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Homology of CP^1, complex projective 1-space

  1. Apr 18, 2010 #1
    Hi again:

    I was trying to use Mayer-Vietoris to compute the homology of CP^1 , the

    complex projective 1-space embedded in C^2-{(0,0)}. ( I know cellular

    homology would have been much easier, but I am trying to practice using M.-V).

    Problem is , I need to get a(n) topological S^1 somewhere , in either A,B or A/\B,

    and a contractible space, but I don't see where I can get the S^1 from.

    The sets A,B I am using I think are the standard charts for CP^1 as a manifold:

    We have :

    1) A= [z1,z2] : z1 not 0 (with the chart map [z1,z2]-->z2/z1

    2) B=[z1,z2] :z2 not 0 (chart map is z1/z2)

    I think A/\B (intersection) consists of the 16 open cuadrants of R^4 ,

    and each of these intersections (implying that each of A,B is contractible)

    , but I do not see how A/\B is a topological S^1 .

    Any Ideas.?
  2. jcsd
  3. Apr 19, 2010 #2
    CP^1 is S^2. Think of the lower hemisphere and the upper hemisphere, thickened so that they intersect on an open set which retracts onto the equator.

    In the language of CP^1, take U=CP^1\[0,1] and V=CP^1\[1,0]. They intersect on C\{0} (map is [z_0,z_1] -> z_0/z_1) which definitely retracts onto S^1.
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