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What does Euler characteristic tell us about topological spaces? For example, what do we find out from the fact that the Euler characteristic of a sphere is 2 and of a torus si 0? Thank you!!
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It's less that the number means something itself and more that the number is an invariant. If two topological spaces have the same Euler characteristic, then they are, for almost all intents and purposes the same "kind" of topological space and whatever you can do with one you can do with the other. By contrast, if they have different characteristics, they are different and you can't necessarily shift from one to the other.

So if you have two topologies that you don't know are different or the same, you can calculate their Euler characteristics and find out.
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The equation χ = 2-2g tells us the genus g of the surface (assuming closed and orientable) if we know the Euler characteristic χ, which is usually defined as χ = V - E + F, where V, E, and F are the number of vertices, edges, and faces, respectively of a polygonal representation of the surface.

You can also relate χ to the curvature of the surface via the Gauss-Bonnet theorem, but that's starting to get fairly advanced.

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