next | previous | forward | backward | up | index | toc

nonPiecewiseLinearSphereComplex(PolynomialRing) -- make a non-piecewise-linear 5-sphere with 18 vertices

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.

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