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!



Regular or h-regular CW-complex structure for the Poincaré homology sphere
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...


