Recall that one way of drawing closed 2-manifolds is to take a disk $D^2$, take a cellular decomposition of $\partial D^2$, pair the vertices in this cellular decomposition so that the pairing preserves edges, and then take $D$ together with this quotient of the boundary.


We can do this in other dimensions as well, for example in dimension 3, every closed 3-manifold can be obtained by a similar procedure where we take $B^3$, take a cellular decomposition of $\partial B^3$, pair the vertices of this cellular decomposition so that the pairing preserves edges and faces, and then look at the quotient of $B^3$ by this pairing.


Threlfall and Seifert did this for the Poincaré homology sphere (see for example [here](http://www.map.mpim-bonn.mpg.de/Poincar%C3%A9%27s_homology_sphere) - which also contains a different such description due to Kreines). In fact, they take the cellulation of $\partial B^3$ to be the dodecahedron. Is there a complete (presumably rather short) list of all of the 3-manifolds obtained in such a way where the cellulation is a Platonic solid? $T^3$, $\mathbb{R}P^3$, and the Seifert-Weber space are other examples that come to mind. I'd guess that the Poincaré homology sphere is maybe the only homology sphere on that list. More generally, I'd like to look through a list of the 3-manifolds that occur in this way using simple cellulations.


This can also be done in a similar way in dimension 4 to produce all smooth closed 4-manifolds. Are there some nice pictures/examples of this being carried out somewhere? I'd love to see such pictures of $S^2 \times S^2, T^4, \mathbb{C}P^2,...$



low dimensional manifolds by gluing the boundary of a ball
This closed orientable 3-manifolds obtained by gluing faces of the Platonic solids were classified by [Everitt](https://doi.org/10.1016/j.topol.2003.08.025).


That is for regular polyhedra with equal dihedral angles, and the gluing is done geometrically. However, it is also possible to do the gluing topologically, and for that problem, I only have a partial answer. There are 3 closed orientable 3-manifolds obtained by gluing faces of the tetrahedron. They are $S^3$, $L(4,1)$, and $L(5,2)$. Explicit gluings can be seen in figure 2 of this paper of [Jaco and Rubinstein](https://projecteuclid.org/euclid.jdg/1090503053).


There are 17 closed orientable 3-manifolds obtained by gluing faces of the octahedron, 13 of which are prime. They are listed in Proposition 4.2 of this paper by [Heard, Pervova, and Petronio](https://projecteuclid.org/euclid.em/1243429960).


Presumably, the closed orientable 3-manifolds obtained from the cube have been enumerated, but I don't know of a reference. They include $\mathbb{R}P^3$, the 3-torus and at least 2 of the other closed orientable Euclidean 3-manifolds. I imagine there are a lot of 3-manifolds obtained from the dodecahedron and icosahedron, but I doubt that anyone has enumerated them all.


As for 4-manifolds, I will leave that for someone else to answer, except to note that there are no 4-manifolds obtained from a single pentachoron (4-simplex), since it has 5 tetrahedra in its boundary and this causes a parity issue.


