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.
The illustrations above show the connected sums of two tori (top figure) and of two pairs of multi-handled tori.
