Description
A piecewise linear (PL) sphere is a manifold which is PL homeomorphic to the boundary of a simplex. All the spheres in dimensions less than or equal to 3 are PL, but there are non-PL spheres in dimensions larger than or equal to 5.
As described in Theorem 7 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 non-PL 5-sphere constructed from the Björner–Lutz homology sphere by taking a double suspension.
i1 : S = ZZ/101[a..s];
|
i2 : Δ = nonPiecewiseLinearSphereComplex S;
|
i3 : matrix {facets Δ}
o3 = | dmopqr bmopqr glopqr dlopqr bgopqr fjmpqr djmpqr fimpqr bimpqr ehlpqr
------------------------------------------------------------------------
dhlpqr eglpqr bikpqr aikpqr egkpqr bgkpqr aekpqr dhjpqr chjpqr cfjpqr
------------------------------------------------------------------------
cfipqr acipqr aehpqr achpqr fjmoqr djmoqr fimoqr bimoqr ehloqr dhloqr
------------------------------------------------------------------------
egloqr bikoqr aikoqr egkoqr bgkoqr aekoqr dhjoqr chjoqr cfjoqr cfioqr
------------------------------------------------------------------------
acioqr aehoqr achoqr hkmopr dkmopr ahmopr abmopr gilopr cilopr cdlopr
------------------------------------------------------------------------
fhkopr cfkopr cdkopr gijopr eijopr bgjopr aejopr abjopr efiopr cfiopr
------------------------------------------------------------------------
efhopr aehopr klmnpr glmnpr hkmnpr ghmnpr iklnpr gilnpr hiknpr ghinpr
------------------------------------------------------------------------
jklmpr fjlmpr eglmpr eflmpr djkmpr efimpr beimpr cghmpr achmpr cegmpr
------------------------------------------------------------------------
bcempr abcmpr ajklpr aiklpr bfjlpr abjlpr acilpr efhlpr bfhlpr bdhlpr
------------------------------------------------------------------------
bcdlpr abclpr dejkpr aejkpr bhikpr bfhkpr cfgkpr bfgkpr cegkpr cdekpr
------------------------------------------------------------------------
ghijpr dhijpr deijpr cghjpr cfgjpr bfgjpr bdhipr bdeipr bcdepr klmnor
------------------------------------------------------------------------
glmnor hkmnor ghmnor iklnor gilnor hiknor ghinor jklmor fjlmor eglmor
------------------------------------------------------------------------
eflmor djkmor efimor beimor cghmor achmor cegmor bcemor abcmor ajklor
------------------------------------------------------------------------
aiklor bfjlor abjlor acilor efhlor bfhlor bdhlor bcdlor abclor dejkor
------------------------------------------------------------------------
aejkor bhikor bfhkor cfgkor bfgkor cegkor cdekor ghijor dhijor deijor
------------------------------------------------------------------------
cghjor cfgjor bfgjor bdhior bdeior bcdeor hkmopq dkmopq ahmopq abmopq
------------------------------------------------------------------------
gilopq cilopq cdlopq fhkopq cfkopq cdkopq gijopq eijopq bgjopq aejopq
------------------------------------------------------------------------
abjopq efiopq cfiopq efhopq aehopq klmnpq glmnpq hkmnpq ghmnpq iklnpq
------------------------------------------------------------------------
gilnpq hiknpq ghinpq jklmpq fjlmpq eglmpq eflmpq djkmpq efimpq beimpq
------------------------------------------------------------------------
cghmpq achmpq cegmpq bcempq abcmpq ajklpq aiklpq bfjlpq abjlpq acilpq
------------------------------------------------------------------------
efhlpq bfhlpq bdhlpq bcdlpq abclpq dejkpq aejkpq bhikpq bfhkpq cfgkpq
------------------------------------------------------------------------
bfgkpq cegkpq cdekpq ghijpq dhijpq deijpq cghjpq cfgjpq bfgjpq bdhipq
------------------------------------------------------------------------
bdeipq bcdepq klmnoq glmnoq hkmnoq ghmnoq iklnoq gilnoq hiknoq ghinoq
------------------------------------------------------------------------
jklmoq fjlmoq eglmoq eflmoq djkmoq efimoq beimoq cghmoq achmoq cegmoq
------------------------------------------------------------------------
bcemoq abcmoq ajkloq aikloq bfjloq abjloq aciloq efhloq bfhloq bdhloq
------------------------------------------------------------------------
bcdloq abcloq dejkoq aejkoq bhikoq bfhkoq cfgkoq bfgkoq cegkoq cdekoq
------------------------------------------------------------------------
ghijoq dhijoq deijoq cghjoq cfgjoq bfgjoq bdhioq bdeioq bcdeoq |
1 269
o3 : Matrix S <-- S
|
i4 : dim Δ
o4 = 5
|
i5 : fVector Δ
o5 = {1, 18, 141, 515, 930, 807, 269}
o5 : List
|
i6 : assert(dim Δ === 5 and isPure Δ)
|
i7 : assert(fVector Δ === {1,18,141,515,930,807,269})
|
This abstract simplicial complex is Cohen-Macaulay.