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

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 |

See also

Ways to use this method:

The source of this document is in SimplicialComplexes/Documentation.m2:4185:0.