What is spacetime topology and geometry and the manipulation of these entities could allow time travel? I have a few questions about spacetime:
1. What is spacetime topology and geometry?
2. Can changing the topology and geometry allow for time travel?
According to the Wheeler's theory  spacetime topology and geometry fluctuates. In quantum foam there might form the wormholes that can connect any two arbitrary points of space-time (possibly distant past or the distant future, and the two ends of the universe), so by manipulating the topology and geometry, one can create a wormhole that allows for travel to any time and any place.
 A: I will try to make this qualitative without a great deal of mathematics. A part of the problem is that these questions tend to be at the frontier of physical understanding. As such I will work with physical thinking primarily.
The first question concerns the topology of the universe, or the spacetime that constitutes the observable universe. This I will answer based on observation, which is that it appears that space is very flat. So far there has been no observation of any curvature to space. However, as noted by the writer Jorges Borges in one of his short stories a sphere with a nearly infinite radius can appear to have a very flat surface. We may only observe a tiny patch on a spherical $3$-sphere, or for that matter a $3$ hyperboloid. The FLRW constraint equation for the scale factor $a~=~a(t)$
$$
\left(\frac{\dot a}{a}\right)^2~=~\frac{8\pi G\rho}{3c^2}~+~\frac{k}{a^2}
$$
determines spherical, flat and hyperbolic geometry for $k~=~1,~0,~-1$.
Changing the topology of space is problematic. I am thinking of the time evolution of a spatial surface, similar to the idea of foliating spacetime with spatial surfaces in ADM relativity, where that changes its topology. We know from mathematics that cohomology groups define certain invariant number, such as Chern numbers, and if the topology of space changes it means that by some means we can change these topological indices. The intersection form,
$$
\omega(\alpha\cup\beta)~=~\int_M\alpha\wedge\beta
$$
defines a “charge” when the two $2$-forms enclose or orbit a hole or some nontrivial topology. 
In a sense a black hole is a nontraversable black hole. Consider Hawking radiation from a Schwarzschild black hole. I include the diagram below which is the Penrose conformal diagram for a black hole. In the regions I and II are hyperbolas that represent the constant radial position of an observer. The red circle is the virtual fluctuation of a particle which an observer on a constant radial path would observe to emerge from the past or white hole horizon and then approach the black hole horizon. The circle determines the $e^{2\pi iH/g}$ for the quantum field with $g~=~c^2/\rho$ on a constant radial path with $\rho$. The black hole with the split horizon represents two entangled blackholes in the region I and II. The emission of Hawking radiation, here diagrammed as the two dots connected by a red segment, transfers some of this entanglement to the two regions. The new event horizon is seen as the two hyperbolic paths in blue. The two black holes are then no longer completely entangled.
This is another illustration of how spacetime is built up from entanglements. Raamsdonk illustrated this in a paper. This presentation does not give a dynamics for how the big bang produces spacetime, but it does illustrate how spacetime is an emergent epiphenomenology of quantum mechanics. I am using the black hole as a sort of theoretical laboratory, which might in some way become more of an experimental object.
Now let us suppose I am in region I and I have the particle emitted by Hawking radiation (red dot on my side region I), and this particle is in the state $\psi~=~\sum_n\chi_n$. I then open up a wormhole into region II, which by the horizons is not accessible by ordinary means, and I grab the other particle and bring it to region I. I now have two particles that because of their entanglement are indistinguishable and are thus duplicates. I have closed the state on my side region I.

The cloning theorem says that if I have a state $\psi$ cloning it so $\psi~\rightarrow~\psi\psi$ results in an inconsistency. Let me write $\psi~=~a|a\rangle~+~b|b\rangle$. For simplicity I am considering it with two basis elements. My duplication then means
$$
\psi~\rightarrow~\psi\psi~=~aa|aa\rangle~+~ab|ab\rangle~+~ba|ba\rangle~+~bb|bb\rangle.
$$
However the duplication can also just mean $a|a\rangle~\rightarrow~aa|aa\rangle$ and $b|b\rangle~\rightarrow~bb|bb\rangle$ and I have no “ab and ba terms.” So there is no consistent way to clone or Xerox states. It would also mean that I can find a hidden variable underlying an entanglement. So it appears that wormholes run into some trouble with quantum physics. In addition the prospect that spacetime is woven together by quantum entanglements appears to suggest that changing spatial topology and cloning a quantum state by unitary means is the same thing.
However we can come up with “close clones” of quantum states. There has been some interest in this. The spacetime correspondence would then be some quantum superposed topology deformation on the base topology. The transition probably can't be complete, but there could be some quantum fluctuations or coherent effects that mean the topology of space can have some small quantum amplitude for a different topology. Changing the topology completely or classically is potentially the same as a perfect clone of a quantum state. This is in a sense a “quantum logical” reason for why the speed of light is a sort of absolute limit.
So the upshot may just be that traversable wormholes do not exist. So far there is no observational data to suggest their existence. It may in part answer the question attributed to Enrico Fermi, “Where are they?”
