dunceHatComplex(PolynomialRing) -- make a realization of the dunce hat with 8 vertices

Synopsis

Function: dunceHatComplex
Usage:

dunceHatComplex S
Inputs:
- S, a polynomial ring, that has at least 8 generators
Outputs:
- an abstract simplicial complex,

Description

The dunce hat is a compact topological space formed by taking a solid triangle and gluing all three sides together, with the orientation of one side reversed. Simply gluing two sides oriented in the same direction would yield a cone much like the dunce cap, but the gluing of the third side results in identifying the base of the cap with a line joining the base to the point.

Following Erik Christopher Zeeman's "On the dunce hat", Topology 2 (1964) 341–358, this method returns non-collapsible but contractible example of an abstract simplicial complex.

i1 : S = ZZ/101[a..h];

i2 : Δ = dunceHatComplex S

o2 = simplicialComplex | fgh agh dfh cdh bch abh cfg bcg abg def aef acf bde bce ace acd abd |

o2 : SimplicialComplex

i3 : dim Δ

o3 = 2

i4 : fVector Δ

o4 = {1, 8, 24, 17}

o4 : List

i5 : assert(dim Δ === 2 and isPure Δ)

i6 : assert(fVector Δ === {1,8,24,17})

Our enumeration of the vertices follows the dunce hat example in Masahiro Hachimori's simplicial complex library.

Ways to use this method: