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CONNECTED SUM OF TWO SURFACES

The connected sum of two connected surfaces
and is
the surface, defined up to homeomorphism, obtained by removing an open
set homeomorphic to the disk to each of the surfaces and by identifying
the boundaries; notation: ,
or even 2*S* for the sum of two homeomorphic surfaces.

Examples:

- the connected sum of two spheres
is a sphere:
(in fact the sphere is the identity element for this operation).

- the connected sum of two real
projective planes is a Klein bottle: .

- the connected sum of three real
projective planes (thus of a Klein bottle and a projective plane) is
topologically equivalent to the sum of a torus
and a real projective plane: ;
we get Dyck's surface.

The classification theorem of compact surfaces without
boundaries states that such a surface is either of the type
(*n* integer), in other words, *n*-torus
(if it has two faces), or
(*n* positive integer), if it has only one face.

We can illustrate this theorem by saying that any compact
surface without boundaries is homeomorphic to a sphere with a certain number
of classic handles (tori) and Möbius handles (projective planes),
with the rule that 2 Möbius handles can be changed for a classic handle,
on the condition that there remains at least another Möbius handle.

See also the Euler characteristic.

Compare the connected sum to the composition
of knots.

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© Robert FERRÉOL 2017