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Fundamental Group of the TorusFigure 8 
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#1
Dec1912, 08:07 AM

P: 91

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 semiexplanations, 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? 


#2
Dec1912, 11:25 AM

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P: 9,484

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



#3
Dec1912, 12:47 PM

P: 91




#4
Dec1912, 05:47 PM

Sci Advisor
P: 1,716

Fundamental Group of the TorusFigure 8
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. 


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