Regular or h-regular CW-complex structure for the Poincaré homology sphere I am looking for a regular (the characteristic maps of the cells are homeomorphisms) or h-regular (the characteristic maps of the cells are homotopy equivalences) CW-complex structure for the Poincaré homology sphere. I would like to find a more economic one than the triangulation having f-vector: [16, 106, 180, 90]. I would like to minimize the number of cells.
Thanks in advance and any idea or potential reference is welcome!
 A: Recently, I (with R. Chirivi and M. Spreafico) have explicitely given a decomposition of the Poincare homology sphere $\mathbb{S}^3/\mathcal{I}$ (with $\mathcal{I}$ the binary icosahedral group of order 120). For this see the Theorem 4.3.2 of https://arxiv.org/abs/2006.14417. In fact, we construct a regular $\mathcal{I}$-equivariant cellular structure on $\mathbb{S}^3$ using orbit polytopes. It has f-vector [1,5,5,1], which could be minimal. Moreover, the homology chain complex associated to it is quite simple, the first and third differentials are zero and the middle one is a sparse invertible circulant matrix. Hence and unfortunately, this decomposition of $\mathbb{S}^3/\mathcal{I}$ is not regular. Hope this still helps a bit though.
A: Henrik Rüping's suggestion (in the comments) decomposes the Poincaré homology three-sphere as 12 pentagonal pyramids.  The resulting face vector is
[0 + 12, 6 + 30, 10 + 20, 5 + 1] = [12, 36, 30, 6]
for a total of 84 cells.  You can reduce the number of three-cells, at the cost of increasing the number of cells overall, as follows.  In Rüping construction, join the pyramids in pairs (along their pentagonal face) to get six bi-pyramids.  This has an face vector of
[6, 30, 30, 6]
but is no longer regular -- every bi-pyramid meets the central vertex twice.  We can fix this by "blowing up" the central vertex to obtain a small dodecahedron.  This truncates the apex and nadir of each bi-pyramid.  Now the face vector is
[6 + 1, 30 + 12, 30 + 30, 6 + 20] = [7, 42, 60, 26]
with sum 135.  At least the number of three-cells is smaller...
