euler -- compute the Euler characteristic of a moment-angle complex

Description

This method computes the Euler characteristic of moment-angle complexes.

The Euler characteristic of a moment-angle complex is $0$ if the underlying simplicial complex is not a full simplex.

i1 : needsPackage "SimplicialComplexes"

o1 = SimplicialComplexes

o1 : Package

i2 : R = QQ[a..d]

o2 = R

o2 : PolynomialRing

i3 : K0 = simplicialComplex {a*b, b*c, c*d, d*a}

o3 = simplicialComplex | cd ad bc ab |

o3 : SimplicialComplex

i4 : Z0 = momentAngleComplex K0

o4 = MomentAngleComplex{MACSimplicialComplex => simplicialComplex | cd ad bc ab |}

o4 : MomentAngleComplex

i5 : euler Z0

o5 = 0

i6 : K1 = simplicialComplex {a*b*c*d}

o6 = simplicialComplex | abcd |

o6 : SimplicialComplex

i7 : Z1 = momentAngleComplex K1

o7 = MomentAngleComplex{MACSimplicialComplex => simplicialComplex | abcd |}

o7 : MomentAngleComplex

i8 : euler Z1

o8 = 1

euler(MomentAngleComplex)
euler(AbstractVariety)
euler(CoherentSheaf) -- Euler characteristic of coherent sheaf
euler(Complex) (missing documentation)
euler(Ideal)
euler(Module)
euler(NormalToricVariety) (missing documentation)
euler(ProjectiveHilbertPolynomial) -- constant term of the Hilbert polynomial
euler(ProjectiveVariety) -- topological Euler characteristic of a (smooth) projective variety
euler(Ring)

The source of this document is in ToricTopology/Documentation.m2:534:0.