Homology of CP^1, complex projective 1-space

[Bacle]

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.?

 

[zhentil]

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.