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poincareSphereComplex(PolynomialRing) -- make a homology 3-sphere with 16 vertices

Description

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];
 
i2 : Δ = poincareSphereComplex S;
 
i3 : matrix {facets Δ}

o3 = | mnop inop jmop ijop kmnp iknp jkmp ijkp lmno hlno gino ghno flmo ajmo
     ------------------------------------------------------------------------
     afmo bhlo bflo ghko bhko dgko bdko eijo cejo acjo egio dego abfo cdeo
     ------------------------------------------------------------------------
     acdo abdo clmn ckmn dhln cdln aikn cekn aekn ghjn dhjn bgjn dfjn bfjn
     ------------------------------------------------------------------------
     bgin abin defn aefn abfn cden fglm cglm djkm bdkm bckm dhjm ahjm ehim
     ------------------------------------------------------------------------
     dhim egim bgim bdim aehm efgm bcgm aefm ijkl fjkl aikl fgkl agkl eijl
     ------------------------------------------------------------------------
     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
 
i4 : dim Δ

o4 = 3
 
i5 : fVector Δ

o5 = {1, 16, 106, 180, 90}

o5 : List
 
i6 : prune HH complex Δ

       ZZ 1
o6 = (---)
      101
      
     3

o6 : Complex
 
i7 : assert(dim Δ === 3 and isPure Δ)
 
i8 : assert(fVector Δ === {1,16,106,180,90})

This abstract simplicial complex is Cohen-Macaulay.

Our enumeration of the vertices also follows the poincare example in Masahiro Hachimori's simplicial complex library.

See also

Ways to use this method:

The source of this document is in SimplicialComplexes/Documentation.m2:1102:0.