kleinBottleComplex(PolynomialRing) -- make a minimal triangulation of the Klein bottle

Synopsis

Function: kleinBottleComplex
Usage:

kleinBottleComplex S
Inputs:
- S, a polynomial ring, that specifies the polynomial ring containing the Stanley–Reisner ideal
Outputs:
- an abstract simplicial complex, corresponding to a minimal triangulation of the Klein bottle

Description

This function accesses Frank Lutz's database of all small triangulated 2-manifolds via the method smallManifold to obtain the minimal triangulation of the Klein bottle.

i1 : S = ZZ[x_0..x_7];

i2 : KleinBottle = kleinBottleComplex S

o2 = simplicialComplex | x_2x_6x_7 x_0x_6x_7 x_2x_5x_7 x_0x_5x_7 x_4x_5x_6 x_3x_5x_6 x_0x_4x_6 x_2x_3x_6 x_1x_4x_5 x_0x_3x_5 x_1x_2x_5 x_2x_3x_4 x_1x_3x_4 x_0x_2x_4 x_0x_1x_3 x_0x_1x_2 |

o2 : SimplicialComplex

From Theorem 6.3 in Chapter one of Munkres' Algebraic Topology states that the first homology of the Klein bottle should have rank one free part, and $\mathbb{Z}/2\mathbb{Z}$ torsion. The second homology should be zero.

i3 : prune HH KleinBottle

o3 = cokernel | 2 |
              | 0 |
      
     1

o3 : Complex

Ways to use this method: