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

## Synopsis

• Function: poincareSphereComplex
• Usage:
poincareSphereComplex S
• Inputs:
• S, , that has at least 16 generators
• Outputs:

## 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 chainComplex Δ o6 = -1 : 0 0 : 0 1 : 0 2 : 0 ZZ 1 3 : (---) 101 o6 : GradedModule 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.