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jgens

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How to compute [itex]\chi(\mathbb{C}\mathrm{P}^2)[/itex]?

This problem is from a class on differential topology, so we have defined the Euler characteristic as the sum of the indices of isolated zeros on a non-vanishing vector field. Off the top of my head, I cannot think of any theorems which really help with this computation, so I am thinking I need to do this by finding a sufficiently nice vector field on [itex]\mathbb{C}\mathrm{P}^2[/itex] and the just calculating the indices of the isolated zeros by hand. Could someone help get me started with this?

This problem is from a class on differential topology, so we have defined the Euler characteristic as the sum of the indices of isolated zeros on a non-vanishing vector field. Off the top of my head, I cannot think of any theorems which really help with this computation, so I am thinking I need to do this by finding a sufficiently nice vector field on [itex]\mathbb{C}\mathrm{P}^2[/itex] and the just calculating the indices of the isolated zeros by hand. Could someone help get me started with this?

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