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# barycentricSubdivision(SimplicialMap,Ring,Ring) -- create the map between barycentric subdivisions corresponding to a simplicial map

## Synopsis

• Function: barycentricSubdivision
• Usage:
barycentricSubdivision(f, R, S)
• Inputs:
• f, , from the simplicial complex $D$ to the simplicial complex $E$
• R, a ring, the ambient ring for the barycentric subdivision of $E$
• S, a ring, the ambient ring for the barycentric subdivision of $D$
• Outputs:
• , from the barycentric subdivision of $D$ to the barycentric subdivision of $E$.

## Description

The vertices of the barycentric subdivision of $D$ correspond to faces of $D$. For every face $F$ in $D$, $\operatorname{barycentricSubdivision} (f,R,S)$ maps the vertex corresponding to $F$ in the barycentric subdivision of $D$ to the vertex corresponding to $f(F)$ in the barycentric subdivision of $E$. We work out these correspondences, and the resulting simplicial map between barycentric subdivisions in the example below.

 i1 : T = ZZ/2[x_0,x_1,x_2]; i2 : Δ = simplicialComplex{T_1*T_2} o2 = simplicialComplex | x_1x_2 | o2 : SimplicialComplex i3 : Γ = simplicialComplex{T_0*T_1} o3 = simplicialComplex | x_0x_1 | o3 : SimplicialComplex i4 : f = map(Γ, Δ, reverse gens T) o4 = | x_2 x_1 x_0 | o4 : SimplicialMap simplicialComplex | x_0x_1 | <--- simplicialComplex | x_1x_2 |

The barycentric subdivisions of $D$, $E$, and $f$ are:

 i5 : R = ZZ/2[y_0..y_2]; i6 : S = ZZ/2[z_0..z_2]; i7 : BΔ = barycentricSubdivision(Δ, R) o7 = simplicialComplex | y_1y_2 y_0y_2 | o7 : SimplicialComplex i8 : BΓ = barycentricSubdivision(Γ, S) o8 = simplicialComplex | z_1z_2 z_0z_2 | o8 : SimplicialComplex i9 : Bf = barycentricSubdivision(f, S, R) o9 = | z_1 z_0 z_2 | o9 : SimplicialMap simplicialComplex | z_1z_2 z_0z_2 | <--- simplicialComplex | y_1y_2 y_0y_2 |

In order to understand the data for $Bf$, we first look at the correspondence between the faces of $\Delta$, $\Gamma$, and the vertices of $B\Delta$, $B\Gamma$, respectively.

 i10 : ΔFaces = flatten for i to dim Δ + 1 list faces(i, Δ) o10 = {x , x , x x } 1 2 1 2 o10 : List i11 : ΓFaces = flatten for i to dim Γ + 1 list faces(i, Γ) o11 = {x , x , x x } 0 1 0 1 o11 : List i12 : netList transpose {for y in vertices BΔ list y => ΔFaces_(index y), for z in vertices BΓ list z => ΓFaces_(index z)} +----------+----------+ o12 = |y => x |z => x | | 0 1 | 0 0 | +----------+----------+ |y => x |z => x | | 1 2 | 1 1 | +----------+----------+ |y => x x |z => x x | | 2 1 2| 2 0 1| +----------+----------+

These correspondences, together the images of each face of $D$ under $f$, will completely determine the map $Bf$.

 i13 : netList transpose {for F in ΔFaces list F => (map f)(F), for v in vertices BΔ list v => (map Bf)(v) } +------------+--------+ o13 = |x => x |y => z | | 1 1 | 0 1| +------------+--------+ |x => x |y => z | | 2 0 | 1 0| +------------+--------+ |x x => x x |y => z | | 1 2 0 1| 2 2| +------------+--------+ i14 : Bf o14 = | z_1 z_0 z_2 | o14 : SimplicialMap simplicialComplex | z_1z_2 z_0z_2 | <--- simplicialComplex | y_1y_2 y_0y_2 |