@DanPetersen Thanks a lot for your comments. I edited my description of gluing in consequence.

@RyanBudney My apologies for the confusion: the "space of contractible loops" is the space of loops which happen to be contractible. In general it won't be homotopy equivalent to the underlying manifold.

@WillieWong: Thanks for the comments. If I understand correctly, you point out that we can solve the IVP for any topology, because the constraint equation can be solved for any topology and Choquet-Bruhat's theorem then gives a solution to the evolution problem? However, I would add that if we impose other conditions on the data which must satisfy the constraints (for instance, that it be CMC), there may be restrictions on the topology. And showing that there are no restrictions can be hard (see for instance this article of Isenberg-Mazzeo-Pollack arxiv.org/abs/gr-qc/0206034)