In a mathematical sense, (differential) topology is imposed on the space-time, it is the "background" of the theory and the metric cannot change the topology of the background manifold. I.e. every metric should come with a "manual of topology" which specifies things such as coordinate ranges and identified coordinate values. However, this is not always the case, coordinates such as $\phi$ are taken as the $(0,2 \pi]$ identification without saying and in many cases there are multiple topological interpretations of a metric even after imposing the conditions sketched below. (btw. see what happens when you do not do the $(0,2 \pi]$ thing.)

Nevertheless, the properties of the metric "feed back" into the manifold topology through the person studying the metric who chooses a different background topology based on physical motivation or convenience.

A few examples. Consider the Schwarzschild metric: $${\rm d}s^2 = -(1-2M/r) {\rm d}t^2 + \frac{1}{1-2M/r} {\rm d}r^2 + r^2 {\rm d}\Omega^2$$

By physical argumentation we find out that there is something fishy about the horizon, it has zero volume, particles which fly through pass through $t=\infty$ but at finite proper time etc. The coordinate transformation known as Kruskalization (see Kruskal-Szekeres coordinates) which makes the horizon regular *actually changes the differential structure of the manifold*. (alternatively see Eddington-Finkelstein coordinates)

Another canonical example would be the Kerr metric in Boyer-Lindquist coordinates representing a spinning black hole. There one finds a singularity at the disc $r=0$ with a field jump in the interior parts of the disc $r=0, \theta \neq \pi/2$ and a genuine metric singularity at the edge of the disc $r=0, \theta=\pi/2$. The field jump on the interior, when confronted with Einstein equations, would mean negative matter density. On the other hand, one can *choose* to avoid the negative matter density by introducing different topology, and by *saying* that by going through the disc one enters a new $r<0$ region. I.e. going through the "top" of the disc $r=0, \theta<\pi/2$ one does not enter the "bottom" of the disc $r=0, \theta>\pi/2$ but a completely new region and vice versa.

In the case of Schwarzschild introducing an $r<0$ region would be contrived; all geodesics end in $r=0$ and there would be no causal communication with $r<0$ whatsoever. However, the nature of the $r=0$ singularity in Kerr allows one to change the topology in a physically natural sense. It is then possible to give physically motivated conditions on the singularities which are "terminal" and which are not allowed at all.

Every mathematician knows from theorems such as the hairy-ball theorem that restrictions on singularities of differential systems restrict the allowed topology. It is also in this sense that the properties of the metric such as asymptotic flatness along with such conditions can restrict the topology of the manifold. The exact-solver of Einstein equations then usually starts with one coordinate patch and tries to solve the equations without encountering the "wrong" singularities - this, on the other hand, already determines the topology and I believe it is exactly in this sense in which you have encountered the interactions of the metric/topology.