Spacetime surgery - why are there unglueable points? In The time travel paradox by S. Krasnikov (2002), Deutsch-Politzer spacetime is constructed by making two cuts and rejoining the manifold by gluing opposite "banks" of the cuts... omitting the "corner" points.
See figure below - cut along dashed lines; corner points are the circles at the ends of the dashed lines; the identification is above upper line to below lower line.
Krasnikov then says "The corner points cannot cannot be glued back into the spacetime..."
Question: why can't they be glued back? What principles are being implicitly invoked?
(Can anyone point to an introductory text that describes what surgeries are possible?)

 A: HOW DOEST ONE MAKE THE DEUTSCH-POLITZER SPACETIME
A good question because it's not 100% the standard procedure of a cut and paste spacetime (a good reference on cutting and pasting manifold is by the way C.T.C. Wall's Differential Topology. There is indeed not a lot of good references for GR on that topic). 
It is best to consider the manifold atlas directly instead of any fancier methods (I think you could probably do it by adding a handle to Minkowski space and some kind of limiting process, but that sounds tricky as it approaches the limit), as it's not that complex. 
Take Minkowski spacetime (ie topologically, $\mathbb{R}^n$). Here are the coordinate patches to consider
First, the obvious one, which covers most of it. Remove the two horizontal slits $D_1, D_2$ from it. Then the first coordinate chart is 
$$(\mathbb{R}^n \setminus (D_1 \cup D_2), \operatorname{Id})$$
That's the easy part. now we need some additional patches to represent the junction of the slits. 
Simplest case is to pick some open product around the slits. For instance, consider the case of $(1+1)$-dimensional Minkowski space, with the slits $D_1 = \{ -1 \} \times [-1, 1]$ and $D_2 = \{ 1 \} \times [-1, 1]$. The first junction can be for instance composed of $A_1 = (-2, -1) \times (-1, 1)$ and $A_2 = (1,2) \times (-1, 1)$, and the second of $B_1 = (-1, 0) \times (-1, 1)$ and $B_2 = (0,1) \times (-1, 1)$. 
To define a manifold, all we need is a set of open sets of $\mathbb{R}^n$ and transition functions [4.1], so let's find that out. Our atlas to $A_1 \cup A_2$ can be made from a set $A = (0,2) \times (-1, 1)$, and our atlas to $B_1 \cup B_2$ can be made from a set $B = (0,2) \times (-1, 1)$. $A$ and $B$ do not overlap, so all we need is the transition functions between the main patch (let's call it $R$) and $A$ and $B$. 
Here we go : for $A$ we consider the subsets 
\begin{eqnarray}
A_{R1} &=& (0,1) \times (-1, 1)\\
A_{R2} &=& (1,2) \times (-1, 1)
\end{eqnarray}
Each mapping to the manifold as $\phi(A_{Ri}) = A_i$), and for $R$ we consider the subsets 
\begin{eqnarray}
R_{A1} &=& (-2, -1) \times (-1, 1)\\
R_{A2} &=& (1,2) \times (-1, 1)
\end{eqnarray}
which simply map to $A_i$ via the identity map. Now we have the following transition maps : 
\begin{eqnarray}
\phi_{A1R1}(t,x) &=& (t-2, x)\\
\phi_{A2R2}(t,x) &=& (t, x)
\end{eqnarray}
Finding the inverse maps isn't terribly hard, and overall these obey the proper transition map properties. The same can be found for the patch $B$ fairly easily.
That's the Deutsch-Politzer spacetime manifold. Consider now a curve lying entirely outside of $(-2, 2) \times (-1,1)$ (to simplify things) heading to the point $(-1, -1)$. In the patch defined by $R$, it is fairly obviously a singularity, as that point is removed from that patch. This wouldn't be a problem if another coordinate patch could continue that curve, but this is not the case here : there exists no point of that curve lying in any other coordinate patch. Hence we have an inextendible curve with finite affine parameter : that point is a singularity. 
Had we left those points in (by say removing open sets $\operatorname{Int}(D)_i$ rather than closed sets), $R$ would not have been an open set, and hence the resulting space would not have been a manifold. 
We are in Minkowski space here, which means that this singularity can only be of two types : either regular or quasi-regular. To make it a "serious" singularity we should probably check that it is not simply a regular point. If you try to extend the manifold to include this point, some open set around it will overlap with regions $A$ and $B$. A bad thing happens here : the tiny region in $R$ that is extended around the singularity contains points from one of the slit $D$. Those slit points will be adjacent to points from the junction (specifically points in $\{1\} \times (-1, 1)$, that is, there are pairs of points such that every neighbourhoods of those points overlap : the manifold isn't Hausdorff anymore, which is a big no-no.
Edit : By the way this kind of gluing may be more in line with the type discussed by Hajicek than the standard manifold gluing, and he discusses the issues in Bifurcate Space‐Times, such as in which conditions a gluing leads to non-Hausdorff spacetimes
A: Physically, the question of what happens precisely at the corner points isn't especially meaningful. Instead, you could think about phenomena in a neighborhood of the points (e.g. flux through a circle or tube enclosing the singularity), and use that to characterize the point itself. 
By definition, a manifold should be locally Euclidean. Any open neighborhood (with the metric topology) containing those endpoints will not look Euclidean, because of 'creases' that develop around them (i.e. curvature singularities). These singularities are apparent from the fact that a small closed circular loop around a corner point has twice the normal circumference. Alternatively, using the Lorentzian metric structure (or causal structure), the singularity can be deduced from observing that a closed loop around a corner point must have at least two components during which time is increasing, instead of one. 
To determine whether the singular corner points can be glued back into the spacetime, one can ask whether the sharply concentrated intrinsic curvature at the end-points can be redistributed so that the manifold is well defined everywhere (in particular, if we sent a massless point-like particle directly at one of the corner points, we should have a mathematically consistent, physically motivated rule to determine what the particle does after it reaches the corner point). If we require spacetime to be asymptotically flat, i.e. Minkowskian far from the altered region, then on physical grounds it's not hard to see that the corner singularities cannot be smoothed out without also removing the handle entirely. The reason for this is essentially the discontinuity in behavior of null geodesics near the corners: whether the trajectory winds $n$ times around the handle before it continues to future null infinity, or $n+1$ times. Removing this discontinuity would require either closing off the ends of the handle, or shrinking all of spacetime to a torus so that all trajectories are periodic in time (up to a boundary), and there is no sharp transition between trajectories with different winding numbers around the handle. 
I am not aware of a general reference for spacetime surgery, but both Wald's General Relativity and Hawking & Ellis The Large Scale Structure of Spacetime provide a more mathematically thorough perspective on GR.
