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Surface


In mathematics (topology), a surface is a two-dimensional manifold. Examples arise in three-dimensional space as the boundaries of three-dimensional solid objects. The surface of a fluid object, such as a rain drop or soap bubble, is an idealisation. To speak of the surface of a snowflake, which has a great deal of fine structure, is to go beyond the simple mathematical definition. For the nature of real surfaces see surface tension, surface chemistry, surface energy.

Definition


In what follows, all surfaces are considered to be second-countable two dimensional manifolds.

More precisely: a topological surface (with boundary) is a Hausdorff space in which every point has an open neighbourhood homeomorphic to either an open subset of E 2 (Euclidean 2-space) or an open subset of the closed half of E 2.
The set of points which have an open neighbourhood homeomorphic to E n is called the interior of the manifold; it is always non-empty.
The complement of the interior, is called the boundary; it is a (1)-manifold, or union of closed curves.

A surface with empty boundary is said to be closed if it is compact, and open if it is not compact.

Classification of closed surfaces


There is a complete classification of closed (i.e compact without boundary) connected, surfaces up to homeomorphism. Any such surface falls into one of three infinite collections:

  • Spheres with n handles attached (called n-tori). These are orientable surfaces with Euler characteristic 2-2n, also called surfaces of genus n.
  • Projective planess with n handles attached. These are non-orientable surfaces with Euler characteristic 1-2n.
  • Klein bottles with n handles attached. These are non-orientable surfaces with Euler characteristic -2n.

Therefore Euler characteristic and orientability describe a compact surfaces up to homeomorphism (and if surfaces are smooth then up to diffeomorphism).

Compact surfaces


Compact surfaces with boundary are just these with one or more removed open disks whose closures are disjoint.

Embeddings in R3


A compact surface can be embedded in R3 if it is orientable or if it has nonempty boundary. It is a consequence of the Whitney embedding theorem that any surface can be embedded in R4.

Some models


To make some models, attach the sides of these (and remove the corners to puncture):
* * B B
v v v ^ *>>>>>* *>>>>>*
v v v ^ v v v v
A v v A A v ^ A A v v A A v v A
v v v ^ v v v v
v v v ^ *<<<<<* *>>>>>*
* * B B

sphere real projective plane Klein bottle torus (punctured: Möbius band) (sphere with handle)

Fundamental polygon

Each closed surface can be constructed from an even sided oriented polygon, called a fundamental polygon by pairwise identification of its edges. This construction can be represented as a string of length 2n of n distinct symbols where each symbol appears
twice with exponent either +1 or -1. The exponent -1
signifies that the corresponding edge has the orientation opposing the one of the fundamental polygon.

The above models can be described as follows:

  • sphere:
  • projective plane:
  • Klein bottle:
  • torus:

    Connected sum of surfaces


    Given two surfaces M and M', their connected sum M # M' is obtained by removing a disk in each of them and gluing them along the newly formed boundary components.

We use the following notation.
  • sphere: S
  • torus: T
  • Klein bottle: K
  • Projective plane: P

    Facts:

    • S # S = S
    • S # M = M
    • P # P = K
    • P # K = P # T

    We use a shorthand natation: nM = M # M # ... # M (n-times) with 0M = S.

    Closed surfaces are classified as follows:


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