Connected sum

In geometry, topology and the formation of related or connected sum is a way to compose new, complex manifolds from given manifolds or vice versa to decompose complex manifolds as connected sum of simpler ones.

Definition

If A and B are two contiguous m- dimensional manifolds, then indicates the sum associated

That manifold, produced by cutting out each of an m- ball of A and B and gluing along the edge of the resulting (M -1) - sphere.

Properties

Well- definedness

If both the original manifolds are oriented, the sum involved is clearly, by requiring that the Verklebeabbildung should be orientation-reversing. For the construction you have to select While each ball, but is the result (up to a homeomorphism ) the same no matter where the ball is cut out.

The total connected can also be transferred to the category of differentiable manifolds by defining the application on a collar around the edge smooth sphere. This uniqueness is obtained up to a diffeomorphism.

The set of all m- dimensional manifolds along with the operation of connected sum forms a semigroup with the m- sphere as a neutral element. The connected sum of with is therefore to be homeomorphic.

Surfaces (2 -manifolds )

In surfaces (2 - dimensional manifolds ) means the above-described construction, the cutting out of each of a disc and bonding the resulting one-dimensional boundary.

The sum associated with a torus is then equivalent to adding a handle, so it increases the genus of the surface by one. The Classification Theorem for 2 -manifolds states that every compact surface homeomorphic to the connected sum of a 2-sphere, a Klein bottle or the projective 2 -dimensional space with zero or more Tori is.

Examples of areas:

• The connected sum of two tori is a sphere with two handles, ie a surface of genus two.
• The connected sum of two projective spaces is a Klein bottle.

3-manifolds

An important result in the three -dimensional topology, the following Primzerlegungssatz Helmut Kneser (1930):

A variety is referred to as prime if it can not be assembled as connected sum except in the trivial way, ie as

If P is a prime 3- manifold, then it is either the non- orientable bundle over or any 2-sphere in P bounded a ball In the latter case P is called irreducible.

The Primzerlegungssatz also applies to non - orientable 3-manifolds, but must be modified for this purpose, the uniqueness statement:

The proof of the theorems used, developed by Kneser normal surface technology.

The Related -sum decomposition plays an important role in connection with the concepts defined by William Thurston geometrization.