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End sums of irreducible open 3-manifolds

Abstract

An end sum is a non-compact analogue of a connected sum. Suppose we are given two connected, oriented nn-manifolds M1M_1 and M2M_2. Recall that to form their connected sum one chooses an nn-ball in each MiM_i, removes its interior, and then glues together the two Snβˆ’1S^{n-1} boundary components thus created by an orientation reversing homeomorphism. Now suppose that M1M_1 and M2M_2 are also open, i.e. non-compact with empty boundary. To form an end sum of M1M_1 and M2M_2 one chooses a halfspace HiH_i (a manifold \homeo\ to Rnβˆ’1Γ—[0,∞){\bold R}^{n-1} \times [0, \infty)) embedded in MiM_i, removes its interior, and then glues together the two resulting Rnβˆ’1{\bold R}^{n-1} boundary components by an orientation reversing homeomorphism. In order for this space MM to be an nn-manifold one requires that each HiH_i be {\bf end-proper} in MiM_i in the sense that its intersection with each compact subset of MiM_i is compact. Note that one can regard HiH_i as a regular neighborhood of an end-proper ray (a 1-manifold \homeo\ to [0,∞)[0,\infty)) \ga_i in MiM_i

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