Connected amount

The connected sum of a sphere with two handles and a torus.

A connected sum is a construction in topology that allows you to build a connected $n$ ${\ displaystyle n}$ $n$ -dimensional manifold by two given connected $n$ ${\ displaystyle n}$ $n$ -dimensional varieties.

Connected sum of varieties $M$ ${\ displaystyle M}$ $M$ and $N$ ${\ displaystyle N}$ $N$ usually indicated $M\#N$ ${\ displaystyle M \ #N}$ ${\ displaystyle M \ #N}$ .

Content

Build

To build a connected sum $M\#N$ ${\ displaystyle M \ #N}$ need to cut out $M$ ${\ displaystyle M}$ and $N$ ${\ displaystyle N}$ over an open ball and glue the obtained spherical edges according to homeomorphism. If both manifolds are orientable, then the orientation is taken into account when gluing.

To determine the bound sum in the smooth category, the collars at the edge are glued together according to diffeomorphism .

This operation is uniquely determined up to a homeomorphism and, accordingly, a diffeomorphism.

Examples

$S^{n}\#M^{n}$ ${\ displaystyle S ^ {n} \ # M ^ {n}}$ homeomorphically $M^{n}$ ${\ displaystyle M ^ {n}}$ .

Properties

The operation of a connected sum is commutative up to a diffeomorphism; i.e, $M\#N$ ${\ displaystyle M \ #N}$ diffeomorphically $N\#M$ ${\ displaystyle N \ #M}$ .
Regarding the operation of a connected sum, smooth structures on a sphere form a group .

Variations and generalizations

Connected Amount of Nodes