Hurwitz–Ran spaces

Abstract

AbstractGiven a couple of subspaces $${\mathcal {Y}}\subset {\mathcal {X}}$$ Y ⊂ X of the complex plane $${\mathbb {C}}$$ C satisfying some mild conditions (a “nice couple”), and given a PMQ-pair $$({\mathcal {Q}},G)$$ ( Q , G ) , consisting of a partially multiplicative quandle (PMQ) $${\mathcal {Q}}$$ Q and a group G, we introduce a “Hurwitz–Ran” space $$\text {Hur}({\mathcal {X}},{\mathcal {Y}};{\mathcal {Q}},G)$$ Hur ( X , Y ; Q , G ) , containing configurations of points in $${\mathcal {X}}\setminus {\mathcal {Y}}$$ X \ Y and in $${\mathcal {Y}}$$ Y with monodromies in $${\mathcal {Q}}$$ 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 $${\mathcal {Q}}$$ Q we prove a homeomorphism between $$\text {Hur} ((0,1)^2;{\mathcal {Q}}_+)$$ Hur ( ( 0 , 1 ) 2 ; Q + ) and the simplicial Hurwitz space $$\text {Hur} ^{\Delta }({\mathcal {Q}})$$ Hur Δ ( Q ) , introduced in previous work of the author: this provides in particular $$\text {Hur} ((0,1)^2;{\mathcal {Q}}_+)$$ Hur ( ( 0 , 1 ) 2 ; Q + ) with a cell stratification in the spirit of Fox–Neuwirth and Fuchs.