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chainComplex(SimplicialMap) -- constructs the associated map between chain complexes

Synopsis

• Function: chainComplex
• Usage:
chainComplex f
• Inputs:
• Outputs:

Description

Given a simplicial map, this constructs the map between the associated chain complexes of the two simplicial complexes.

 i1 : S = ZZ[x_0..x_5]; i2 : Δ = simplicialComplex monomialIdeal(x_0*x_5, x_1*x_4, x_2*x_3) o2 = simplicialComplex | x_3x_4x_5 x_2x_4x_5 x_1x_3x_5 x_1x_2x_5 x_0x_3x_4 x_0x_2x_4 x_0x_1x_3 x_0x_1x_2 | o2 : SimplicialComplex i3 : Γ = simplicialComplex monomialIdeal(x_1*x_2) o3 = simplicialComplex | x_0x_2x_3x_4x_5 x_0x_1x_3x_4x_5 | o3 : SimplicialComplex i4 : f = map(Γ, Δ, {x_0,x_0,x_1,x_2,x_3,x_3}) o4 = | x_0 x_0 x_1 x_2 x_3 x_3 | o4 : SimplicialMap simplicialComplex | x_0x_2x_3x_4x_5 x_0x_1x_3x_4x_5 | <--- simplicialComplex | x_3x_4x_5 x_2x_4x_5 x_1x_3x_5 x_1x_2x_5 x_0x_3x_4 x_0x_2x_4 x_0x_1x_3 x_0x_1x_2 | i5 : F = chainComplex f 1 1 o5 = -1 : ZZ <--------- ZZ : -1 | 1 | 6 6 0 : ZZ <------------------- ZZ : 0 | 1 1 0 0 0 0 | | 0 0 1 0 0 0 | | 0 0 0 1 0 0 | | 0 0 0 0 1 1 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | 14 12 1 : ZZ <------------------------------- ZZ : 1 | 0 1 0 0 1 0 0 0 0 0 0 0 | | 0 0 1 0 0 1 0 0 0 0 0 0 | | 0 0 0 1 0 0 1 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 1 1 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 1 1 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 | 16 8 2 : ZZ <----------------------- ZZ : 2 | 0 0 1 0 1 0 0 0 | | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 | | 0 0 0 1 0 1 0 0 | | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 | o5 : ChainComplexMap

The inclusion of a face induces an inclusion of chain complexes.

 i6 : S' = ZZ[y_0..y_5]; i7 : fish = simplicialComplex {y_0*y_1*y_2, y_1*y_2*y_3, y_3*y_4*y_5} o7 = simplicialComplex | y_3y_4y_5 y_1y_2y_3 y_0y_1y_2 | o7 : SimplicialComplex i8 : S'' = ZZ[z_0,z_1,z_2]; i9 : fishface = simplicialComplex {z_0*z_1*z_2} o9 = simplicialComplex | z_0z_1z_2 | o9 : SimplicialComplex i10 : f = map(fish,fishface,{y_0,y_1,y_2}); o10 : SimplicialMap simplicialComplex | y_3y_4y_5 y_1y_2y_3 y_0y_1y_2 | <--- simplicialComplex | z_0z_1z_2 | i11 : F = chainComplex f 1 1 o11 = -1 : ZZ <--------- ZZ : -1 | 1 | 6 3 0 : ZZ <------------- ZZ : 0 | 1 0 0 | | 0 1 0 | | 0 0 1 | | 0 0 0 | | 0 0 0 | | 0 0 0 | 8 3 1 : ZZ <------------- ZZ : 1 | 1 0 0 | | 0 1 0 | | 0 0 1 | | 0 0 0 | | 0 0 0 | | 0 0 0 | | 0 0 0 | | 0 0 0 | 3 1 2 : ZZ <--------- ZZ : 2 | 1 | | 0 | | 0 | o11 : ChainComplexMap i12 : kernel F == 0 o12 = true