1. Compute the fundamental group of the space obtained from 2 two-dimensional tori S(adsbygoogle = window.adsbygoogle || []).push({}); ^{1}x S^{1}by identifying a circle S^{1}x {0} in one torus with the corresponding circle S^{1}x {0} in the other torus.

Am I right in thinking that the resulting space is S^{1}x S^{1}x S^{1}, and hence the fundamental group is Z x Z x Z?

2. Let p and q be integers and let f: S^{1}x S^{1}----> S^{1}x S^{1}be given by f(x,y)=(x^{p},x^{q}) (for x and y complex numbers of magnitude 1).

Let X = D^{2}x S^{1}and Y = S^{1}x D^{2}. Then both spaces have S^{1}x S^{1}as a boundary. Form the adjunction space by identifying the boundaries of X and Y using f as the attaching map.

Compute the homology of the resulting space.

I could easily compute the homology here if I knew the cell structure of the resulting space, but I am having trouble visualizing it.

Any help much appreciated!

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# Two torus identification problems

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