightarrow}[/itex] [itex]\:[/itex][itex]Z×Z[/itex]
And then we can do something from there.
Is that going somewhere, or not at all?
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,
[itex]\pi[/itex]1(Figure 8) [itex]\stackrel{i*}{\longrightarrow}[/itex] [itex]\pi[/itex]1(Torus)
[itex]\:\:\:\:\:\:\:[/itex][itex]f\downarrow[/itex][itex]\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:[/itex][itex]g\downarrow[/itex]
[itex]\:\:\:\:\:\:\:\:[/itex][itex]Z[/itex]*[itex]Z[/itex][itex]\:\:\:\:[/itex][itex]\stackrel{j*}{\longrightarrow}[/itex] [itex]\:[/itex][itex]Z×Z[/itex]
And then we can do something from there.
Is that going somewhere, or not at all?
Last edited: