Macaulay2 » Documentation
Packages » SimplicialComplexes :: realProjectiveSpaceComplex(ZZ,PolynomialRing)
next | previous | forward | backward | up | index | toc

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



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


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.

See also

Ways to use this method: