End sums of irreducible open 3-manifolds

An end sum is a non-compact analogue of a connected sum. Suppose we are given two connected, oriented $n$-manifolds $M_1$ and $M_2$. Recall that to form their connected sum one chooses an $n$-ball in each $M_i$, removes its interior, and then glues together the two $S^{n-1}$ boundary components thus created by an orientation reversing homeomorphism. Now suppose that $M_1$ and $M_2$ are also open, i.e. non-compact with empty boundary. To form an end sum of $M_1$ and $M_2$ one chooses a halfspace $H_i$ (a manifold \homeo\ to ${\bold R}^{n-1} \times [0, \infty)$) embedded in $M_i$, removes its interior, and then glues together the two resulting ${\bold R}^{n-1}$ boundary components by an orientation reversing homeomorphism. In order for this space $M$ to be an $n$-manifold one requires that each $H_i$ be {\bf end-proper} in $M_i$ in the sense that its intersection with each compact subset of $M_i$ is compact. Note that one can regard $H_i$ as a regular neighborhood of an end-proper ray (a 1-manifold \homeo\ to $[0,\infty)$) $\ga_i$ in $M_i$.

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