Connected sum

In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classification of closed surfaces.

More generally, one can also join manifolds together along identical submanifolds; this generalization is often called the fiber sum. There is also a closely related notion of connected sum on knots, called the knot sum or composition of knots.

Connected sum of manifolds at a point

A connected sum of two $m$ -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. Although the construction uses the choice of the balls, the result is unique up to homeomorphism. One can also make this operation work in the smooth category, and then the result is unique up to diffeomorphism. (This uniqueness 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; that is, $M \# S^m$ is homeomorphic (or diffeomorphic) to $M$ .

Connected sum of manifolds along a submanifold

Let $M_1$ and $M_2$ be two smooth, oriented manifolds of equal dimension and $V$ a smooth, closed, oriented manifold, embedded as a submanifold into both $M_1$ and $M_2$ . Suppose furthermore that there exists an isomorphism of normal bundles

\psi: N_{M_1} V \to N_{M_2} V

that reverses the orientation on each fiber. Then $\psi$ induces an orientation-preserving diffeomorphism

N_1 \setminus V \cong N_{M_1} V \setminus V \to N_{M_2} V \setminus V \to N_{M_2} V \setminus V \cong N_2 \setminus V,

where each normal bundle $N_{M_i} V$ is diffeomorphically identified with a neighborhood $N_i$ of $V$ in $M_i$ , and the map

N_{M_2} V \setminus V \to N_{M_2} V \setminus V

is the orientation-reversing diffeomorphic involution

v \mapsto v / |v|^2

on normal vectors. The connected sum of $M_1$ and $M_2$ along $V$ is then the space

(M_1 \setminus V) \bigcup_{N_1 \setminus V = N_2 \setminus V} (M_2 \setminus V)

obtained by gluing the deleted neighborhoods together by the orientation-preserving diffeomorphism. The sum is often denoted

(M_1, V) \# (M_2, V).

Its diffeomorphism type depends on the choice of the two embeddings of $V$ and on the choice of $\psi$ .

Loosely speaking, each normal fiber of the submanifold $V$ contains a single point of $V$ , and the connected sum along $V$ is simply the connected sum described the preceding section, performed along each fiber. For this reason, the connected sum along $V$ is often called the fiber sum. The special case of $V$ a point recovers the connected sum of the preceding section.

Codimension-two submanifolds

Another important special case occurs when the dimension of $V$ is two less than that of the $M_i$ . Then the isomorphism $\psi$ of normal bundles exists whenever their Euler classes are opposite:

e(N_{M_1} V) = -e(N_{M_2} V).

Furthermore, in this case the structure group of the normal bundles is the circle group $SO(2)$ ; it follows that the choice of embeddings can be canonically identified with the group of homotopy classes of maps from $V$ to the circle, which in turn equals the first integral cohomology group $H^1(V)$ . So the diffeomorphism type of the sum depends on the choice of $\psi$ and a choice of element from $H^1(V)$ .

A connected sum along a codimension-two $V$ can also be carried out in the category of symplectic manifolds; this elaboration is called the symplectic sum.

Local operation

The connected sum is a local operation, meaning that it alters the summands only in a neighborhood of $V$ . This implies, for example, that the sum can be carried out on a single manifold $M$ containing two disjoint copies of $V$ , with the effect of gluing $M$ to itself. For example, the connected sum of a two-sphere at two distinct points of the sphere produces the two-torus.

Connected sum of knots

There is a closely related notion of the connected sum of two knots. In fact, if one regards a knot merely as a one-manifold, then the connected sum of two knots is just their connected sum as a one-manifold. However, the essential property of a knot is not its manifold structure (all knots are circles) but rather its embedding into the ambient space. So the connected sum of knots has a more elaborate definition that produces a well-defined embedding, as follows.

Sum_of_knots2.png Sum_of_knots3.png

This procedure results in the projection of a new knot, the connected sum (or knot sum, or composition) of the original knots.

Under this operation, knots in 3-space form a commutative monoid with prime factorization, which allows us to define what is meant by a prime knot. Proof of commutativity can be seen by letting one summand shrink until it is very small and then pulling it along the other knot. The unknot is the unit. The trefoil knot is the simplest prime knot. Higher dimensional knots can be added by splicing the $n$ -spheres.

In three dimensions, the unknot cannot be written as the sum of two non-trivial knots. This fact follows from additivity of knot genus; another proof relies on an infinite construction sometimes called the Mazur swindle. In higher dimensions, it is possible to get an unknot by adding two nontrivial knots.

References

Robert Gompf: A new construction of symplectic manifolds, Annals of Mathematics 142 (1995), 527-595

Differential topology | Geometric topology | Knot theory | Binary operations

Verbundene Summe