Two torus identification problems

[mrbohn1]

1. Compute the fundamental group of the space obtained from 2 two-dimensional tori S¹ x S¹ by identifying a circle S¹ x {0} in one torus with the corresponding circle S¹ x {0} in the other torus.

Am I right in thinking that the resulting space is S¹ x S¹ x S¹, and hence the fundamental group is Z x Z x Z?

2. Let p and q be integers and let f: S¹ x S¹ ----> S¹ x S¹ be given by f(x,y)=(x^p,x^q) (for x and y complex numbers of magnitude 1).
Let X = D² x S¹ and Y = S¹ x D². Then both spaces have S¹ x S¹ 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!

 

[pat_connell]

The resulting space is a connected sum of two tori, which is homeomorphic to a sphere. Therefore, its homology is H_0(X) = Z, H_1(X) = 0, and H_2(X) = Z.