Partially multiplicative quandles and simplicial Hurwitz spaces

Abstract

This is the first article in a series about Hurwitz spaces. We introduce the notion of partially multiplicative quandle (PMQ), which is a generalisation of the classical notions of partial abelian monoid and of quandle. We give a first, simplicial definition of Hurwitz spaces of configurations of points in the plane with monodromies in a PMQ.

Moduli spaces of Riemann surfaces as Hurwitz spaces

“…. t2d−2 in S geo d , with t 1 = t, see [Bia21a,Proposition 7.11]. The group completion theorem ensures that the map Ξ HM+, lc d ;e : Tel( HM + , lc d ; e ) → Ω 0 B HM + is a homology equivalence.…”

“…The aim of this article is to use the theory of generalised Hurwitz spaces from [Bia21a,Bia21b,Bia21c] to give a combinatorial model for the moduli space M g,n of closed, connected Riemann surfaces of genus g ≥ 0 with n ≥ 1 ordered and directed poles (marked points endowed with a non-zero tangent vector).…”

“…As in [EVW16], the "C" indicates that we restrict to connected branched covers, i.e. we require S to be connected; the conjugacy class of transpositions in the symmetric group S d , where local monodromies take value, is understood in the notation Chur k (d) ; and we use the small letters "hur" to distinguish this classical Hurwitz space from the generalised ones, which we will denote by "Hur" as in [Bia21a].…”

“…The second result is Theorem 4.1, asserting that S geo d is a Poincare partially multiplicative quandle. The notion of partially multiplicative quandle (PMQ) was introduced in [Bia21a, Section 2]; for an augmented PMQ Q (see [Bia21a,Definition 4.9]), a simplicial Hurwitz space Hur ∆ (Q) was introduced in [Bia21a, Definition 6.13]. The augmented PMQ S geo d was introduced in [Bia21a, Definition 7.1], and will play a central role in this article.…”

“…where at the beginning the factor (1, 2) is repeated in total 2g + 1 times. The element ((d)) g can be also expressed in the notation of [Bia21a,Proposition 7.13] as ((d)) g = (lc d ; {1, . .…”