Milnor fibrations and oriented matroids
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915454859083776 |
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| author | Mücksch, Paul Yoshinaga, Masahiko |
| author_facet | Mücksch, Paul Yoshinaga, Masahiko |
| contents | We introduce a combinatorial model for the Milnor fibration of a complexified real arrangement using oriented matroids. It is a poset quasi-fibration, a notion recently introduced by the first author, whose domain is a subdivision of the Salvetti complex stemming from a natural subdivision of the dual oriented matroid complex. This yields a concrete finite regular CW complex which is homotopy equivalent to the Milnor fiber of the complexified real arrangement and implies that the homotopy type of the Milnor fiber of a complexified real arrangement only depends on the underlying combinatorial structure given by its oriented matroid. Moreover, our construction works for any oriented matroid, disregarding realizability, so we obtain a notion of a combinatorial Milnor fibration for any oriented matroid. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_15331 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Milnor fibrations and oriented matroids Mücksch, Paul Yoshinaga, Masahiko Algebraic Geometry Algebraic Topology Combinatorics Geometric Topology Primary: 32S55, 52C40, 52C35, Secondary: 32S22, 55R10, 57Q70 We introduce a combinatorial model for the Milnor fibration of a complexified real arrangement using oriented matroids. It is a poset quasi-fibration, a notion recently introduced by the first author, whose domain is a subdivision of the Salvetti complex stemming from a natural subdivision of the dual oriented matroid complex. This yields a concrete finite regular CW complex which is homotopy equivalent to the Milnor fiber of the complexified real arrangement and implies that the homotopy type of the Milnor fiber of a complexified real arrangement only depends on the underlying combinatorial structure given by its oriented matroid. Moreover, our construction works for any oriented matroid, disregarding realizability, so we obtain a notion of a combinatorial Milnor fibration for any oriented matroid. |
| title | Milnor fibrations and oriented matroids |
| topic | Algebraic Geometry Algebraic Topology Combinatorics Geometric Topology Primary: 32S55, 52C40, 52C35, Secondary: 32S22, 55R10, 57Q70 |
| url | https://arxiv.org/abs/2508.15331 |