- #1
PsychonautQQ
- 784
- 10
The Euler Characterist of the projective plane and sphere is given by V - E + F. V is vertices, E is edges, F is faces.
A presentation of the projective plane is {a | aa} and a presentation of the sphere is {b | bb^1}
Yet the Euler characteristic is 2 for the sphere and 2-n for the connected sum of n protective planes. So one projective plane should have euler characteristic of 1. Looking at these manifolds as equivalences on the closed disk, it seems that their Euler characteristic should be the same. They both have 2 vertices, 2 edges, and 1 face. Or perhaps since the edges are identified together there is only 1 edge, and perhaps for the sphere you have to count both side of the disc as a face...
Can somebody clarify this?
A presentation of the projective plane is {a | aa} and a presentation of the sphere is {b | bb^1}
Yet the Euler characteristic is 2 for the sphere and 2-n for the connected sum of n protective planes. So one projective plane should have euler characteristic of 1. Looking at these manifolds as equivalences on the closed disk, it seems that their Euler characteristic should be the same. They both have 2 vertices, 2 edges, and 1 face. Or perhaps since the edges are identified together there is only 1 edge, and perhaps for the sphere you have to count both side of the disc as a face...
Can somebody clarify this?