Causal structure is not only explained by the domain of
influence. There is also the dual notion, all the space
at one time, which is a slice. The slice (definition 
page ) will
be used for the later definition of the domain of
dependence. But now we will study some general features of
slice and how it is linked with the underlying manifold and the
causality relations that hold on it.
It can be interesting, in our research of the topology of our universe, to know what kind of spacetimes admit slices and what is then the allowed topology of that slice.
Not all the spacetimes admit slices. One result is that every stably causal spacetime admits slices, even more there is a slice passing through every point p. It is easy to prove by using the time-function on the spacetime, that the t=constant sub-manifolds are the slices. Sadly the converse is false and having a slice does not imply that the spacetime is stably causal.
Without more condition all topologies are allowed for the
slices. In the same way there is no general result for saying
when two slices must have the same topology.
An important notion for the next part is the idea of achronal slice .
Every stably causal spacetime admits achronal slices. Indeed any slice of constant time-function must be achronal. But the relation between the two is not so easy; some stably causal spacetimes admit slice which are not achronal and some spacetimes which are not stably causal can have achronal slices.
One can conclude that the link is rather weak between slice in one hand and topology and causal structure in the other hand.
