Non-Hausdorff Incidence Completions of Finite Coverings: Monodromy, Transport, and Defect Posets

Abstract

Let X be a connected Hausdorff surface, let Sigma be a finite subset of X, and let pi from Y to X minus Sigma be a finite sheeted covering. Peripheral monodromy at each marked point determines a finite set of peripheral component germs together with their local degree labels. We introduce incidence completions by adjoining an exceptional fibre at each marked point and prescribing a closed many to many incidence relation between peripheral component germs and exceptional points. The resulting tail saturated topology realizes the prescribed endpoint relation intrinsically. A fixed regular lift extends through a defect precisely when its exceptional value lies in the intersection of the endpoint sets associated with all accumulated peripheral germs. For finite discrete exceptional fibres, a bipartite incidence graph determines the local fundamental group and low dimensional homology. Power probes recover peripheral degrees, while multi arm probes recover higher endpoint intersections and reconstruct the discrete incidence relation by Mobius inversion. For finite exceptional fibres, the completion data are equivalently described by a weighted specialization poset with an antitone incidence map. This yields a structured monodromy classification, automorphism and quotient results, and relation valued and multiplicity valued transport through isolated defects. Varying the exceptional topology and incidence produces a finite defect poset described by a Grothendieck construction. Its total order complex forgets the incidence direction, while vertical incidence complexes and their survival under topology degeneration retain incidence dependent information.