The connected sum 
of 
-manifolds 
and 
is formed by deleting the interiors
of 
-balls 
in 
and attaching the resulting punctured manifolds 
to each other by a homeomorphism 
, so
 |
is required to be interior
to 
and 
bicollared in 
to ensure that the connected sum is a manifold.
Topologically, if 
and 
are pathwise-connected, then the connected
sum is independent of the choice of locations on 
and 
where the connection is glued.
For closed surfaces 
and 
that are connected, their
Euler characteristic satisfies
 |
(Hatcher 2002, p. 257). To verify this, choose a triangulation of each surface and remove the interior
of a triangle from each. Gluing the triangular boundaries
identifies three pairs of vertices and three pairs
of edges. Thus the total numbers of vertices,
edges, and faces change
by 
, 
, and 
, respectively, so 
decreases by 2.
Since a Klein bottle has Euler characteristic 0, the connected sum of 
Klein bottles has Euler characteristic 
. This sum is nonorientable,
so its genus is 
. In particular, the connected sum of two Klein
bottles has genus 4. In contrast, the connected sum
of 
tori has the
same Euler characteristic but is orientable
and has genus 
.
The illustrations above show the connected sums of two tori (top figure) and of two pairs of multi-handled tori.
