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Two torus identification problems

  1. May 18, 2009 #1
    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)=(xp,xq) (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!
  2. jcsd
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