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# euler(CentralArrangement) -- compute the Euler characteristic of the projective complement

## Synopsis

• Function: euler
• Usage:
euler A
• Inputs:
• Outputs:
• an integer, equal to the Euler characteristic

## Description

For any topological space, the Euler characteristic is the alternating sum of its Betti numbers (a.k.a. the ranks of its homology groups). For a central hyperplane arrangement, the associated topological space is the projectivization of its complement.

The Euler characteristic for the hyperplane arrangements defined by root systems are described by simple formulas.

 i1 : A2 = typeA 2 o1 = {x - x , x - x , x - x } 1 2 1 3 2 3 o1 : Hyperplane Arrangement  i2 : euler A2 o2 = -1 i3 : assert all(5, n -> euler typeA (n+1) === (-1)^(n) * n!) i4 : B2 = typeB 2 o4 = {x , x - x , x + x , x } 1 1 2 1 2 2 o4 : Hyperplane Arrangement  i5 : euler B2 o5 = -2 i6 : assert all(4, n -> euler typeB (n+1) === (-1)^(n) * 2^n * n!)

Given a flat, this method computes the Euler characteristic of the subarrangement indexed by the flat.

 i7 : A4 = typeA 4 o7 = {x - x , x - x , x - x , x - x , x - x , x - x , x - x , x - x , x - x , x - x } 1 2 1 3 1 4 1 5 2 3 2 4 2 5 3 4 3 5 4 5 o7 : Hyperplane Arrangement  i8 : F = flat(A4, {0,7}) o8 = {0, 7} o8 : Flat of {x - x , x - x , x - x , x - x , x - x , x - x , x - x , x - x , x - x , x - x } 1 2 1 3 1 4 1 5 2 3 2 4 2 5 3 4 3 5 4 5 i9 : euler F o9 = 0 i10 : assert(euler A4_F === euler F) i11 : euler flat(A4, {2,3,9}) o11 = -1 i12 : euler flat(A4, {0,1,2,4,5,7}) o12 = 2 i13 : euler flat(A4, {2,4,6,8}) o13 = 0

The Euler characteristic of the empty arrangement is just the Euler characteristic of the ambient projective space. For instance, the Euler characteristic of the complex projective plane is $3$.

 i14 : assert (euler arrangement({}, ring A2) === 3)

• typeA -- make the hyperplane arrangement defined by a type $A_n$ root system
• typeB -- make the hyperplane arrangement defined by a type $B_n$ root system