An end sum is a non-compact analogue of a connected sum. Suppose we are given
two connected, oriented n-manifolds M1β and M2β. Recall that to form
their connected sum one chooses an n-ball in each Miβ, removes its
interior, and then glues together the two Snβ1 boundary components thus
created by an orientation reversing homeomorphism. Now suppose that M1β and
M2β are also open, i.e. non-compact with empty boundary. To form an end sum
of M1β and M2β one chooses a halfspace Hiβ (a manifold \homeo\ to Rnβ1Γ[0,β)) embedded in Miβ, removes its interior, and then
glues together the two resulting Rnβ1 boundary components by an
orientation reversing homeomorphism. In order for this space M to be an
n-manifold one requires that each Hiβ be {\bf end-proper} in Miβ in the
sense that its intersection with each compact subset of Miβ is compact. Note
that one can regard Hiβ as a regular neighborhood of an end-proper ray (a
1-manifold \homeo\ to [0,β)) \ga_i in Miβ