Classification Theorem of Surfaces

In summary, the theorem is clear, Every closed Surface is homeomorphic to a sphere, the connected sum of g tori, or the connected sum of g Projective plane. Closed surface means Every closed 2 dimensional manifold that is embeddable in R^n . So S^3 or in fact S^n is homeomorphic to one of the above. Another point is that S^n \cong R^{n+1} so by transitivity we get R^4 \cong S^3 \cong S^2 \cong R^3 , which is false by the invariance of dimensions. Given A,B, and C, the conclusion which i know to be wrong is that R^4 \
  • #1
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I am having difficulties grasping the consequences of this theorem, would really appreciate a little enlightenment.

A: Well, the statement of the theorem is clear, that Every closed Surface is homeomorphic to:
1) a sphere,
2) the connected sum of g tori
3) or the connected sum of g Projective plane.

B: Closed surface means Every closed 2 dimensional manifold that is embeddable in [tex]R^n [/tex]. So [tex]S^3[/tex] or in fact [tex]S^n [/tex] is homeomorphic to one of the above.

C: Another point is that [tex]S^n \cong R^{n+1}[/tex]

So by transitivity we get [tex]R^4 \cong S^3 \cong S^2 \cong R^3 [/tex], which is false by the invariance of dimensions.

Given A,B and C i come to this conclusion which i know to be wrong, so can someone explain to me where i went wrong.

The only conscious leap i have made is B, which i haven't read from any book, but given the definition i don't see why it should be otherwise.
 
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  • #2
I just realized my mistake, my third point is wrong. The Euler characteristic is different ...
 
  • #3
I was wondering why you would even think so! S1, a circle, is obviously not homeomorphic to R2, a plane. S2, the surface of a a sphere is not homeomorphic to R3. In both cases, the first is compact and the second isn't.
 
  • #4
I had a perfectly valid reason at the time :tongue2:, f(x)= x/|x| , if you discount to consider the point x=0 which i didn't.
 
  • #5
Also, S^3 is clearly not a 2 dimensional manifold.
 
  • #6
i assumed this was about classifying complex surfaces, by kodaira dimension.

i.e. rational, ruled surfaces, elliptic surfaces, abelian surfaces, K3 surfaces, surfaces of general type.
 
  • #7
mathwonk said:
i assumed this was about classifying complex surfaces, by kodaira dimension.

i.e. rational, ruled surfaces, elliptic surfaces, abelian surfaces, K3 surfaces, surfaces of general type.
That's what I thought about too when I saw the title. :smile:
 
  • #8
I only know the basic theorem:
every topological 2-manifold (surface by some authors) is homeomorphic to the connected sum of either a sphere or some finite combination of tori and projective planes. I also only know the basic proof which involves building your surface from simplexes, noticing that by labeling the edges you can represent each surface as an element in some free group, and then showing that each of these elements reduce to either 1 or some finite "product" of the element which corresponds to tori and the element which corresponds to the projective plane. Seems like there should be a more advanced proof via homology though.
 
  • #9
for oriented surfaces at least, i like the morse theory proof, namely there is a smooth function on the surface that has a finite number of non degenerate critical points, which can be arranged to be one max, one min, and 2g saddle points.

(all surfaces have a smooth structure.)
 

1. What is the Classification Theorem of Surfaces?

The Classification Theorem of Surfaces, also known as the Uniformization Theorem, is a mathematical theorem that states that any closed, connected, and orientable surface is topologically equivalent to either a sphere, a connected sum of tori, or a connected sum of projective planes.

2. Who first proved the Classification Theorem of Surfaces?

The Classification Theorem of Surfaces was first proved by German mathematician Felix Klein in 1871.

3. How is the Classification Theorem of Surfaces used in mathematics?

The Classification Theorem of Surfaces is an important result in topology and geometry, as it allows for the classification of all possible surfaces into a few simple categories. This allows for a better understanding of the properties and relationships between different surfaces.

4. What are the implications of the Classification Theorem of Surfaces?

The Classification Theorem of Surfaces has many important implications in mathematics and other fields. For example, it is used in the study of manifolds, which are higher-dimensional surfaces. It also has applications in physics, as it helps in the understanding of the topology of spacetime.

5. Are there any exceptions to the Classification Theorem of Surfaces?

Yes, there are some exceptions to the Classification Theorem of Surfaces. For example, there are non-orientable surfaces, such as the Möbius strip, which cannot be classified using this theorem. Additionally, there are surfaces with non-constant curvature, such as the pseudosphere, which also do not fit into the categories outlined in the Classification Theorem.

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