smallManifold(ZZ,ZZ,ZZ,PolynomialRing) -- get a small manifold from the Frank Lutz database

Synopsis

Function: smallManifold
Usage:

smoothManifold(d, v, i, S)
Inputs:
- d, an integer, equal to dimension of the manifold
- v, an integer, equal to the number of vertices
- i, an integer, indexing a small $d$-manifold in the database
- S, a polynomial ring, that specifies the polynomial ring containing the Stanley–Reisner ideal
Outputs:
- an abstract simplicial complex, corresponding to a triangulation of a $d$-manifold

Description

This function accesses a database of all small triangulated 2- or 3-manifolds with at most ten vertices. The enumeration of these abstract simplicial complex follows Frank H. Lutz's classification in "Triangulated Manifolds with Few Vertices: Combinatorial Manifolds", arXiv:math/0506372v1. Alternative formats, further references, and more details may be found at the manifold page.

There is a $1$ surface with four vertices, $1$ with five vertices, $3$ with six vertices, $9$ with seven vertices, $43$ with eight vertices, $655$ with nine vertices, and $42,426$ with ten vertices. There is $1$ threefold with five vertices, $2$ with six vertices, $5$ with seven vertices, $39$ with eight vertices, and $1,297$ with nine vertices.

i1 : S = ZZ[a..j];

i2 : Γ = smallManifold(3, 8, 21, S);

i3 : matrix {facets Γ}

o3 = | dfgh afgh cdgh bcgh abgh bdfh abfh bcdh defg aefg cdeg bceg abeg adef
     ------------------------------------------------------------------------
     abdf acde abce abcd |

             1      18
o3 : Matrix S  <-- S

i4 : prune HH Γ

o4 = -1 : 0  

      0 : 0  

      1 : 0  

      2 : 0  

            1
      3 : ZZ

o4 : GradedModule

i5 : assert(isWellDefined Γ and dim Γ === 3 and # vertices Γ === 8)

i6 : assert (simplicialComplex faces(3, simplexComplex(4, S)) === smallManifold(3,5,0,S))

Frank Lutz also includes some information of the topological type of the manifolds on his webpage. The 2-sphere, 2-torus, real projective plane and Klein Bottle can all be accessed from the database. Here are the smallest instances which appear in the database for real projective plane and the Klein Bottle (note that our index begins counting at zero instead of one):

i7 : RP2 = smallManifold(2, 6, 1, S)

o7 = simplicialComplex | bef aef cdf adf bcf cde bde ace abd abc |

o7 : SimplicialComplex

i8 : KleinBottle = smallManifold(2, 8, 12, S)

o8 = simplicialComplex | cgh agh cfh afh efg dfg aeg cdg bef adf bcf cde bde ace abd abc |

o8 : SimplicialComplex

From Theorems 6.3 and 6.4 in Chapter one of Munkres' Algebraic Topology, both real projective plane and the Klein Bottle should have zero homology except in the first position.

i9 : RP2H = prune HH RP2

o9 = -1 : 0             

      0 : 0             

      1 : cokernel | 2 |

      2 : 0             

o9 : GradedModule

i10 : KBH = prune HH KleinBottle

o10 = -1 : 0             

       0 : 0             

       1 : cokernel | 2 |
                    | 0 |

       2 : 0             

o10 : GradedModule

Ways to use this method: