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