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7 Milnor fibers

The Milnor fiber is the typical fiber \(f^{-1}(1)\) of the Milnor fibration

\[f:\mathbb{C}^\ell \setminus \cup_{H \in \mathcal{A}}H \to \mathbb{C}^\times, z\mapsto \prod_{H \in \mathcal{A}}\alpha_H(z)\]

of the hyperplane arrangement \(\mathcal{A}\), where \(\alpha_H\) are defining linear forms.

The following describes functions to construct a regular cell complex having the homotopy type of the Milnor fiber from [MY25] and thus enables computation of homotopy invaraints using e.g. the HAP package [Ell26].

7.1 Complexes

7.1-1 MilnorFiberComplex

Returns: FacePoset

Computes the face poset of the regular cell complex of the combinatorial Milnor fiber of the oriented matroid OM.

gap> O:=OrientedMatroid([[1,0],[0,1],[1,1]]);
<OrientedMatroid: 3 elements, rank 2>
gap> MCpx:=MilnorFiberComplex(O);
<FacePoset of dimension 1 with f-vector [ 3, 6 ]>

7.1-2 MilnorFiberComplex

Returns: FacePoset

Computes the face poset of the regular cell complex having the homotopy type of Milnor fiber of the arrangement A.

gap> A:=AGpql(2,2,3);
<HyperplaneArrangement: 6 hyperplanes in 3-space>
gap> MCpx:=MilnorFiberComplex(A);
<FacePoset of dimension 2 with f-vector [ 12, 60, 60 ]>

7.1-3 FPtoCWCpx

Returns: HAP CW Complex

Converts a HypArr FacePoset into a HAP RegularCWComplex for further computations of topological invariants.

gap> A:=AGpql(2,2,3);
<HyperplaneArrangement: 6 hyperplanes in 3-space>
gap> MCpx:=MilnorFiberComplex(A);
<FacePoset of dimension 2 with f-vector [ 12, 60, 60 ]>
gap> MCW := FPtoCWCpx(MCpx);
Regular CW-complex of dimension 2
gap> Homology(MCW,0);
[ 0 ]
gap> Homology(MCW,1);
[ 0, 0, 0, 0, 0, 0, 0 ]

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