So I'm revamping the question I had posted here, after a bit of work.(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Fundamental Group of the Torus-Figure 8

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