The connectedness of the moduli space $\mathcal{M}\_g$ of complex algebraic curves of genus $g$ can be proven by showing that it is dominated by a Hurwitz space of simply branched d-fold covers of the line, which in turn can be shown to be connected by proving the transitivity of the the natural action of the braid group on n-tuples of transpositions in $S\_n$ with product 1, which generate $S\_n$: in this action, a generator $\sigma\_i$ of the braid group acts as


$$(g\_1, ... g\_n) \to (g\_1, ... g\_{i+1}, g\_i^{g\_{i+1}}, g\_{i+2}, ..., g\_n)$$


This argument is often referred to as "a theorem of Clebsch (1872 or 1873), Luroth (1871), and Hurwitz (1891)." Does anyone know the history of this argument more precisely, and in particular which parts are due to Luroth, which to Clebsch, and which to Hurwitz? 



A historical question: Hurwitz, Luroth, Clebsch, and the connectedness of $\mathcal{M}_g$
I was able to find the resources online (6 years after this question was asked):


* 1871 <https://eudml.org/doc/156527> Lüroth - 4 pages
* 1873 <https://eudml.org/doc/156610> Clebsch - 16 pages
* 1891 <https://eudml.org/doc/157563> Hurwitz - 61 pages


I think it's pretty clear Luroth was first, but Hurwitz developed this material much further. 




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I get thrown off because we way Riemann surface is just a polygon with edges glued together, and that's pretty much the picture of moduli space painted here.


