The Poincaré homology sphere is a homology 3-sphere; it has the same homology groups as a 3-sphere. Following Theorem 5 in Anders Björner and Frank H. Lutz's "Simplicial manifolds, bistellar flips and a 16-vertex triangulation of the Poincaré homology 3-sphere", Experimental Mathematics 9 (2000) 275–289, this method returns a Poincaré homology sphere with 16 vertices.
i1 : S = ZZ/101[a..q];
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i2 : Δ = poincareSphereComplex S;
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i3 : matrix {facets Δ}
o3 = | mnop inop jmop ijop kmnp iknp jkmp ijkp lmno hlno gino ghno flmo ajmo
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afmo bhlo bflo ghko bhko dgko bdko eijo cejo acjo egio dego abfo cdeo
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acdo abdo clmn ckmn dhln cdln aikn cekn aekn ghjn dhjn bgjn dfjn bfjn
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bgin abin defn aefn abfn cden fglm cglm djkm bdkm bckm dhjm ahjm ehim
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dhim egim bgim bdim aehm efgm bcgm aefm ijkl fjkl aikl fgkl agkl eijl
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bfjl bejl ehil dhil adil behl acgl acdl dfjk aghk behk aehk dfgk bcek
------------------------------------------------------------------------
aghj bcgj acgj bcej abdi defg |
1 90
o3 : Matrix S <-- S
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i4 : dim Δ
o4 = 3
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i5 : fVector Δ
o5 = {1, 16, 106, 180, 90}
o5 : List
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i6 : prune HH chainComplex Δ
o6 = -1 : 0
0 : 0
1 : 0
2 : 0
ZZ 1
3 : (---)
101
o6 : GradedModule
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i7 : assert(dim Δ === 3 and isPure Δ)
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i8 : assert(fVector Δ === {1,16,106,180,90})
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