Classification of areas

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:

• a 2-sphere ,
• a connected sum of tori ,
• a connected sum of projective planes .

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 ${\displaystyle aa^{-1}}$
4. Identify all corners of the polygon as one point
5. Bring edges and neighbors ${\displaystyle a}$ ${\displaystyle a}$
6. Construct edge sequences ${\displaystyle aba^{-1}b^{-1}}$
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. ${\displaystyle n}$ ${\displaystyle m}$ ${\displaystyle n\neq m}$

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

${\displaystyle \chi (S_{1}\#S_{2})=\chi (S_{1})+\chi (S_{2})-2}$

This gives the following Euler characteristics:

• connected sum of tori: 2 - 2n ${\displaystyle n}$
• connected sum of projective planes: 2 - n ${\displaystyle n}$

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. ${\displaystyle n}$ ${\displaystyle m}$ ${\displaystyle n\neq m}$

literature

• Wladimir G. Boltjanski , Vadim A. Efremovitsch: Illustrative combinatorial topology. With a foreword by SP Novikov . Translated from Russian and with a foreword by Detlef Seese and Martin Weese. Mathematical student library 129. Deutscher Verlag der Wissenschaften, Berlin, 1986, ISBN 3-326-00008-1 .