elementaryCollapse(SimplicialComplex,RingElement) -- construct the elementary collapse of a free face in a simplicial complex

Synopsis

• Function: elementaryCollapse
• Usage:
elementaryCollapse(Delta, F)
• Inputs:
• Delta, ,
• F, , Corresponding to a free face of $\Delta$
• Outputs:
• , the simplicial complex where the face $F$ and the unique facet containing $F$ are removed

Description

A free face of a simplicial complex $\Delta$ is a face that is a proper maximal subface of exactly one facet. The elementary collapse is the simplicial complex obtained by removing the free face, and the facet containing it, from $\Delta$. A simplicial complex that can be collapsed to a single vertex is called collapsible. Every collapsible simplicial complex is contractible, but the converse is not true.

 i1 : R = ZZ/103[x_0..x_3]; i2 : Δ = simplicialComplex{R_0*R_1*R_2,R_0*R_2*R_3,R_0*R_1*R_3} o2 = simplicialComplex | x_0x_2x_3 x_0x_1x_3 x_0x_1x_2 | o2 : SimplicialComplex i3 : C1 = elementaryCollapse(Δ,R_1*R_2) o3 = simplicialComplex | x_0x_2x_3 x_0x_1x_3 | o3 : SimplicialComplex i4 : C2 = elementaryCollapse(C1,R_2*R_3) o4 = simplicialComplex | x_0x_2 x_0x_1x_3 | o4 : SimplicialComplex i5 : C3 = elementaryCollapse(C2,R_1*R_3) o5 = simplicialComplex | x_0x_3 x_0x_2 x_0x_1 | o5 : SimplicialComplex i6 : C4 = elementaryCollapse(C3,R_1) o6 = simplicialComplex | x_0x_3 x_0x_2 | o6 : SimplicialComplex i7 : C5 = elementaryCollapse(C4,R_2) o7 = simplicialComplex | x_0x_3 | o7 : SimplicialComplex i8 : C6 = elementaryCollapse(C5,R_3) o8 = simplicialComplex | x_0 | o8 : SimplicialComplex