# How many compactifications of Minkowski space are possible?

Given an $n$-dimensional Minkowski space, how many spacetime manifolds can one obtain by some compactification of it?

In two dimensions, I can think of those, at least :

• The spacelike cylinder $\Bbb R_t \times S_x = \Bbb R^2 / \Bbb Z$
• The timelike cylinder $S_t \times \Bbb R_x = \Bbb R^2 / \Bbb Z$
• The torus $S_t \times S_x = \Bbb R^2 / \Bbb Z^2$
• The Klein bottle $\mathbb K = \Bbb R^2 / \sim$, $\sim$ the usual identification (I'm guessing it comes in both timelike and spacelike flavors)
• Misner space $\Bbb R^2 / B$, $B$ a Lorentz boost
• The timelike Moebius strip
• The spacelike Moebius strip
• The non-time orientable cylinder obtained by $(\Bbb R^2 / \Bbb Z) / (I \times T)$, $I$ some involution and $T$ time reversal

There are probably a bunch more, but those are the ones that pop to mind. The list seems fairly long, but I'm not really sure how to generalize it to any dimension.

I'm guessing that since they all can be expressed as a quotient, that is the thing to look into, the full list of discrete proper free groups that leave Minkowski space invariant, but there are of course infinitely many so. What classes can be found that will lead to distinct spacetimes?

There seems to be at least the translations along spacelike and timelike directions (and null, I suppose), the same but with change of orientation, and some involutions for the already compactified groups. Are those all the possible ones?

• I think you might want to be more precise/explicit in specifying when you consider compactifications as equivalent (Riemannian structure? smooth structure?) – Danu Jun 11 '17 at 19:26
• Being isometric, I suppose. Maybe up to finitely many continuous parameters (corresponding to the characteristic dimension of that compactification) – Slereah Jun 11 '17 at 19:30