Given a couple of subspaces Y ⊂ X of the complex plane C satisfying some mild conditions (a “nice couple”), and given a PMQ-pair ( Q , G), consisting of a partially multiplicative quandle (PMQ) Q and a group G, we introduce a “Hurwitz–Ran” space Hur( X , Y ; Q , G), containing configurations of points in X \ Y and in Y with monodromies in Q and in G, respectively. We further introduce a notion of morphisms between nice couples, and prove that Hurwitz–Ran spaces are functorial both in the nice couple and in the PMQ-group pair. For a locally finite PMQ Q we prove a homeomorphism between Hur((0, 1) 2 ; Q + ) and the simplicial Hurwitz space Hur  ( Q ), introduced in previous work of the author: this provides in particular Hur((0, 1) 2 ; Q + ) with a cell stratification in the spirit of Fox–Neuwirth and Fuchs.
Bianchi, A. (2023). Hurwitz–Ran spaces. GEOMETRIAE DEDICATA, 217(5), 1-56 [10.1007/s10711-023-00820-z].
Hurwitz–Ran spaces
Bianchi, Andrea
2023
Abstract
Given a couple of subspaces Y ⊂ X of the complex plane C satisfying some mild conditions (a “nice couple”), and given a PMQ-pair ( Q , G), consisting of a partially multiplicative quandle (PMQ) Q and a group G, we introduce a “Hurwitz–Ran” space Hur( X , Y ; Q , G), containing configurations of points in X \ Y and in Y with monodromies in Q and in G, respectively. We further introduce a notion of morphisms between nice couples, and prove that Hurwitz–Ran spaces are functorial both in the nice couple and in the PMQ-group pair. For a locally finite PMQ Q we prove a homeomorphism between Hur((0, 1) 2 ; Q + ) and the simplicial Hurwitz space Hur ( Q ), introduced in previous work of the author: this provides in particular Hur((0, 1) 2 ; Q + ) with a cell stratification in the spirit of Fox–Neuwirth and Fuchs.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
