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Categories: Topology | Differential topology | 3-manifolds

Connected sum

In geometric topology, a connected sum of two connected $m$ -dimensional manifolds is a manifold formed by deleting a ball inside each manifold and gluing together the resulting boundary spheres.

If both manifolds are oriented, there is a unique connected sum defined by having the gluing map reverse orientation. The construction uses the choice of the balls but the result is unique up to homeomorphism. One can make this operation work in a smooth category and then the result is unique up to diffeomorphism. The well-definedness of this operation depends crucially on the annulus theorem , which is not at all obvious.

The operation of connected sum is denoted by $\#$ , for example $A \# B$ denotes the connected sum of $A$ and $B$ .

The operation of connected sum has the sphere, $S m$ , as an identity, so $M \# S^m$ is homeomorphic (diffeomorphic) to $M$ .

Surfaces

For surfaces, i.e., 2-dimensional manifolds, connected sum with a torus is equivalent to adding a handle. Every compact surface is the connected sum of one of the sphere, projective plane, or Klein bottle with zero or more tori. Examples:

The connected sum of two projective planes is the Klein bottle.
The connected sum of two tori is a sphere with two handles.

3-manifolds

Prime decomposition theorem. Every compact, orientable 3-manifold is the connected sum of a unique (up to homeomorphism) collection of prime 3-manifolds.

The manifold is prime if it can not be presented as a connected sum in a non-trivial way, where the trivial way is

$P=P\#S^3.$

If $P$ is an prime 3-manifold then either it is $S^2\times S^1$ or the non-orientable $S 2$ bundle over $S 1$ , or any embedded 2-sphere in $P$ bounds a ball, i.e. is irreducible. So the theorem can be restated to say that there is a unique connected sum decomposition into irreducible 3-manifolds and $S^2 \times S^1$ 's.

The prime decomposition holds also for non-orientable 3-manifolds, but the uniqueness statement must be modified slightly: every compact, non-orientable 3-manifold is a connected sum of irreducible 3-manifolds and non-orientable $S 2$ bundles over $S 1$ . This sum is unique as long as we specify that each summand is either irreducible or a non-orientable $S 2$ bundle over $S 1$ .

The proof is based on normal surface techniques originated by Kneser.

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