Classification of areas

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The classification theorem for 2-manifolds from the mathematical sub-area of topology states into which classes connected 2-manifolds (also called surfaces) can be divided. In addition, it also indicates how to create representatives of these classes and how to check whether two 2-manifolds belong to the same class. The classification theorem itself is:

Every closed, connected surface is homeomorphic to exactly one of the following three spaces:

The first two rooms indicate the possibilities for orientable areas. You can think of them as spheres with handles attached. Non-orientable areas are covered by the third class.

A modification of this theorem, in which the Euler characteristic is used, is:

Two compact surfaces are homeomorphic if and only if they have the same Euler characteristic and both are orientable or both are not orientable.

In order to classify a surface, one only has to calculate its Euler characteristic and determine whether it is orientable or not.

proof

The proof of the theorem takes place in several steps:

  1. Triangulation of the area
  2. Construction of a fundamental polygon
  3. Removal of edge sequences
  4. Identify all corners of the polygon as one point
  5. Bring edges and neighbors
  6. Construct edge sequences
  7. Connected sum of projective plane and torus ⇒ Connected sum of three projective planes
  8. Non-equivalence of the classes using the Euler characteristic

Step eight of the proof is detailed here. So far it has been shown that every surface is homeomorphic to a 2-sphere, a connected sum of tori, or a connected sum of projective planes. But it is still possible that the connected sum of tori to the connected sum of tori ( ) is homeomorphic. The same applies to the connected sum of projective planes.

In order to rule out this, the Euler characteristic is used . This is a topological invariant . If the two connected sums have different Euler characteristics, they are not homeomorphic.

The Euler characteristic of the connected sum of two areas is calculated as follows

This gives the following Euler characteristics:

  • connected sum of tori: 2 - 2n
  • connected sum of projective planes: 2 - n

As a result, it is impossible for the connected sum of tori to be homeomorphic to the connected sum of tori ( ). The same applies to the connected sum of projective levels.

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