Associated sum

In geometry and topology , the formation of connected or connected sums is a possibility to assemble new, more complicated manifolds from given manifolds or, conversely, to decompose complicated manifolds as connected sums of simpler ones.

Combined total of A and B .

definition

If A and B are two connected m -dimensional manifolds, then denotes the connected sum

{\ displaystyle A \ #B}

the manifold that is created by cutting out an m - ball from A and B and gluing it together along the resulting edge (m-1) spheres .

properties

Well-definition

If both original manifolds are oriented , the connected sum becomes unambiguous by requiring that the bonded image should be orientation- reversing . For the construction you have to choose a ball, but the result (except for a homeomorphism ) is the same regardless of where the ball is cut out.

The connected sum can also be transferred to the category of differentiable manifolds by smoothly defining the bonding on a collar around the edge sphere. Thereby one obtains uniqueness except for a diffeomorphism . ${\ displaystyle S ^ {m-1} \ times [0,1]}$

The set of all m -dimensional manifolds together with the operation of the connected sum forms a semigroup with the m -sphere as the neutral element. The connected sum of with is therefore homeomorphic to . ${\ displaystyle M}$ ${\ displaystyle S ^ {m}}$ ${\ displaystyle M}$

Surfaces (2-manifolds)

In the case of surfaces (2-dimensional manifolds), the construction described above means cutting out one pane each and gluing it to the one-dimensional edge that has been created.

The sum connected with a torus is then equivalent to adding a handle , i.e. it increases the gender of the area by one. The classification theorem for 2-manifolds says that every compact surface is homeomorphic to the connected sum of a 2-sphere, a Klein bottle or the projective 2-dimensional space with zero or more tori.

Examples of areas:

The connected sum of two tori is a sphere with 2 handles, i. H. a plane of gender two.
The combined sum of two projective spaces is a Klein bottle.

3-manifolds

An important result in 3-dimensional topology is the following prime decomposition theorem by Helmut Kneser (1930):

Each compact, orientable 3-manifold is the connected sum of a unique collection of prime 3-manifolds.

A manifold is said to be prime if it cannot be put together as a connected sum except in the trivial way, i.e. H. as

{\ displaystyle P = P \ # S ^ {3}}

.

If P is a prime 3-manifold, then it is either the non-orientable -bundle above or every 2-sphere in P borders a ball. In the latter case, P is called irreducible . ${\ displaystyle S ^ {2} \ times S ^ {1}}$ ${\ displaystyle S ^ {2}}$ ${\ displaystyle S ^ {1}}$

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

Every compact, non-orientable 3-manifold is the connected sum over a collection of irreducible 3-manifolds and non-orientable- bundles . This sum is clearly if one demands that each summand either irreducible or a non-orientable -bundle about is ${\ displaystyle S ^ {2}}$ ${\ displaystyle S ^ {1}}$ ${\ displaystyle S ^ {2}}$ ${\ displaystyle S ^ {1}}$ .

The proof of the two theorems uses the normal surface technique developed by Kneser.

The Related-sum decomposition plays an important role in connection with that of William Thurston established Geometrization .

literature

Allen Hatcher: Notes on Basic 3-Manifold Topology (PDF; 385 kB) 2000 (English).