realProjectiveSpaceComplex(ZZ,PolynomialRing) -- make a small triangulation of real projective space

Synopsis

Function: realProjectiveSpaceComplex
Usage:

realProjectiveSpaceComplex(n, S)
Inputs:
- n, an integer, that specifies the dimension of real projective space
- S, a polynomial ring, that specifies the polynomial ring containing the Stanley–Reisner ideal
Outputs:
- an abstract simplicial complex, corresponding to a triangulation of real projective space

Description

This method implements some of the minimal triangulations of real projective space found in the literature. For $n = 0, 1$, these are just the obvious point and 1-sphere. For $n = 2$, the minimal triangulation is provided by Frank H. Lutz's small manifold database. Frank Lutz has also provided minimal triangulations for $n = 3$ and $4$, in "Triangulated Manifolds with Few Vertices: Combinatorial Manifolds", arXiv:math/0506372v1.

i1 : S = ZZ[x_0..x_10]

o1 = S

o1 : PolynomialRing

i2 : Δ = realProjectiveSpaceComplex(3, S)

o2 = simplicialComplex | x_3x_4x_5x_10 x_0x_4x_5x_10 x_1x_3x_5x_10 x_0x_1x_5x_10 x_2x_3x_4x_10 x_0x_2x_4x_10 x_1x_2x_3x_10 x_0x_1x_2x_10 x_2x_5x_8x_9 x_1x_5x_8x_9 x_2x_4x_8x_9 x_1x_4x_8x_9 x_1x_4x_7x_9 x_0x_4x_7x_9 x_1x_3x_7x_9 x_0x_3x_7x_9 x_3x_5x_6x_9 x_2x_5x_6x_9 x_0x_3x_6x_9 x_0x_2x_6x_9 x_1x_3x_5x_9 x_0x_2x_4x_9 x_2x_5x_7x_8 x_0x_5x_7x_8 x_2x_3x_7x_8 x_0x_3x_7x_8 x_3x_4x_6x_8 x_1x_4x_6x_8 x_0x_3x_6x_8 x_0x_1x_6x_8 x_0x_1x_5x_8 x_2x_3x_4x_8 x_4x_5x_6x_7 x_2x_5x_6x_7 x_1x_4x_6x_7 x_1x_2x_6x_7 x_0x_4x_5x_7 x_1x_2x_3x_7 x_3x_4x_5x_6 x_0x_1x_2x_6 |

o2 : SimplicialComplex

Caveat

Since no minimal or small triangulations of real projective space have been constructed for $n > 4$, we haven't implemented the triangulations for higher projective space yet. Due to the exponential growth of the number of vertices, computations quickly become intractable.

Ways to use this method: