Topology change with Morse functions: progress on the Borde-Sorkin conjecture

Topology change is considered to be a necessary feature of quantum gravity by some authors, and impossible by others. One of the main arguments against it is that spacetimes with changing spatial topology have bad causal properties. Borde and Sorkin proposed a way to avoid this dilemma by considering topology changing spacetimes constructed from Morse functions, where the metric is allowed to vanish at isolated points. They conjectured that these Morse spacetimes are causally continuous (hence quite well behaved), as long as the index of the Morse points is different from $1$ and $n-1$. In this paper, we prove a special case of this conjecture. We also argue, heuristically, that the original conjecture is actually false, and formulate a refined version of it.Comment: Mistake in proof of old Lemma 3 fixed. Additional assumption added in Theorem 1. Numbering of Lemmas has shifted. Section 2.2 is completely redone, in the other sections there are some minor changes. 23 pages, 8 figures, comments welcome