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added pointer also to
which is another review of flat compact 2-dimensional orbifolds as homotopy quotients of the 2-torus by wallpaper groups.
Does anyone ever discuss this in higher dimensions? Such as flat 4d orbifolds as homotopy quotients of the 4-torus by one of its finite subgroups? I understand that the full classification will tend to get out of control as the dimension increases, but there should still be something to say. But I just can’t find anyone discussing this. If anyone has a pointer, please let me know.
There is a MathSE question here in the same direction (and still without reply): Is there for $d \geq 3$ an orbifold of the form $T^d \sslash G$ whose homotopy type is that of $S^d$? The MathSE question wants to know this specifically for $G = \mathbb{Z}/2$, but I’d be happy already to know the converse: For which finite group actions $G$ is this the case?
Well, they’ve classified all space groups up to dimension 6.
Did you see this MO question? And your question in #3 was taken over to MO here.
Thanks! Hadn’t seen these yet. This is useful.
So according to Piergallini 95 every 4-manifold may be realized as a branched cover of the 4-sphere. In particular hence the 4-torus may be realized as a branched cover of the 4-sphere, just not as a “cyclic branched cover” according to Hirsch-Neumann 75.
I’d like to conclude that the 4-sphere is indeed the homotopy type of a flat 4-dimensional orbifold, just not of a global quotient by a cyclic group. Unfortunately, there is fine print in all the definitions (“manifold”, “branched cover”) which is probably as crucial to the statement as it seems notoriously hard to extract from these references.
And indeed I said this incorrectly:
That every $n$-manifold is a branched cover of $S^n$ is already due to Alexander 20. What Piergallini 95 and followups add to this is extra nicety properties of the branching for $n = 4$. Need to understand these in terms of the corresponding orbifolding…
added pointer to
for classification of 6-dimensional toroidal (hence flat) orbifolds
How does ’symmetric’ qualify ’toroidal orbifold’?
Right, they never say this, do they? Usually, by “the symmetric orbifold” people mean just the quotient by the symmetric group $\Sigma_n$ of the $n$-fold Cartesian product of some orbifold with itself, such as in equations (2.9) to (2.11) in “Higher spins in the symmetric orbifold of K3” (arXiv:1504.00926).
added pointer to
for classification of flat (more generally: locally homogeneous) compact oriented 3-dimensional orbifolds.
Many of them have coarse underlying topological space the 3-sphere!
Added the following brief comment on realizing the 4-sphere as the topological space underlying a flat orbifold. (Is there a more canonical citation for the first part of the statement?)
The complex projective space $\mathbb{C}P^2$ is an orbifold quotient of the 4-torus (see this MO comment), in generalization of how the Weierstrass elliptic function exhibits the Riemann sphere as $\mathbb{T}^2/\mathbb{Z}_2$.
A further $\mathbb{Z}_2$-quotient of $\mathbb{C}P^2$ gives the 4-sphere (see at 4-sphere as quotient of complex projective plane).
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