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# SimplicialComplex * SimplicialComplex -- make the join for two abstract simplicial complexes

## Synopsis

• Operator: *
• Usage:
Delta * Gamma
• Inputs:
• Delta, ,
• Gamma, ,
• Outputs:
• , that is the join of $\Delta$ and $\Gamma$

## Description

The join of two simplicial complexes $\Delta$ and $\Gamma$ is a new simplicial complex whose faces are disjoint unions of a face in $\Delta$ and a face in $\Gamma$.

If $\Gamma$ is the simplicial complex consisting of a single vertex, then the join $\Delta \mathrel{*} \Gamma$ is the cone over $\Delta$. For example, the cone over a bow-tie complex.

 i1 : S = QQ[a..e]; i2 : Δ = simplicialComplex {a*b*c, c*d*e} o2 = simplicialComplex | cde abc | o2 : SimplicialComplex i3 : R = QQ[f]; i4 : Γ = simplicialComplex {f}; i5 : Δ' = Δ * Γ o5 = simplicialComplex | cdef abcf | o5 : SimplicialComplex i6 : assert (dim Δ' === dim Δ + 1)

If $\Gamma$ is a $1$-sphere (consisting of two isolated vertices), then the join $\Delta \mathrel{*} \Gamma$ is the suspension of $\Delta$. For example, the octahedron is the suspension of a square.

 i7 : S = QQ[a..d]; i8 : Δ = simplicialComplex {a*b, b*c, c*d, a*d} o8 = simplicialComplex | cd ad bc ab | o8 : SimplicialComplex i9 : R = QQ[e,f]; i10 : Γ = simplicialComplex {e, f} o10 = simplicialComplex | f e | o10 : SimplicialComplex i11 : Δ' = Δ * Γ o11 = simplicialComplex | cdf cde adf ade bcf bce abf abe | o11 : SimplicialComplex i12 : assert (dim Δ' === dim Δ + 1) i13 : assert (apply(2+dim Δ', i -> #faces(i-1,Δ')) == {1,6,12,8})

The join of a hexagon and a pentagon is a 3-sphere.

 i14 : S = ZZ[a..f]; i15 : Δ = simplicialComplex {a*b, b*c, c*d, d*e, e*f, a*f} o15 = simplicialComplex | ef af de cd bc ab | o15 : SimplicialComplex i16 : R = ZZ[g..k]; i17 : Γ = simplicialComplex {g*h, h*i, i*j, j*k, g*k} o17 = simplicialComplex | jk gk ij hi gh | o17 : SimplicialComplex i18 : Δ' = Δ * Γ o18 = simplicialComplex | efjk efgk efij efhi efgh afjk afgk afij afhi afgh dejk degk deij dehi degh cdjk cdgk cdij cdhi cdgh bcjk bcgk bcij bchi bcgh abjk abgk abij abhi abgh | o18 : SimplicialComplex i19 : prune HH Δ' o19 = -1 : 0 0 : 0 1 : 0 2 : 0 1 3 : ZZ o19 : GradedModule i20 : assert (dim Δ' === 3)

## Caveat

When the variables in the ring of $\Delta$ and the ring of $\Gamma$ are not disjoint, names of vertices in the join may not be intelligible; the same name will be used for two distinct variables.